# Error of the estimated mean for time-series data


A common analysis task is determining the mean	value of a metric in a population - here we consider the metric to be the arithmetic mean. Because we don't have access to the entire population, this is almost always done by taking some sample data from the broader population, measuring the mean in the sample, then taking that measurement as an estimation of the true value (hence 'estimator'). Because the estimator doesn't return the exact true value, it should be accompanied by error bars. For sample data that is independent, calculating the standard error of the estimated mean is trivial using a formula. Some other data, for example time series, are highly correlated and thus need some adjustments to determine the error. 

Despite the broad range of fields that measure time series data - ecology, econometrics, physical simulations, astronomy, etc... - there's no single, accepted approach to calculating the standard error of the mean, motiviating this article.  


This article demonstrates four different techniques for determing the standard error of the estimated mean:
- Block averaging
- Using a correction factor
- Autoregressive (AR(1)) maximum likelihood fit
- Autoregressive (AR(1)) bayesian fit


------------

First, to demonstrate the problem, below shows examples of independent data (top panel) and highly correlated data time series data (bottom panel):



```python
import matplotlib.pyplot as plt
import numpy as np
#np.random.seed(2572367)
np.random.seed(257367)


def gen_correlated_curve(ac, num=100, innovation=1):
    y = np.zeros((num,))
    for i in range(1,num):
        y[i] = ac * y[i-1] + np.random.normal()*innovation
    return y

uncorrelated = np.random.randn(100)
correlated = gen_correlated_curve(0.90)

fig, ax = plt.subplots(ncols=1, nrows=2); fig.set_figheight(10); fig.set_figwidth(10)
ax[0].plot(uncorrelated, '-o', mfc='white'); ax[0].set_title('Independent sample'); ax[0].axhline(0, c='k', linestyle='--')
ax[1].plot(correlated, '-o', mfc='white'); ax[1].set_title('Highly correlated sample'); ax[1].axhline(0, c='k', linestyle='--')

```

The key difference is that in the case of highly correlated data, each new point is very similar to the previous one. This is known as autocorrelation. For independent data this is not the case - each new point has no memory of the previous points even though all points are still centred on a common mean value. 

This causes difficulty when calculating error bars. Normally, for independent data, when estimating the mean value you would calculate the standard error of that estimate using:

\begin{align}
SEM = \frac{s}{\sqrt{n}}
\end{align}

which in english says "the standard error of the (estimated) mean, called $SEM$, is equal to the standard deviation ($s$) of the sample divided by the square root of the number of samples". The formula isn't so special, it can be found on the wikipedia entry for [standard error of the mean](https://en.wikipedia.org/wiki/Standard_error). 

The problem with using this formula for highly correlated data is the $\sqrt{n}$ in the denominator - it can become very large for time series data, especially if you take samples at high frequency. Consider an example: measure your body temperature every hour for four weeks. It fluctuates over the day/night cycle, but doesn't change too much hour by hour. In that case, $n=672$. Now imagine measuring it every minute for a month, so $n=40320$. The $n$ has multiplied by $60$, meaning the $SEM$ becomes smaller by $\sqrt{60}$. Yet in both cases you sampled for the same total time. Your actual body temperature would have varied by about the same amount, meaning you would expect the error bars to be about the same.


Put another way, high frequency sampling of a time-varying process gives you lots of unnecessary data points because each point is similar to its neighbours, and this makes the sample look very precise when it isn't. 



# Autocorrelation

This correlation between adjacent points can be visualized using an `autocorrelation plot`. Autocorrelation plots range from `-1 to 1` and measure how correlated adjacent points are, with `1` being total positive correlation, `0` being no correlation at all, and `-1` being total negative correlation. Autocorrelation plots measure how correlated a sample is to another point at some time lag. When the time lag is zero, autocorrelation must be `1` because each point must be completely positively correlated with itself! 

The below shows an example of three time series with different levels of positive correlation. When the autocorrelation plot reaches zero, it's safe to assume that the time lag is big enough that any data point has no memory of another data point separated by that lag. This value of the time lag is characteristic to an autocorrelated process, and is called the 'autocorrelation time' ($\tau$, tau). In practice, there's a lot of noise around the autocorrelation of zero which appears even using independent data. 

`??? See Detecting Signals from Data with Noise: Theory and Applications` for significance testing of autocorrelation vs. white noise 



```python
plt.figure(figsize=(10, 5))
for ac, label in zip([0, 0.8, 0.99], ['Not autocorrelated at all', 'A bit autocorrelated', 'Very autocorrelated']):
    curve = gen_correlated_curve(ac, num=500)
    #calculate autocorrelation:
    co = np.correlate(curve-curve.mean(),curve-curve.mean(), mode='same')
    co = co/co.max()
    co = co[int(co.shape[0]/2):]
    
    plt.plot(co,label=label)

plt.legend()
plt.xlabel('Lag time', fontsize=16)
_=plt.ylabel('Autocorrelation', fontsize=16)

```

# Approach 1: Block averaging
'Undoing' the autocorrelation allows us to calculate $SEM$ as if the sample was independent. One way to do this is block averaging. First, a baseline value for the $SEM$ is calculated using all points. To do one round of block averaging, every second point and it's adjacent point forward in time are averaged, creating 'blocks' of size 2 and resulting in a dataset with $ \frac{n}{2} $ points. 

Here's an example:


```python

correlated = gen_correlated_curve(0.99, num=25)
x = np.arange(len(correlated))

plt.figure(figsize=(10,5))
plt.title('One round of block averaging')
plt.plot(x,correlated,mew=1, linewidth=0.5, label='Autocorrelated data')

#plot pairs:
for i in range(0,13):
    plt.plot(x[i*2:i*2+2],correlated[i*2:i*2+2],'-o', c='C0', mfc='white', mew=2, linewidth=3, zorder=0)

#do one round of block averaging, blocksize of 2
x_ = x[:len(x)-(len(x)%2)].reshape(-1,2).mean(1)
y_ = correlated[:len(x)-(len(x)%2)].reshape(-1,2).mean(1)

plt.scatter(x_, y_, facecolor='white', edgecolor='red', linewidth=2,zorder=2, label='Block size=2')
plt.xlabel('Time',fontsize=16)
plt.ylabel('Value',fontsize=16)
_=plt.legend()
```

By increasing the size of the blocks from 2,3,4..., the hope is that eventually adjacent blocks are far enough apart in time that they are totally uncorrelated and the data resemble an independent sample. This occurs when the blocks are bigger than the autocorrelation time $\tau$. It appears as an asymptote in the $SEM$, and would also correspond to reaching zero in the autocorrelation plot. 

To demonstrate, the below shows blocks up to `size=200` using 5000 points from an autocorrelated time series: 


```python
from scipy.stats import sem

def do_block_averaging(corr):
    sems = list()
    x=np.arange(len(corr))
    #for blocksize in range(1, int(len(corr)//25)+1):
    for blocksize in range(1, int(len(corr)**(2/3))+1):
        #this is the blocking bit:
        x_ = x[:len(x)-(len(x)%blocksize)].reshape(-1,blocksize).mean(1)
        y_ = corr[:len(x)-(len(x)%blocksize)].reshape(-1,blocksize).mean(1)
        sems.append(sem(y_))
    return sems

        
correlated = gen_correlated_curve(0.6, num=5000)
x = np.arange(correlated.shape[0])

blocked_sems = do_block_averaging(correlated)
```


```python
#plot:
fig, ax = plt.subplots()
fig.set_figwidth(10)
fig.set_figheight(5)

ax.plot(blocked_sems, label='Blocked SEM')
ax.set_xlabel('Block Size', fontsize=14)
ax.set_ylabel('SEM',fontsize=14)
_=ax.legend()
```

There are a few problems with this approach:

- **Wasting data**. In order to maintain identically distributed blocks, each block should be exactly the same size. So, what if the block size doesn't evenly divide the total number of measurements? The overhanging data has to be discarded, which is a waste of perfectly good measurements
- **Noise** - as seen above, the $SEM$ values become noisy after the asymptote is reached, rather than converging towards the true value. This is a consequence of cutting data - as the block size grows, larger and larger overhanging chunks get removed until the blocksize is an even divisor of the number of samples. The changes in the amount of data being used leads to jumps in the calculated $SEM$.
- **Arbitrary choice of asymptote**. By eye it seems pretty obvious what region the $SEM$ lies in, but how do you choose a consensus value? One way to remedy this is to automate the choice by fitting some curve that also grows to an asymptote like the $SEM$. There doesn't appear to be a known curve to use for the fit, but curves based on `y=arctan(x)` work. However, on the edge cases where the optimum block size is only 2 or 3, or it's nearly as many samples as you have, the fit can be (wildly) inaccurate.


Because we have a good smooth curve above, the below demonstrates fitting the curve `y= a*arctan(b*(x-c))` to find the asymptote of the $SEM$.



```python
# fitting to arctan
from scipy import optimize

def block_avg_fit(x, a, b, c):
    return a * np.arctan(b*(x-c))

popt, pcov = optimize.curve_fit(block_avg_fit, np.arange(len(blocked_sems)), np.array(blocked_sems))

```


```python
#plot:
ax.plot(np.arange(len(blocked_sems)),
        block_avg_fit(np.arange(len(blocked_sems)), popt[0], popt[1],popt[2]), 
        label='Line of best fit')
ax.axhline(popt[0] *np.pi/2, label='SEM from block avg. asymptote', linestyle='--', c='C1')
ax.legend()
fig
```

So block bootstrapping works well if you have lots of data and a smooth curve. But what to do in other cases? Below demonstrates some alternatives that use all the available data and give definite answers without requiring a good block averaging plot.

# Approach 2: Estimating $n_{eff}$ from the autocorrelation function

Block averaging works by reducing the number of data points $n$ until the $SEM$ converges. At this point, we expect adjacent points to look independent. The value of $n$ that achieves this is known as the 'effective $n$', or $n_{eff}$, and it reflects how many independently sampled data points you actually have in the sample. There are other ways to calculate $n_{eff}$ that are more direct than block averaging, and interestingly they can result in fractional values.  

Adjusting the wikipedia equation for $SEM$, we now have:

\begin{align}
SEM = \frac{s}{\sqrt{n_{eff}}}
\end{align}

Calculating $n_{eff}$ normally happens by first calculating the autocorrelation time $\tau$. The autocorrelation time is the time it takes for the process to forget a past value - meaning  the autocorrelation plot should be close to zero. This time, of course, doesn't need to be an integer value, which is how $n_{eff}$ can be fractional. 

\begin{align}
n_{eff} = \frac{n}{\tau}
\end{align}

So how to find $\tau$? The two techniques given below, called here Sokal and Chodera (see refs below), estimate $\tau$ from the autocorrelation function and have reasonable agreement with block averaging if you sampled for many multiples of $\tau$.



>Chodera: see https://github.com/choderalab/pymbar for code, and `Use of the Weighted Histogram Analysis Method for the Analysis of Simulated and Parallel TemperingSimulations`
>
>Sokal: see https://dfm.io/posts/autocorr/ for code, [here](https://pdfs.semanticscholar.org/0bfe/9e3db30605fe2d4d26e1a288a5e2997e7225.pdf) for Sokal notes



```python
import statsmodels.api as sm
def next_pow_two(n):
    #lew: complicated one-liner: int(2**(np.ceil(np.log2(number))))
    i = 1
    while i < n:
        i = i << 1
    return i

def autocorr_func_1d(x):
    n = next_pow_two(len(x))
    # Compute the FFT and then (from that) the auto-correlation function
    f = np.fft.fft(x - np.mean(x), n=2*n)
    acf = np.fft.ifft(f * np.conjugate(f))[:len(x)].real
    acf /= 4*n 
    acf /= acf[0]
    return acf

# Automated windowing procedure following Sokal (1989)
def auto_window(taus, c):
    m = np.arange(len(taus)) < c * taus
    if np.any(m):
        return np.argmin(m)
    return len(taus) - 1

def sokal_autocorr_time(corr):
    f = autocorr_func_1d(corr)
    taus = 2.0*np.cumsum(f)-1.0
    c=5
    window = auto_window(taus, c)
    return taus[window]

def statineff(corr, mintime=3):
    N = corr.size
    C_t = sm.tsa.stattools.acf(corr, fft=True, unbiased=True, nlags=N)
    t_grid = np.arange(N).astype('float')
    g_t = 2.0 * C_t * (1.0 - t_grid / float(N))
    ind = np.where((C_t <= 0) & (t_grid > mintime))[0][0]
    g = 1.0 + g_t[1:ind].sum()
    return max(1.0, g)


#chodera:
chodera_autoc_time = statineff(correlated)

#sokal:
sokal_autoc_time = sokal_autocorr_time(correlated)

n=len(correlated)
chodera_sem = np.std(correlated) / np.sqrt(n/chodera_autoc_time)
sokal_sem = np.std(correlated) / np.sqrt(n/sokal_autoc_time)
```


```python
#plot:
ax.axhline(chodera_sem, label='SEM from Chodera', c='green')
ax.axhline(sokal_sem, label='SEM from Sokal', c='red')
ax.legend(fontsize=14)
fig
```

# Approach 3: Autoregressive processes
A third way to estimate the $SEM$ is by treating it as an autoregressive process. Here, it helps to understand what it is that makes time series correlated with themselves. The simplest model of autocorrelation is called an `autoregressive model` where we only consider the time lag at lag=1, commonly just called `AR(1)`. [Wikipedia](https://en.wikipedia.org/wiki/Autoregressive_model) has some more details on these, but here's the short version. In an AR(1) process, each new point is equal to the previous point multiplied by a number $\rho$ (rho), plus or minus some white noise (i.e. random noise).

In maths, using $X_t$ to mean 'point $X$ at time $t$' and $\epsilon$ (epsilon) to mean 'random noise':

\begin{align}
X_t = \rho X_{t-1} + \epsilon
\end{align}

When $\rho$ is less than 1, each step pulls the point back towards zero (hence auto*regressive*), meaning the time series will maintain a mean of zero. Moving average models are also commonly used, but a stationary mean makes things a bit easier. If your time series isn't centred on zero, the AR(1) model can just add a constant that shifts the whole sample back to zero, and then still use the model above. 

First, let's estimate the $\rho$ parameter. This can be done using linear regression, where you calculate the slope of the line that goes between any point and its adjacent neighbour. Remember, the true $\rho$ in this case was 0.6.



```python
from statsmodels.tsa.ar_model import AutoReg

#find estimated rho using ordinary least squares
result = AutoReg(correlated-correlated.mean(), lags = [1]).fit()
estimated_rho = result.params[1]
print('###################################')
print(f'Estimated rho is {estimated_rho}, with 95% CI of {result.conf_int()[1]}')
print('###################################')
```

    ###################################
    Estimated rho is 0.6111243087797814, with 95% CI of [0.58917639 0.63307223]
    ###################################


Not a bad estimate for $\rho$. Next, we can use a handy formula, which is a consequence of assuming an autoregressive AR(1) process, to get an estimate for the $SEM$. First the $SEM$ gets calculated naively, giving the wrong answer. Then it's multiplied by a correction factor $k$, which is:

\begin{align}
k = \sqrt{\frac{1+\rho}{1-\rho}}
\end{align}

So now we have:

\begin{align}
SEM = \frac{ks}{\sqrt{n}}
\end{align}


```python
correction = np.sqrt((1+estimated_rho)/(1-estimated_rho))

naive_sem = sem(correlated)
corrected_sem = naive_sem*correction

ax.axhline(corrected_sem, label='SEM from correction', c='purple')
ax.legend()
fig
```

Before moving on, it's also neat to see how big this correction factor can get. At $\rho$ close to 0, i.e. less than about 0.2, $k$ is barely greater than 1, meaning the naive $SEM$ doesn't change much. But as $\rho$ approaches 1, the $k$ can grow very large. Because you might be multiplying small numbers by big numbers here, you would probably need very large sample sizes to get precise estimates if the process is highly autocorrelated.  


```python
def correction_factor(rho, n):
    d = ((n-1)*rho - n*rho**2 + rho**(n+1)) / (1-rho)**2
    k = np.sqrt( (1 + (2*d)/n) / ( 1 - (2*d)/(n*(n-1))  ) )
    return k

x = np.linspace(0,0.9,100)
plt.figure(figsize=(10,5))
plt.ylabel('Correction factor, $k$',fontsize=14); plt.xlabel('Autocorrelation at lag 1, $\\rho$',fontsize=16)
_ = plt.plot(x, correction_factor(x, 100))

```

So, to take stock: 
Block averaging clumps the data together in order to reduce $n$, but doesn't make use of all data. Other approaches estimate $n_{eff}$ from the autocorrelation plot. Alternatively you can assume an AR(1) process, fit with ordinary least squares to find $\rho$, then use a correction factor to fix the naively-calculated $SEM$.


The final technique here also uses an AR(1) model, but instead of using a correction equation, it simulates how probable a range of different mean values are,then reports the region of highest probability.


# Approach 4: AR(1) via Bayesian estimation
The above approaches report what is known as the `maximum likelihood` estimate of the $SEM$. It's called maximum likelihood simply because it's the single most likely value after you factor in any model assumptions that were made (like the AR(1) assumption, or normally-distributed noise). 

An alternative is Bayesian estimation. There are, of course, still model assumptions - this still uses an AR(1) model. The trick here is that we try and find the $\rho$ parameter by also estimating where the mean value is. Afterwards, we can throw away the $\rho$ estimate and just look at the most probable values for the mean. These most probable values are encompassed in a 'highest posterior density' which can be directly interpreted as a probability (unlike confidence intervals). 

The below demonstrates a Bayesian fit of an AR(1) model, using the `PyMC3` library, which jointly estimates the value of the mean and the $\rho$, and can then report a full spectrum of likely values for the mean (or for $\rho$ if you want). The 95% highest posterior density is used instead of the $SEM$.


```python
import pymc3 as pm
import logging
logger = logging.getLogger('pymc3')
logger.setLevel(logging.ERROR)


def do_bayes_sem(corr):
    with pm.Model() as ar1:
        k_ = pm.Uniform('k',-1,1) #we assume process is stationary, so -1<k_<1 
        tau_ = pm.Gamma('tau',mu=1,sd=1)
        center = pm.Normal('center', mu=corr.mean(), sigma=5) #set the prior for the mean to be centred onthe population mean
        likelihood = pm.AR1('likelihood', k=k_, tau_e=tau_, observed=corr-center)
        trace = pm.sample()
    return trace
#slower:
#     with pm.Model() as ar1:
#         rho = pm.Uniform('rho', -1,1)
#         center = pm.Normal('center', mu=corr.mean(), sigma=5)
#         likelihood = pm.AR('likelihood', rho, sigma=1.0, observed=corr-center)
#         trace = pm.sample(500, chains=1, progressbar=True)
#     return trace

trace = do_bayes_sem(correlated)

```

    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1488.18draws/s]



```python
import pymc3 as pm
with pm.Model() as ar1:
    k_ = pm.Uniform('k',-1,1) #we assume process is stationary, so -1<k_<1 
    tau_ = pm.Gamma('tau',mu=1,sd=1)
    center = pm.Normal('center', mu=correlated.mean(), sigma=5) #set the prior for the mean to be centred onthe population mean
    likelihood = pm.AR1('likelihood', k=k_, tau_e=tau_, observed=correlated-center)
    
    
with ar1:
    trace = pm.sample()

kdeplot(trace['center'])

```


```python

```


```python
hpd = pm.stats.hpd(trace['center'])
hpd_sem = (hpd[1]-hpd[0]/2)/1.96

ax.axhline(hpd_sem, c='black', linestyle='--', linewidth=3,
            label='SEM from Bayesian estimation')

ax.legend()
fig


```

There's a lot more analysis one might do with the results here. As a taste, you can plot the posterior density of the mean value, along with the 95% probability bounds (equivalent to the $SEM$). In lay terms, this means it can tell you what the 95% limits are, as well as how probable all the intermediate values are for the mean, which arguably is much more useful than just the error bounds. 

For example:



```python
from seaborn import kdeplot
plt.figure(figsize=(10,5))
kdeplot(trace['center'], shade=True, label='Posterior density of mean value')

hpd = pm.stats.hpd(trace['center'])
plt.errorbar([hpd.mean()], [0], xerr = np.abs(hpd.mean()-hpd[:,None]),c='C3',
             linewidth=4,fmt='o', mfc='white', label='95% highest posterior density')
plt.axvline(hpd.mean(), c='C3', linestyle='--')
plt.legend()
```


```python
rpts = 100
bavg_results = np.zeros([rpts,5])
chodera_results = np.zeros([rpts,5])
sokal_results = np.zeros([rpts,5])
correction_results = np.zeros([rpts,5])
bayes_results = np.zeros([rpts,5])
estimated_means = np.zeros([rpts,5])
```


```python
import tqdm
for row in tqdm.tqdm_notebook(range(rpts)):
    for col, datasize in enumerate([30,100,300,1000,3000]):
        cor = gen_correlated_curve(0.9, datasize)
        estimated_means[row,col]=cor.mean()
    
        #block avg:
        s = do_block_averaging(cor)

        popt, pcov = optimize.curve_fit(block_avg_fit, np.arange(1, int(datasize**(2/3))+1), np.array(s), maxfev=25000)

            
        block_avg_sem = popt[0] *np.pi/2
        bavg_results[row,col]=block_avg_sem

    
        #chodera, sokal:
        chodera_autoc_time = statineff(cor)
        sokal_autoc_time = sokal_autocorr_time(cor)
        n=len(cor)
        chodera_sem = np.std(cor) / np.sqrt(n/chodera_autoc_time)
        chodera_results[row,col] = chodera_sem
        sokal_sem = np.std(cor) / np.sqrt(n/sokal_autoc_time)
        sokal_results[row,col] = sokal_sem
    
        #correction factor:
        result = AutoReg(cor-cor.mean(), lags = [1]).fit()
        estimated_rho = result.params[1]
        correction = np.sqrt((1+estimated_rho)/(1-estimated_rho))
        naive_sem = sem(cor)
        corrected_sem = naive_sem*correction
        correction_results[row,col]=corrected_sem

        #bayesian:
        trace = do_bayes_sem(cor)
        hpd = pm.stats.hpd(trace['center'])
        hpd_sem = (hpd[1]-hpd[0]/2)/1.96
        bayes_results[row,col]=hpd_sem
        print(block_avg_sem, chodera_sem, sokal_sem, corrected_sem, hpd_sem)
        
        
```


    HBox(children=(IntProgress(value=0), HTML(value='')))


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1851.53draws/s]


    0.1775414728967347 0.19671575297310862 0.1797101018984732 0.2189091263458386 0.5779104872256262


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1634.64draws/s]


    0.8487580687296291 0.5635396371588551 0.46256010109946066 0.6452426294895117 0.9449296367739065


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1429.49draws/s]


    0.7812688822268117 0.6761636200117869 0.5681957738929955 0.5369913706919309 0.7200098519976401


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1668.58draws/s]


    0.3276931042827803 0.26486819214843976 0.26409952889847493 0.27079087828239534 0.3944548200842321


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1351.44draws/s]


    0.23444000911178473 0.20670619940512897 0.21531067116984165 0.1793941923625477 0.24540785753258015


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1423.01draws/s]


    1.8797646169129345 1.0830444565533806 0.6395117742959521 1.5649789450572582 1.7376939873283463


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1373.17draws/s]


    0.3020403302113065 0.4186975049217826 0.3575071864920809 0.4991838434482843 0.7479348826867971


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1500.66draws/s]


    0.987430340980957 0.4418952753887394 0.3871474623619931 0.38462318950287167 0.6198815601663963


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1319.67draws/s]


    0.31014704684753275 0.2866006959203905 0.2765998090111328 0.28223147022169465 0.4685805531188349


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1121.48draws/s]


    0.18791701329417504 0.16187597088810807 0.18226752041104963 0.1679963423489255 0.2007758812617746


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1295.43draws/s]


    1.421327306180503 0.9164583109070755 0.5472141643378722 1.4336836801629325 1.2271149699996962


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1183.38draws/s]


    0.6303436824684806 0.567907795358388 0.3918483700179168 0.6093122456556989 0.8619864870655317


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1336.40draws/s]


    0.33281528088701057 0.3665012699482179 0.2623880800485338 0.47333506737740905 0.8430633902762817


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1682.49draws/s]


    0.3555086939651867 0.32941397267906053 0.29116258577988285 0.2782886805323881 0.37718194822854595


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1474.24draws/s]


    0.1608144734175902 0.16376685459668583 0.1516121164900192 0.16683998262322117 0.25098103345733785


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1315.28draws/s]


    0.6106608736672231 0.5371546818564391 0.35039409675720334 0.599763590350409 0.937513347361418


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1596.03draws/s]


    0.7246356994563761 0.5117427280807051 0.36942849535610395 0.48873505758730157 0.9424169888453606


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1418.92draws/s]


    0.8195240274711044 0.691933087207279 0.5872607554068986 0.7974996112427798 0.8634191470387202


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1572.61draws/s]


    0.21648771992521815 0.25301629201235265 0.19993309285861968 0.28192757498796994 0.4327137516780463


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1456.86draws/s]


    0.15577280238844252 0.15209114524846587 0.14082196762020832 0.17124672306022698 0.2127464489219193


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1687.56draws/s]


    1.1560952934091397 0.7827368067954318 0.47103561046590997 0.6686363869865056 1.4022412177144998


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1699.38draws/s]


    0.4710981571585974 0.37557676703462306 0.30050099891073234 0.3833615706778162 0.7236797518418807


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1703.31draws/s]


    0.4413957116718882 0.4096180231085905 0.3508956619018821 0.4558939832794241 0.6316383480931923


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1501.46draws/s]


    0.28894304464649617 0.27161959374089645 0.2572922995529945 0.25527433405173927 0.5029241689301307


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1428.15draws/s]


    0.19434326229567186 0.16482589767762723 0.18433218989474096 0.17111645430291023 0.29520044028388076


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1647.04draws/s]


    0.53631145710182 0.4466926155498495 0.27677036221187123 0.5185189516218905 0.7164117474709938


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1525.27draws/s]


    0.7231093922688712 0.6548280733162444 0.5123332998043512 0.6099662062508809 0.924354450754671


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1662.02draws/s]


    0.5901440971463442 0.4946281228057564 0.5894002914474423 0.6497599140411283 0.837141198395588


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1677.37draws/s]


    0.21975423092503013 0.21364648940576067 0.20822049695113515 0.2346151792479692 0.12965670662647402


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1489.56draws/s]


    0.1721778265882496 0.18744368950825688 0.16644580688403365 0.17162149852597366 0.2731091182530847


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1441.18draws/s]


    1.2681939228731067 0.8780075863787076 0.39302952704382205 1.3169205224550498 1.1091849323562313


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1802.61draws/s]


    2.0013187223043687 0.352980948975431 0.3050413667770752 0.3250630091511701 0.4853386810105004


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1643.30draws/s]


    0.9244341270529758 0.6325134232758949 0.47391227284235193 0.5172670932720482 0.6731410163937966


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1780.69draws/s]


    0.2885107397705406 0.27445493608491306 0.23398488749021673 0.24526313731322785 0.2669209125405718


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1388.88draws/s]


    0.23312292189456252 0.21690351682515205 0.21755289699947006 0.19089457951731406 0.21262959817761187


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1699.33draws/s]


    0.9291379558867516 0.637428289068169 0.382436521605295 0.6352791378717834 0.857065751342666


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1682.86draws/s]


    0.44578223494469593 0.38599254636537905 0.3780288114990627 0.40617199128235565 0.7507161207216687


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1628.90draws/s]


    0.6231276677077319 0.5561102228485523 0.4745956726095261 0.461613801328974 0.7385055020091371


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1529.81draws/s]


    0.4115688128280707 0.3278604309365738 0.3208242365615976 0.38573310099290686 0.4757332586460508


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1505.35draws/s]


    0.18658266034493504 0.19377050078452138 0.17497065272271808 0.17508443166002804 0.22064720671952495


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1641.26draws/s]


    173.19801503588167 0.675896217657182 0.40156682188004034 0.8442722616885184 0.9673884565043052


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1565.56draws/s]


    0.818160604051244 0.7880077289916549 0.5882140932332987 0.8708625875564393 1.0436254925216377


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1216.30draws/s]


    0.3267336853658787 0.3001610101064714 0.28674576725236184 0.3356353820187165 0.6753030289282935


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1707.77draws/s]


    0.27229511750103685 0.2768967185479453 0.24558030111889584 0.27695164449045345 0.36724174058149217


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1379.78draws/s]


    0.21514309472491874 0.22807978214043592 0.22722690196078987 0.16884397264350112 0.281663017861012


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1665.42draws/s]


    360.5026679366411 0.5449651979153485 0.3343241272650903 0.6049947066100152 1.084382433129715


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1603.19draws/s]


    0.6227917873152269 0.5511301564377343 0.5039485610517975 0.5856197793142386 1.0556934320064653


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1739.15draws/s]


    0.3483725766008895 0.3645606420657617 0.3250655964913254 0.31148539363905176 0.6250526487159975


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1541.13draws/s]


    0.43337917474074317 0.4098767746767121 0.38575637519426587 0.32757871225431967 0.666963476849956


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1302.79draws/s]


    0.2896818881558292 0.26579260463804355 0.27069536920834486 0.1817865334760727 0.1915486627692452


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1763.24draws/s]


    79.19150976235451 0.19623343562571044 0.14571289177243898 0.22240166516829352 0.483369420694841


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1565.17draws/s]


    1.3443488292231245 0.9637787472354654 0.7237026117127379 1.307009561755553 1.401041173861969


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1656.90draws/s]


    0.45326385490209264 0.40391932591414975 0.3318642462943938 0.441750694221773 0.5153843642811916


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1503.31draws/s]


    0.4106355863466697 0.3841662515084069 0.3170399411836984 0.38679033062655627 0.6730693990096012


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1508.14draws/s]


    0.192291561517978 0.19836307090289532 0.190825027396967 0.1941974396395701 0.32599544048251194


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1406.93draws/s]


    1.9117771932475165 0.7944908486603602 0.4626750537674421 0.9507711512315455 1.2561852923814139


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 840.33draws/s] 


    0.49872168425414093 0.37323957865674573 0.24075766790862888 0.40327094847994255 0.4878683875190469


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1400.26draws/s]


    0.6535226611909777 0.547682349711481 0.40385736812475254 0.6146118358503144 0.7514594479276312


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1175.84draws/s]


    0.21605429712850957 0.2166125484001333 0.20115529541282906 0.2499901702464266 0.39333896254253187


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1359.98draws/s]


    0.26384373230518127 0.24903786864015803 0.2483312472333524 0.17779297675713696 0.30371202885938114


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1484.80draws/s]


    1.7121117012971698 1.2492398120106527 0.8576369760994573 2.0357207323789175 1.9047236972864718


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1370.05draws/s]


    1.840412112209282 0.9676997471234289 0.7156703551603909 0.9256524683967283 1.0388848311669054


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1600.89draws/s]


    0.4933863234964045 0.47454675774332344 0.43042106388565055 0.5424320205668556 0.7264905491694701


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1680.14draws/s]


    0.21504061094353535 0.24228215457077543 0.20975379291824517 0.28226880971175694 0.380648792891134


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1466.48draws/s]


    0.19288537166755 0.16009881361043124 0.18478389049739263 0.1737793031074399 0.1883171646813524


    Sampling 2 chains, 1 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1554.99draws/s]
    There was 1 divergence after tuning. Increase `target_accept` or reparameterize.


    8.540543471000753 1.8455219851413869 1.0523048192097402 2.834418947013593 1.593353662374582


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1146.58draws/s]


    0.9190593788061777 0.5450504954855633 0.4300784985513087 0.8069817916290619 1.1568990168227045


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1229.01draws/s]


    0.5219380472919848 0.489339372010533 0.35270581511543014 0.5833245141265452 0.9144200149443639


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1746.84draws/s]


    0.23860900105433627 0.2594694285954483 0.19935895631978043 0.26677963592495385 0.31095148750910917


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1353.09draws/s]


    0.18338998311113808 0.1796493348192239 0.17848348615601212 0.17089590204361382 0.25791263997488256


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1619.53draws/s]


    0.33455198922761387 0.3140310217218537 0.3078159717920874 0.4831549733420813 0.8133040265124879


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1641.42draws/s]


    0.5795955876558782 0.6156798963439587 0.369569095902985 0.7184906910195662 0.8886354234966465


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1674.81draws/s]


    0.6004273015721544 0.47458525648168426 0.4650955535854338 0.4126553187650169 0.5263260868741524


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1372.55draws/s]


    0.21724328445662652 0.23143467405981646 0.19670398498668487 0.2618322546749613 0.38988993197464994


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1125.12draws/s]


    0.18612845294635302 0.18974359006647554 0.17839228734352372 0.20160175580171558 0.2321390497905428


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1783.77draws/s]


    154.3834699906577 0.8497097748964748 0.5101525325211327 1.2882786087707274 1.1345679280657048


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1459.03draws/s]


    1.77830493349841 1.0929264067756181 0.7823498694993822 1.2584487618872164 1.0899482955564332


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1943.71draws/s]


    0.7909878955296342 0.6905416836379026 0.5989671745389781 0.47678699639136557 0.7092996769748772


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1693.58draws/s]


    0.2946896174193129 0.3032100296989333 0.2339653411170021 0.33905438397453064 0.39901950719740825


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1059.49draws/s]


    0.19006149505384856 0.20233519866416746 0.18677991762829202 0.19083420891670758 0.20053186965347616


    Sampling 2 chains, 4 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1324.05draws/s]
    There were 4 divergences after tuning. Increase `target_accept` or reparameterize.


    4.563591076843098 1.0320908283480839 0.658690252115624 1.7325835163809131 1.0101755465252034


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1487.34draws/s]


    0.481749621513137 0.3810866321392145 0.2528644708771754 0.36898954016180285 0.6045527713651835


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1375.76draws/s]


    0.42750267226110233 0.3977451168510441 0.3016742074327644 0.4487686844936874 0.6218975036561399


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1593.20draws/s]


    0.2995555705212828 0.27079744512998805 0.25924657772638066 0.2858040500346717 0.4103315726087869


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 680.40draws/s]


    0.19619296597192415 0.18278133897797333 0.1805727732347605 0.17413678447207928 0.25388692128992085


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1390.73draws/s]


    358.4968826561746 0.4644066048428525 0.28739270477335005 0.5534046684350421 1.0691906112237421


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 836.42draws/s] 


    0.5466226785036372 0.46528843685270777 0.4036150899321927 0.6138849754403157 1.3531531398523982


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1412.12draws/s]


    0.6247060585231431 0.47465779634920513 0.3960105895124672 0.4624852066790612 0.5137872254102354


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1670.45draws/s]


    0.4015581435198943 0.3851203904737729 0.3446174422897079 0.41360326333213737 0.5355217856751064


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1589.10draws/s]


    0.1905239301049696 0.18980289228670763 0.18439928945612552 0.1669445066079741 0.30434605706141277


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 962.92draws/s] 
    The estimated number of effective samples is smaller than 200 for some parameters.


    0.19726501716849218 0.2622813575667614 0.16671540034110266 0.3355892225645495 1.5048370072680646


    Sampling 2 chains, 1 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1371.62draws/s]
    There was 1 divergence after tuning. Increase `target_accept` or reparameterize.


    1.2843021759449713 1.2183714379455681 0.8436930205108114 1.4768466254586556 1.191773667178807


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1252.43draws/s]


    0.5912675769579252 0.7318867487286662 0.6771991906127026 0.6131397611742141 0.688256986988249


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1108.21draws/s]


    0.38671572221907957 0.4141486891100327 0.37645077258130516 0.3687082482831306 0.5053084617622078


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 811.43draws/s] 


    0.20887692240475172 0.1976073721051388 0.20803265760968462 0.186970170188265 0.24921025580898787


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1034.08draws/s]


    1.4889291342272304 0.6278183839237806 0.3939704095549215 0.7704972200502216 1.4072498738502275


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1412.34draws/s]


    1.4093832873287249 0.4725972049762015 0.4660480841686638 0.6601747044078089 0.7996044692774277


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1180.61draws/s]


    0.40184773053927975 0.4105251945395977 0.3137965410837633 0.5998886255686711 1.0007265241033005


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 847.79draws/s] 


    0.30816211370826246 0.30415745475701667 0.26213872413854294 0.3539809520810611 0.6156555580473873


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1044.95draws/s]


    0.2679045269614124 0.273190321493701 0.2729695591358273 0.18723262605006039 0.2944057849715867


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1184.32draws/s]


    1.2659121937389208 0.6478401148491082 0.38920840521198363 0.7543419218470068 1.5200223173755945


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1351.18draws/s]


    0.4419450573801067 0.49641148693338094 0.4650561049264574 0.4011308787213311 0.9943185218974966


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1815.21draws/s]


    0.5591379145392515 0.43303897454206736 0.39524615521975087 0.42796626562873685 0.7223975202131276


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1600.05draws/s]


    0.3588401340633063 0.3310360539382048 0.34178407886223594 0.3581807688158834 0.47688159381538914


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1391.06draws/s]


    0.20022039177948225 0.16493726016137675 0.2000972241763042 0.1756194944841159 0.2763968697383641


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1964.52draws/s]


    0.2041017447853818 0.21993310077676503 0.14395114624083874 0.19822118145545034 0.5160171257212106


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1922.96draws/s]


    0.680309295068929 0.539226895853981 0.4452367682082974 0.362590412091373 0.5039808949393073


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1762.21draws/s]


    0.48597579896874943 0.43313528336172197 0.4855507267819351 0.5470070888714293 0.7408518799627896


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1801.47draws/s]


    0.3030011323965275 0.2687733351548186 0.26088782439900426 0.2839253404793374 0.4044370617023494


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1272.30draws/s]


    0.20842543444242292 0.20243965103554015 0.20244361502234098 0.1834628029939518 0.2652569842643416


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1610.29draws/s]


    3.8274874793650664 0.5894449608328447 0.38620060115891314 0.7551648062300655 1.2469129091931357


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1509.17draws/s]


    0.9910325122746797 0.6775129584656501 0.5995814147482373 0.7698828703364603 0.9543776484652121


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1487.43draws/s]


    0.5558643573946324 0.4704774274029255 0.5226369056351018 0.609793290275381 0.738136681026318


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1532.78draws/s]


    0.35378215258415496 0.31343676977453555 0.3197402673191953 0.3404522144233756 0.4872476362598709


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 736.85draws/s]


    0.19820911406625727 0.19513480765607313 0.1873232646190607 0.17107971145838635 0.16961833122179473


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1800.76draws/s]


    147.11482782715737 0.28318311193567797 0.22629086394941825 0.27004463847281057 0.6508686393458939


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1225.47draws/s]


    1.3811532240626214 1.1013777481824176 0.7923475394754144 1.129136455051499 1.1131254414815253


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1606.61draws/s]


    0.6660722998219081 0.5184056828666241 0.4465879633187022 0.4202335804895746 0.5986261698113285


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1188.20draws/s]


    0.45444213862064664 0.4209246709664384 0.4101948482896576 0.4053527370669045 0.5336976263141857


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1139.91draws/s]


    0.20202252675060567 0.20622360200066955 0.1930586061778553 0.20736478396179525 0.18313892614506447


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 675.86draws/s] 


    0.9899957401665841 0.7636638547757854 0.5902676257536649 1.07147480695186 1.1204736136722842


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1285.58draws/s]


    0.945098677465371 0.7589222559766532 0.6477314534062839 0.8548162698216838 1.040790763280735


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1037.80draws/s]


    0.5235830622436211 0.5028098700276498 0.3841130533508262 0.5130815269662722 0.5797495918381462


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1307.80draws/s]


    0.27931968210514985 0.2872320386289162 0.22641402364511956 0.3077951338690144 0.41229560504852


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 676.67draws/s]


    0.2524345323853646 0.23399673835832532 0.22186516007002538 0.17841150308320441 0.21863959005115646


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1345.97draws/s]


    0.33536227689896464 0.33413754404258905 0.3571543336006664 0.4646763788390154 1.1852288492024654


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1376.06draws/s]
    The estimated number of effective samples is smaller than 200 for some parameters.


    0.9271198629348014 0.826873655846912 0.5746323483583211 1.0509135055700571 1.1471692921915313


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1486.04draws/s]


    0.4198972355380126 0.36622183736851666 0.3947574592086428 0.37827663198989 0.5523487475859271


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1697.29draws/s]


    0.2730812564200423 0.2796040055127889 0.23254210870793066 0.28613461809545193 0.4551097358091561


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1475.84draws/s]


    0.20030022730542363 0.16308517683004686 0.2078163267622064 0.1622633163832274 0.21551959874302898


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1471.59draws/s]


    2.4417226067019566 0.9807085308606699 0.5562038305646809 1.40413851582019 1.4902172918641696


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1636.09draws/s]


    0.8108897965264645 0.7138854971042031 0.5771361937334535 1.3267899524879716 0.9932900777438841


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1835.98draws/s]


    0.25198194873769275 0.27906949768597966 0.2039710784127089 0.30814199827776784 0.40640639521430383


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1383.96draws/s]


    0.2864982877837504 0.2865212952121241 0.2489624301691889 0.29697599579481104 0.3043247018912048


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1310.43draws/s]


    0.20826580838728362 0.21547585562545546 0.194242388562797 0.1840457738986967 0.16565166830215944


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 857.86draws/s]


    287.08734492158527 0.7230642664846537 0.4727101015339873 0.9339972230487603 1.2913219240321938


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 853.25draws/s] 


    0.5384991689552325 0.6008750702036707 0.6528418152884932 1.0358898675002939 1.0226034524634282


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1637.44draws/s]


    0.6656319272127031 0.5979551896315545 0.5172140064535692 0.723584423026869 0.7256655490228741


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 920.91draws/s] 


    0.2036864689059524 0.23175776981608423 0.1810068225688533 0.2557568360494944 0.2517756888057576


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 675.79draws/s]


    0.22163685327514357 0.2031279853642988 0.2085207558532791 0.18164332196316121 0.23973213703386245


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 550.94draws/s]


    3.0720280001602447 0.9994326654756305 0.639206931050541 1.2951643335217053 1.2664702199988374


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1383.40draws/s]


    1.376429379007051 0.5592692539922549 0.3774514521618829 0.6452366924522013 0.7910178919492


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1397.82draws/s]


    0.4143407902947615 0.40127047906068836 0.2867673848790794 0.44721459930959706 0.738701514891038


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 712.57draws/s]


    0.35976559729905927 0.3000756907038594 0.26783171949492224 0.26323622203215025 0.43656387332113655


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 695.90draws/s] 


    0.1519143492385616 0.15741895465171563 0.13811833374383062 0.15149821678584824 0.14424263398817683


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 572.99draws/s]


    1.2598242419196826 0.8345993896367866 0.49413811286961673 1.2826629603078326 1.2389682104779753


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1479.46draws/s]


    0.2557978767575349 0.30491682110147744 0.20078805232669028 0.39048801180256615 0.8268106329379613


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1330.31draws/s]


    0.32864957758526475 0.33962840927541543 0.24907034839622758 0.36477317116488783 0.5498565526715551


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1570.97draws/s]


    0.23763551056355958 0.2241208181016884 0.21760202812599294 0.24194134354981062 0.32043852658942795


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1118.76draws/s]


    0.17953656522806785 0.1825567397435639 0.1574253431575793 0.19362812004868676 0.25875020543303395


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1496.88draws/s]


    1.7529761726324231 0.5212172425294663 0.436069483960598 0.7511841346436218 1.4550854135103286


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1491.88draws/s]


    0.6589667121912225 0.4615193910694155 0.4908417857325215 0.48947393168461656 0.6199177173705849


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1364.04draws/s]


    0.5737269332695069 0.4901070681549263 0.37139834367093505 0.594140156725316 0.7922607410179565


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1183.97draws/s]


    0.2625340191695283 0.2725728462452484 0.23047379203713822 0.3262468282805317 0.662350751538325


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1053.48draws/s]


    0.17701237866188171 0.18901153530517575 0.16842801069141028 0.18938046764651809 0.22319310735123898


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1528.06draws/s]


    6.268380274000467 1.0748111228840571 0.8037338030334407 1.2920309762784645 1.916908843382388


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1064.03draws/s]


    0.7710038171561459 0.8374444837319471 0.6205596342944154 1.2534295399287316 1.2014541175986941


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1672.66draws/s]


    0.537252870025627 0.46481285823951235 0.39049342377094415 0.47636191079765494 0.6139271447281618


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1337.23draws/s]


    0.23413809142231834 0.23057129266167406 0.22034964933474172 0.25332118616105426 0.4253190782032551


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1206.43draws/s]


    0.19953003072689054 0.18312609117379372 0.19952362256955042 0.18611260680825556 0.33259446822154065


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1843.68draws/s]


    0.1810863851684402 0.1536092026108118 0.15402584814275574 0.24536764946119555 0.37910384081075404


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 660.07draws/s]


    0.3109070659762947 0.39147098791942053 0.2555615730960482 0.5260037742878223 0.7493862710608926


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 742.75draws/s]


    0.38082749300251584 0.33415916470872525 0.3082596999880366 0.36344128267961956 0.6127343979369966


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1441.16draws/s]


    0.40518805861372775 0.2942399793733618 0.3337946849551994 0.33834261697481055 0.49938088756967225


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 587.46draws/s]


    0.1703688336596339 0.1708846294650206 0.16310541970523093 0.1503495909663888 0.28888942731793876


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1299.10draws/s]


    0.6472066352676058 0.4521702024238416 0.3184915833950223 0.6782671685823919 0.8708965399180957


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1440.47draws/s]


    1.2934527896136483 0.5876134614698448 0.421328361942313 0.5646368288967054 0.710499799603674


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1407.76draws/s]


    0.48629690354436406 0.42625957417102833 0.32700221472229646 0.5315325322165153 0.7982758330801394


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 565.96draws/s]


    0.33479280074844275 0.30002413211433193 0.29495980659296084 0.3212204843703616 0.46763024038294854


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1245.21draws/s]


    0.18218872515175108 0.18221165949812226 0.17287415542238105 0.1952697073773093 0.21466773118686927


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1368.72draws/s]


    0.48680950504453724 0.5317022394507648 0.21336236714806467 0.6684842522755219 0.8921369261397304


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1312.15draws/s]


    0.3116064995745823 0.3252768493086109 0.22400421252015834 0.3737468677540795 0.6908845692961338


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1288.13draws/s]


    0.6188157178313378 0.5417343305360218 0.40074372730184094 0.6244796276821929 0.9046107302226761


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1496.79draws/s]


    0.34823912558885206 0.32978210449548123 0.3904455348534413 0.3435304369952803 0.5040518436393044


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1193.29draws/s]


    0.20824929256833985 0.21205614620816865 0.21184271206435867 0.17444279559178033 0.19854545045508676


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1402.96draws/s]


    1.5413099588610415 0.9662286230339514 0.5492771497524488 1.6343240172649345 0.9805882006780958


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1446.01draws/s]


    0.9674222881671486 0.8480830077498644 0.5383544639479549 1.1454030290606894 0.9650500313006036


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1735.62draws/s]


    0.4708971257058685 0.3974917540134921 0.3902425904470175 0.43262350798649035 0.4911762066605434


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1713.28draws/s]


    0.24729146066914048 0.2452099118761947 0.2376655266960935 0.2879224158478229 0.4424337330768339


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1415.70draws/s]


    0.1758803628884714 0.18120691712969486 0.16224281784924882 0.19024950624688544 0.305754258042933


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1690.93draws/s]


    82.65317093677776 0.33797230434172537 0.29025063457577 0.4339150492813813 0.8238813309800764


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1690.13draws/s]


    0.8217349888756038 0.5617514210024148 0.4864517002624683 0.5682694942217452 0.5930115047816756


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1808.96draws/s]


    0.4499638500467926 0.45860213892492124 0.33799213119291627 0.5849133426851202 0.6787069882809058


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1428.92draws/s]


    0.33081731362810735 0.3327066694897861 0.2514909202021081 0.355080122049177 0.3621736021854554


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1204.88draws/s]


    0.1921531223808153 0.18297550595351855 0.18528396484650173 0.1741313813859037 0.20221312730232407


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:04<00:00, 449.09draws/s]


    2.7809716531360316 1.003395633965294 0.6027266397866925 1.2922293563359768 1.1885728894166356


    Sampling 2 chains, 2 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1325.74draws/s]
    There were 2 divergences after tuning. Increase `target_accept` or reparameterize.


    1.7666845027799485 1.0870155773206063 0.7075201409594946 1.1155235914318113 1.0491005099624955


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1702.55draws/s]


    0.4745421794461975 0.4241477346939336 0.4531204244562684 0.488488331254526 0.5314537000011111


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1528.87draws/s]


    0.2770850363826961 0.26185211698694355 0.26740414064370854 0.2716658271719474 0.36865460540628403


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1030.44draws/s]


    0.17889818855616935 0.17065944989494272 0.17843776712614445 0.18055200540861352 0.26182326546652285


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1278.97draws/s]


    0.3599074733769311 0.4020006487074424 0.2798037034188761 0.5654715761961456 0.8210469436608812


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1689.72draws/s]


    0.4835449687407226 0.33134929536488084 0.3249684223637112 0.3296358902034251 0.3961766806933809


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:04<00:00, 477.30draws/s]


    0.5465941680119567 0.4621329195226847 0.552022184309512 0.6018624787654616 0.8905309411664333


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 661.69draws/s]


    0.3427617472579805 0.3046407749773335 0.31173265842218095 0.3156853991319039 0.3478240534801444


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 834.64draws/s] 


    0.20197280587866515 0.20946320982908218 0.19177055662192496 0.19304820800656738 0.2407407537232123


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1567.48draws/s]


    0.8939199896388605 0.7468738382644616 0.5156238021517442 0.9620547611651055 1.1306130261703944


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1523.59draws/s]


    0.8842015176568322 1.0021736911706374 0.7608274735579457 1.1343622652443748 1.3137000903855762


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1340.63draws/s]


    0.48354367439921525 0.4758652626253213 0.5187252492165311 0.6205427670514299 1.0375831151335126


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1255.12draws/s]


    0.37466427765785376 0.36011276886765475 0.3588763390802071 0.29736386140313337 0.5127563048177253


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1272.21draws/s]


    0.12553727009795634 0.1602263457561981 0.13306223709416126 0.18160963617527318 0.2559775492794088


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 851.70draws/s] 


    0.33778516915970513 0.4175354472070236 0.35297557634333854 0.45606551100989795 0.752710850371752


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1385.51draws/s]


    1.016196971537362 0.7586847603878923 0.511131238939506 1.0374863678657522 1.2671305720638988


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1610.38draws/s]


    0.280187989474519 0.29563283695878284 0.21982016845492902 0.31248370068141695 0.41382241739833153


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1657.76draws/s]


    0.3662995045861432 0.3792079090800446 0.3616551603635595 0.25358690791962174 0.41177109197002093


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1458.82draws/s]


    0.1710302466868382 0.19650061624956747 0.17141153375948776 0.1877354071559833 0.23778953872541522


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1213.58draws/s]


    0.5354533631021944 0.5198325181508283 0.382106541304986 0.7250463079929861 1.183948331061097


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1317.15draws/s]


    0.9546924211065564 0.7137258742089267 0.4423944812910726 0.953798037495527 1.1936973337051338


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1572.66draws/s]


    0.7157923796508395 0.641638930583854 0.5273087476257624 0.6404187288157243 0.8084023458114652


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1656.68draws/s]


    0.19296158706707034 0.22325090813288015 0.15564281357016418 0.2678930539795419 0.3736410179883518


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1383.84draws/s]


    0.19868776589124604 0.18951563645004144 0.1937801540966416 0.20053523335230067 0.3029677347587762


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1705.99draws/s]


    1.5192609765940415 1.4161110979411138 0.9894886775002569 nan 1.4177371780251946


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1283.40draws/s]


    1.0377863319786707 0.6317020733396935 0.5058844895833547 0.900464724315951 0.9773548637159328


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1718.93draws/s]


    0.4364400094355081 0.40661336317060925 0.2976695709252044 0.42333679538612484 0.5698074227897886


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 882.07draws/s] 


    0.3056078270997437 0.2811024907664927 0.23367365780631863 0.2853549503019566 0.3335385339946843


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1354.90draws/s]


    0.14302233775857834 0.1568668523071928 0.13313014817399643 0.17028407050438635 0.21656288695266457


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1752.90draws/s]


    233.8960967729914 0.3872601472916355 0.15531764325409333 0.4021433953317353 0.5881570898852037


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1797.68draws/s]


    0.3719445976141488 0.29780588656407636 0.28427858350417495 0.3515923154735593 0.5191297871767344


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1729.08draws/s]


    0.4078401147422451 0.3896734515771892 0.34375971247143683 0.39455719202094974 0.4748796173205546


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 542.90draws/s]


    0.40943151710074477 0.3445586744739726 0.3969779663434278 0.3379279045158298 0.5179412178344903


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1321.13draws/s]


    0.21609795071508234 0.1899617722736251 0.20158354012984522 0.1886283039125503 0.33609740789702075


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1502.26draws/s]


    0.6101807169209651 0.7432078730370708 0.46561065212838926 1.0142841927340387 1.2874944542235016


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1624.87draws/s]


    0.5697239367923056 0.42145184120551105 0.362394462249855 0.4834935610221621 0.954635005895625


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1146.09draws/s]


    0.4994589081584294 0.4346940550858951 0.3828563846298375 0.4057472451849258 0.5471927597525845


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1760.31draws/s]


    0.4059892329199012 0.35545698472888587 0.4998175653245916 0.3161784553286392 0.5104775242426641


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1528.07draws/s]


    0.25094946125468903 0.24523224121728007 0.22999527091013797 0.18404167000929822 0.2664071998886278


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1730.24draws/s]


    0.28152364709405076 0.36055329389627255 0.24589469211775153 0.34813648619127513 0.6021393091654456


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1602.42draws/s]


    0.6903244182292614 0.681462601117773 0.501874449829312 1.0007750023691944 1.219699934520798


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1828.03draws/s]


    0.5498924600664882 0.5186581861579478 0.530713827274929 0.5648507188607971 0.8141384992991307


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1873.73draws/s]


    0.22588840768965782 0.23864254477163602 0.20753912877951888 0.25324323176431834 0.472155073778334


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1519.45draws/s]


    0.12625078311183174 0.15388027919149366 0.12272938890192084 0.17750474107808026 0.1902568071359213


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1832.93draws/s]


    0.7924499788010555 0.8371087467551014 0.5209529581524165 1.4415934984695007 1.1680387654681121


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1878.02draws/s]


    0.4741114960544898 0.4520613867789225 0.24897738643865744 0.5876256745621029 0.6966334448976169


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1695.65draws/s]


    0.5718360593338581 0.5094315785908263 0.4199490499210815 0.6130460364012456 0.8930637658688747


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1771.30draws/s]


    0.3487224073261339 0.3378811303079358 0.32365646243820484 0.30905505421758994 0.5000069581613191


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1564.77draws/s]


    0.16396098833474876 0.16982179934086888 0.1468798626778368 0.1845487670386748 0.3542240697005147


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1959.22draws/s]


    0.1918646164242473 0.2187942253721317 0.12747989356897124 0.25027725599700124 1.2413604871453459


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1872.56draws/s]


    0.6764819035563948 0.761863763307099 0.6255827213267464 0.7233795323525849 1.0138021000302377


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1751.37draws/s]


    0.3535221588518116 0.3220790008698524 0.32583733799673575 0.34206101285819734 0.4110972780610992


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1829.02draws/s]


    0.26772386236503937 0.2462038421054629 0.2374513114920732 0.28362911352937303 0.377782072090016


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1573.79draws/s]


    0.182792927215819 0.1816829515289737 0.17541071397336408 0.15926568669200977 0.3092012112219338


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1742.44draws/s]


    626.3170397024869 1.0581157290046752 0.6184223313572978 1.208264352595088 1.1807890033238888


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1914.75draws/s]


    0.4511090036639681 0.4147793318432422 0.29464101344333987 0.4049364523328442 0.6546834932126114


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1878.00draws/s]


    0.35130513935057833 0.3198294723943006 0.27634488147512837 0.3862422132953187 0.6379791825941935


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1641.26draws/s]


    0.26441501349716867 0.2758038432763218 0.2279300813378305 0.2848125692668828 0.37160072905893743


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1506.21draws/s]


    0.2010033118852553 0.19366357245437035 0.1977637424524717 0.18891721137967707 0.23284515632943


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1243.34draws/s]


    0.3557219252917656 0.4363150293590977 0.3073189519202698 0.49370100383394117 1.0896853165795124


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1219.38draws/s]


    0.6782721646722398 0.6645534882121019 0.399423104246022 0.8235432482242723 0.8926070484166573


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1666.63draws/s]


    0.7751229407354204 0.6290128426843278 0.5680899268227707 0.6814804156396631 0.9262145624021408


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1498.05draws/s]


    0.40076736148834236 0.304982280085825 0.29028129620454446 0.28994820140122657 0.4982898144584494


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1266.42draws/s]


    0.19797975721425157 0.1992654138192461 0.19414328103716993 0.20251009349617718 0.26603297985962476


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 920.30draws/s]


    3.7567940560516155 1.1672178018792765 0.6804549734593154 2.1935225061342494 1.5667800923728177


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1642.06draws/s]


    0.6810565261448842 0.5646907158357047 0.3610236598558526 0.7367550461338747 1.1793500907790246


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1497.25draws/s]


    0.9413296082604445 0.40619846263568776 0.3773441691067639 0.35799637400566714 0.46906108267360275


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1568.75draws/s]


    0.36238300978107835 0.3251388346127607 0.2787860502144926 0.29213112938577396 0.5245333531235202


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1118.14draws/s]


    0.17771329966709815 0.17943535272663777 0.16686744422226765 0.17454056874189156 0.2998748332726931


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1585.54draws/s]


    2.2139115706636323 1.1947142289255754 0.7088274701171158 1.8354443836020031 1.3755148560958281


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1449.10draws/s]


    0.5132163570277994 0.33407568577927094 0.3781798648520408 0.43010801168534063 0.8160299096685516


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1247.62draws/s]


    0.5369196948308221 0.5378456949068218 0.47563310019794125 0.7074309770954731 1.0199908386712766


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1453.56draws/s]


    0.3257187216771365 0.3104674218261809 0.28400285763751637 0.31942907636367646 0.37280924492340334


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1126.46draws/s]


    0.2150846565746109 0.1965214412704586 0.20253629759444583 0.18350726037882398 0.23604531468138476


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1279.91draws/s]


    0.2136200293106617 0.2551487694261103 0.14072422659023337 0.3234182996709902 0.7348588706196294


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1208.41draws/s]
    The estimated number of effective samples is smaller than 200 for some parameters.


    2.361548852126914 0.9066323301256748 0.660983330230962 1.1337096795200547 1.538270075419331


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1281.51draws/s]


    0.6561616212438184 0.5103763387157126 0.3867834474185106 0.5091218256421597 0.5415102910707936


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1660.71draws/s]


    0.3424940698421242 0.3000784630426187 0.306349829522102 0.2815862726225892 0.38594066992265896


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1551.02draws/s]


    0.20291411686647717 0.19800321162007828 0.18189249672263796 0.16776100024910665 0.2015068992602774


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1250.34draws/s]


    1.0219062699196422 0.6199341626399384 0.41940178249242227 0.868582477268564 1.1347390672258102


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 894.44draws/s] 


    0.5765479817079417 0.44535600064773484 0.4321298352822118 0.5770410697752758 0.7872469682586233


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1477.64draws/s]


    0.7921705063279575 0.6239998262981019 0.5321878038892173 0.563777797573698 0.8307564802119765


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1308.84draws/s]


    0.3340525831968138 0.29696319277534106 0.3290532803498224 0.3598839452709763 0.5433091829284884


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1369.35draws/s]


    0.13676236300027755 0.15585319256659644 0.13833550955177393 0.16712323309976784 0.1878980000202824


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1476.54draws/s]


    1.9592034896321908 0.8073162265426902 0.6364165357720257 1.641457926361816 1.1283702618255456


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1512.93draws/s]


    0.8512650532112656 0.5704464898375484 0.44111728761719854 0.6048555264527343 0.9253013037967143


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1531.96draws/s]


    0.5581716560868278 0.5637257787652715 0.46572498599282935 0.642520410340414 0.8556775321797482


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1605.34draws/s]


    0.4552914197964751 0.3903931262560442 0.3976415275622676 0.2726208806899546 0.5038404660266158


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1372.61draws/s]


    0.1662904090375172 0.1619242636490825 0.16023516193851908 0.16881370611595303 0.30454950015987714


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1066.13draws/s]


    0.5147175156131937 0.4693497683239484 0.3980775585395831 0.5903766071243449 1.0681693865769188


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1526.82draws/s]


    0.5102379581965739 0.565190727052818 0.39421639979146256 0.8099201028337835 0.966884784599871


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1636.88draws/s]


    0.6602716584562255 0.5042004758091537 0.3974620380590587 0.5438045514674347 0.7263710284238798


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1656.32draws/s]


    0.30992420637770357 0.30303581722786965 0.27981492703070737 0.2592367556390106 0.29014220725332757


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1141.03draws/s]


    0.18170932740850043 0.19659541217879783 0.1728786753593973 0.18582089079696038 0.2168505604351358


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1627.99draws/s]


    1.285676241318278 0.9658042017390962 0.5753479496620629 1.1073874939070845 1.430558088258342


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 798.29draws/s]


    0.7204322096712458 0.7929375540398867 0.5857155973315928 0.705763625901068 0.9678172962308713


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1213.40draws/s]


    0.5567684866222842 0.49399143281783225 0.3719101950202836 0.6544096974007282 0.9952415595341063


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1819.34draws/s]


    0.22643583709033266 0.24329503913645137 0.1972237709060733 0.2634261486313758 0.32771266201634636


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1288.85draws/s]


    0.1585690528308405 0.16539219658104642 0.16284647494947968 0.18165199385743755 0.2646696135152358


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1469.07draws/s]


    141.10976266709645 0.3029966642522538 0.24347235524977842 0.3097310489684796 0.598248797294143


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 683.23draws/s] 


    0.9315176413857881 0.6118832097290884 0.5535520286551604 0.6356880742994764 0.8333073403669456


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1583.92draws/s]


    0.6014195927355637 0.47083787684733 0.34682182958477537 0.5218203312463028 0.8026349190645694


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1227.43draws/s]


    0.25302674254438234 0.26201982404291174 0.230232136851878 0.24647061272238188 0.38507807177668735


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1383.34draws/s]


    0.19996129627369735 0.17367479497381924 0.19728693141906448 0.1799573364507872 0.26237115268116423


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 934.50draws/s] 
    The estimated number of effective samples is smaller than 200 for some parameters.


    0.31730360245774697 0.287120059019806 0.19646663283568655 0.32224376168387747 1.318507925798663


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1853.23draws/s]


    0.4718318312325056 0.34082818855611363 0.274691794539753 0.21768337705643834 0.19919051393458292


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1669.98draws/s]


    0.5274765667400342 0.3710929062097665 0.3125027957599142 0.37201526454063727 0.4396069979954516


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1698.75draws/s]


    0.31651279915682934 0.3031299303034757 0.34339237682097196 0.3352506108154917 0.4067992252751218


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1502.38draws/s]


    0.24713917286751175 0.22703764451038214 0.22175360079939693 0.1851591361375257 0.23536639679332264


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1675.33draws/s]


    0.8069384585010463 0.5688111972756261 0.24158044872840956 0.7932192883492022 0.8207610680611659


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1549.64draws/s]


    0.5939108584604381 0.4648244631867834 0.34052732870215535 0.4327065316559554 0.6757714668552696


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1790.21draws/s]


    0.7992725838409476 0.6495592990901021 0.5805438915805129 0.7446522499604469 1.042983413933045


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1800.26draws/s]


    0.31091198286545685 0.3094654143281795 0.2608450141028106 0.3102269311144452 0.4265811174193536


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1005.74draws/s]


    0.1462008366081577 0.16259365045809696 0.14758560054372263 0.17492165478201796 0.29573613348100763


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1587.14draws/s]


    0.5735063446732471 0.5438935495801908 0.2967784818402497 0.6712334482143403 1.0365161468183774


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1780.11draws/s]


    1.5272526758500846 0.8593468790643222 0.5930505421265655 0.786459608528951 1.0012970561290884


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1716.82draws/s]


    0.5822591924930709 0.5481279474581713 0.38301066114107046 0.5680822502270865 0.6971689674433208


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1798.67draws/s]


    0.2601492792743275 0.2426320933197687 0.22630886435645386 0.2732580925294871 0.4082383629675712


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1513.98draws/s]


    0.1769903329450572 0.18107973896630714 0.17123630160245756 0.16216492520451967 0.211671765506225


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1550.04draws/s]


    0.8214420852583011 0.8390567659797054 0.6190555782888256 1.2748080261030856 1.340527722123027


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1362.63draws/s]


    0.6862113194735062 0.5528732647157169 0.41632101025460094 0.6382416918892715 0.8185073345449237


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1393.72draws/s]


    0.6035475347306246 0.4988450537045423 0.5663876928892421 0.6172753361272786 0.8090183370530005


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1518.41draws/s]


    0.3747694560090121 0.3162345908597507 0.31001344732832264 0.29586235662015237 0.46518699970486194


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1320.58draws/s]


    0.1803531275490575 0.18566003862295358 0.16460428758743587 0.17495610840848447 0.16689058993766498


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1465.06draws/s]


    0.38728821096479904 0.40537758592185713 0.23832325181564357 0.623230032021631 1.1860487327559948


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1423.81draws/s]


    1.0730199081434206 0.8807952174732584 0.7751670049539255 1.0277167948171646 1.1793128835809288


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1628.64draws/s]


    0.6652328611017315 0.8014611989779846 0.5836666408335768 0.7244673512459013 0.8620669603342546


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1547.71draws/s]


    0.2599263077255234 0.23078392665837455 0.234103187421326 0.24393621264676316 0.31230761834589793


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 645.25draws/s]


    0.15728013411261707 0.15919287435000243 0.15910393717933838 0.16154762326216682 0.1692299537836156


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1149.05draws/s]


    1.0705120284124021 0.6885783623200472 0.4383985490222038 0.7518739432818294 1.0574308584198426


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1609.21draws/s]


    1.5761651810740942 0.8167932360699918 0.5126302378065192 0.9947382221566052 1.2657375665882473


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1454.26draws/s]


    0.5016561880912966 0.47538682364903934 0.48439193135015446 0.5378817455716182 0.8391439527298495


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1682.43draws/s]


    0.24115742791659162 0.2460471556126554 0.204963303462093 0.23760483279145783 0.43456206503840766


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1056.50draws/s]


    0.23529685261145572 0.20668081827643278 0.2331876597783781 0.1964653397495552 0.2788315419811274


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1692.48draws/s]


    11.836002107895958 0.2754002912956425 0.2504808880996196 0.3383551001505874 0.8036383340051508


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1486.47draws/s]


    1.1103495317346437 0.8563243492940326 0.5504431254000848 0.9584311355271768 1.080647667508869


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1787.19draws/s]


    0.4701969575356075 0.4530328714483414 0.3161137139539585 0.48587388497606315 0.618827715183955


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1550.67draws/s]


    0.3198361619927883 0.30546716800068435 0.2775372517804048 0.3161584065701286 0.4049727981968063


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1148.60draws/s]


    0.21485066491259766 0.20989557710757134 0.2004809980579448 0.21400233754540693 0.3552603518364715


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1443.44draws/s]


    0.8464436073877766 1.0297111310845044 0.7174362438458796 2.483260713207272 1.263694648206908


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1737.87draws/s]


    0.6039090325048726 0.44954903990854106 0.3569065269834823 0.5152901456769726 0.9616080668724478


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1408.94draws/s]


    0.3622114118240459 0.32559973398767045 0.26665555861990475 0.3620780352780893 0.4483015978497335


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1509.41draws/s]


    0.3772577934459329 0.351700875015502 0.3058250435617292 0.35009617541076476 0.46597904659725764


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1426.99draws/s]


    0.17672397954935665 0.1714799303218345 0.16275163797024994 0.18870377926725931 0.30535569823401304


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1441.27draws/s]


    0.46811378226854167 0.4226152577544994 0.15294852145895385 0.6214197756255592 0.8719852442523516


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1317.92draws/s]


    0.7839911996725352 0.5106164834254917 0.40577696026015275 0.5037484265031454 1.2070993229779958


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1449.75draws/s]


    0.6730504536959336 0.540233214253042 0.5026977390382374 0.5386686421520386 0.9258498345674273


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1397.79draws/s]


    0.36154803621927706 0.3155189901269337 0.3298555037887246 0.3456741554269557 0.47723275513494245


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1059.96draws/s]


    0.19001477931455774 0.19497750027910635 0.18820825718280002 0.18579943853214406 0.2270803948110535


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1277.60draws/s]


    608.4734645726838 0.8014457551860436 0.5154418859010031 1.004705912658659 1.0938918035152465


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1459.19draws/s]


    1.1768892047893826 0.6812812963363265 0.5175788952047058 0.6682865883007336 0.8057916964354904


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1541.12draws/s]


    0.49437307777793543 0.45430108424999366 0.2955675787043118 0.5193428182533433 0.5988674653831044


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1401.59draws/s]


    0.41658382346139566 0.3303659240927114 0.2914562427426072 0.26805238298022227 0.41991943239434326


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1519.50draws/s]


    0.15528166045417133 0.16929495725868837 0.14442827293749458 0.1892613622578549 0.24222802231655025


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1635.10draws/s]


    1.2509641618709075 0.9942025599134039 0.5800110103973718 1.8645322820499124 1.383500378581444


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1915.62draws/s]


    0.6751872170963691 0.5736389022837863 0.3822581224893913 0.7544926142152822 0.9164539907477621


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1695.00draws/s]


    0.7205501515781517 0.5849893027503626 0.3617950831917529 0.7213837603513498 0.8447466665812928


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1620.97draws/s]


    0.3671882729621907 0.3041227357118025 0.29018480932927165 0.32822228450629587 0.5718055445429351


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1478.59draws/s]


    0.19462968048549711 0.18829802618040026 0.19951698137900636 0.17833566354016903 0.23202880574989276


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1277.91draws/s]


    0.6163275818543261 0.4058849704518103 0.2764764913043075 0.48965272154851747 0.8864051766989721


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1737.29draws/s]


    0.6396073764981794 0.4959131017307134 0.6509760301796433 0.7302383326198931 0.9521435688959753


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1558.94draws/s]


    0.8698664150447359 0.5811903720214245 0.4876550699997371 0.6080605827450642 0.7265374439762464


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1471.04draws/s]


    0.30050494461190164 0.28908739428941027 0.3202637377192739 0.3291410554732779 0.5193737516249713


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 855.53draws/s] 


    0.20781171162602505 0.20122423168682915 0.20871342329285064 0.199201350451296 0.31992005055878114


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1349.88draws/s]


    165.45803892791002 0.4215500559792748 0.3362586960103542 0.43741644633209104 0.897267009067749


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1241.29draws/s]


    0.5234436201585222 0.49446503120018054 0.3185437945887008 0.5825458299336713 0.8054336160263112


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1387.31draws/s]


    0.43242737496051353 0.40593910560717883 0.2915548843312067 0.3599813162903552 0.2634566164612401


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1274.97draws/s]


    0.3680718103816974 0.3095713504727558 0.305014700179895 0.3341887598901436 0.4544336841223466


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1210.18draws/s]


    0.179716534122872 0.16919294097046403 0.16461506552613878 0.1581199069740349 0.23473447450893614


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1220.53draws/s]


    2.3776389682399723 0.7329718630990368 0.5459027106233623 0.9664219527882993 1.3533101827559337


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1720.77draws/s]


    0.584590708601321 0.5048947167220431 0.3431893490034834 0.688882249653039 0.7997195440887483


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1671.50draws/s]


    0.7671855896264975 0.7611265063564951 0.501853412060168 0.9287277948054077 1.038747068274499


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1813.81draws/s]


    0.32489401874410423 0.2864089350474448 0.24298557693332176 0.282156414708289 0.314023791980402


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1516.41draws/s]


    0.1913062889024614 0.19255676964507942 0.18946099000811634 0.1834995393381278 0.24068571579202344


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1756.00draws/s]


    0.2449803473390821 0.18835286237993668 0.11126932199798574 0.30550317088438367 1.0506749984841635


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1703.21draws/s]


    1.0487157711842428 0.6905675835846923 0.4695233178795432 0.6221293251383287 0.6709310320415605


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1883.70draws/s]


    0.3895553019598965 0.296044339019491 0.33154565406956427 0.3040384746462891 0.6548270883769555


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1658.91draws/s]


    0.44166117949493855 0.4098173717011523 0.333234209649124 0.3768188379140158 0.4542116541311364


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1624.88draws/s]


    0.2455824210003044 0.19912871153712308 0.26943232591186866 0.19467106144756394 0.3243709210075716


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1813.47draws/s]


    433.5447582809109 0.6624961058304237 0.4451448601283276 0.6168144725620424 0.9253254177148906


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1698.12draws/s]


    0.8984171846133673 0.7090472059533648 0.573357424842342 1.075576465169414 1.2468806171236964


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1550.76draws/s]


    0.3359722589747646 0.3272985291095343 0.3232178575025588 0.4250416598523873 0.6485693638402861


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1726.95draws/s]


    0.4312991565376275 0.32233690056461994 0.29220238484018224 0.2725524349163412 0.48013062838679554


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1376.20draws/s]


    0.13877770102185896 0.16080975678168233 0.1348125744883022 0.1677939602874564 0.2600211204556163


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1672.30draws/s]


    1.0191335028138684 0.4142892693850085 0.2561283303936861 0.5627111481700542 1.3743888824549875


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1900.25draws/s]


    0.3825813644915471 0.3183735132459687 0.2848506135740502 0.28651382503825445 0.4967228688752372


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1560.83draws/s]


    0.8831725260652439 0.7266962524308438 0.5633929927915268 0.6869353323547567 0.6978702012865944


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1732.38draws/s]


    0.2969856046730033 0.31905091292459525 0.2550466191659442 0.34114641438391463 0.48049306824573146


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1452.70draws/s]


    0.15801692372325493 0.16811979133024468 0.14238436894665574 0.17444474022055362 0.3060603786622361


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1796.69draws/s]


    1.2514724328435087 1.5814365820040952 1.0953532612770482 nan 1.4267645742357065


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1756.43draws/s]


    0.584201188680839 0.4930920635706783 0.5961176154829342 0.6395296674834331 1.0001401466556645


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1841.25draws/s]


    0.49559732174364046 0.4093118182407001 0.4749610951933755 0.35017364478263474 0.39759803136916466


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1749.92draws/s]


    0.2466020117358207 0.2536877436998962 0.22312767887785528 0.29983740000828174 0.49503407562888063


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1529.95draws/s]


    0.179584668783667 0.14947792019506045 0.17891565228148287 0.16512242983192582 0.15412127469195827


    Sampling 2 chains, 1 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1748.44draws/s]
    There was 1 divergence after tuning. Increase `target_accept` or reparameterize.


    1.48677638470185 1.0253545152937336 0.6329642255196053 1.7965598355079353 1.2523875448381139


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1813.50draws/s]


    1.1785332269617028 0.8095870914343115 0.613103041543841 0.6662123665500682 0.7894442116761001


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1489.81draws/s]


    0.8382560259458736 0.7370435650331758 0.595179142287563 0.8755399218996447 0.9196876654876989


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1670.53draws/s]


    0.2797721118773678 0.2998571545907997 0.227902978919773 0.3434866845236107 0.47426661875839626


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:02<00:00, 905.41draws/s] 


    0.21117259787012588 0.21455874384166312 0.20429593268042295 0.18156595849836596 0.2471436770596736


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1754.76draws/s]


    1.046801457477692 0.6088580651425571 0.39459126351916657 0.7012091276709547 1.087521094643549


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1717.01draws/s]


    0.6880677815502588 0.47496390209403927 0.3381537870158586 0.6131177112686975 1.2544350573065994


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1904.07draws/s]


    0.5605332580023086 0.5721689007601858 0.5454854665804177 0.39358445911025947 0.7320599660727765


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1638.81draws/s]


    0.358503116245085 0.35570686039705424 0.2778007873635013 0.3471506077534159 0.5246527372464388


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1584.42draws/s]


    0.17861084592637552 0.17466247178985975 0.17589265330215823 0.17233056644927683 0.30286843861008783


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1853.00draws/s]


    0.7421688574223684 0.7982497548498825 0.596677383659232 1.3109715168045062 1.1492334761041083


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1800.47draws/s]


    0.5212889202453276 0.4401273268222602 0.4817079572518464 0.5273077615639574 0.58466163153202


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1894.84draws/s]


    0.39206712964079443 0.3827107161740717 0.4035822465952709 0.42043231023786165 0.44427373402950776


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1793.59draws/s]


    0.30605276038959245 0.2898143303555889 0.25734471946550685 0.30219923453283043 0.3877314867323616


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1596.57draws/s]


    0.19944982085829385 0.1746436559175934 0.17668038021828245 0.17630988351882249 0.2001596052451558


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1702.43draws/s]


    0.7468358170135551 0.9142644633828715 0.6143786263877342 1.1970959659728615 1.2984751767829237


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1686.68draws/s]


    0.42834511681915777 0.45670351092351635 0.2549909054342725 0.5687929992404002 0.9122331306699671


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1760.40draws/s]


    0.3479378731081015 0.3582446777235609 0.32161964492091205 0.42785777957520627 0.3852052958631081


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1751.00draws/s]


    0.5622516398056133 0.4798796158762636 0.4707801539074668 0.3682108015527769 0.4490699807703312


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1487.59draws/s]


    0.1584631237292638 0.1759215359045859 0.1535250161375873 0.19114768923811182 0.20467589283681942


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1801.18draws/s]


    155.83505221853045 0.5910394767068641 0.4173014806534653 0.7944281102386316 1.480941403051361


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1731.60draws/s]


    0.8388947153705786 0.5799176355904916 0.3288629598600907 0.7239220317976828 0.933721311259185


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1723.79draws/s]


    0.6226772544519376 0.5282857340841496 0.42050441854496584 0.5700804148874664 0.8427923299662232


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1682.27draws/s]


    0.27111593291573827 0.2530784974033884 0.23204501822723725 0.27650278782174487 0.48915931471369006


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1514.93draws/s]


    0.2546523544405917 0.17826188857532982 0.2170999149158956 0.19120238622563918 0.3016475959293051


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1852.49draws/s]


    444.4221758235653 0.7078057947576553 0.48058553752915484 0.8406432754595885 1.1916150507376173


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1824.84draws/s]


    1.4674392772570337 0.9219552365450305 0.5736767043606916 1.0065619877390215 1.2473492884083046


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1719.13draws/s]


    0.5912940312110159 0.5606606694685639 0.5278249039525797 0.691703764940783 1.0456037865938954


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1744.89draws/s]


    0.3730110697040794 0.35734819896226183 0.3140576299515886 0.3253089422133498 0.35259704277819454


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1316.66draws/s]


    0.15058030009682888 0.17169178319400255 0.15030786755392794 0.17744822010080866 0.3281012180460741


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1833.56draws/s]


    0.26021566118326794 0.3094803182942032 0.16796553324941674 0.37849821118384075 0.8069267645314004


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1764.16draws/s]


    1.1144005363620797 0.8690073428353323 0.5497109720813133 0.986541783946693 1.0802350607081403


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1806.32draws/s]


    0.584698945268846 0.5356347357795815 0.4874043672614381 0.5149702836977342 0.7711427609944977


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1735.40draws/s]


    0.263672178112541 0.26721758340362506 0.24200650831275655 0.29508129273365974 0.5076265475548


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1493.94draws/s]


    0.21771021287674475 0.20553967407972867 0.20619356724475607 0.18739535104735056 0.25891150528326756


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1739.47draws/s]


    0.6671779091235042 0.553960018532983 0.33210059702552047 0.7844399138996092 0.9863124693403572


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1843.09draws/s]


    0.4173616929283062 0.36543896100100604 0.2714213017307824 0.4005039545574698 0.5270334866336062


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1768.84draws/s]


    0.6924110671636546 0.5813626472368847 0.3959316710911607 0.6077698862149481 0.6446656503547302


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1795.15draws/s]


    0.3405227169625162 0.33130664237894286 0.2896219103304798 0.34948517342291024 0.5297243058660022


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1547.06draws/s]


    0.18611768164811224 0.16126239433879822 0.17169134175386308 0.15321322163149953 0.20439890496866736


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1699.77draws/s]


    374.22644386840204 0.9162028231930884 0.5534419364501526 1.0187197189755093 1.2529037344843281


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1613.87draws/s]


    0.8230812705971329 0.6246127107026267 0.35836125971651434 0.7038819118776382 0.9503492332724554


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1873.65draws/s]


    0.5517925436262445 0.5973970163291117 0.5751583904036793 0.414305603318535 0.5481838290543112


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1818.97draws/s]


    0.27751281527718247 0.258304549278955 0.2821440311231073 0.279502338669972 0.5194074687310549


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1622.54draws/s]


    0.2196836805097076 0.21418531486407777 0.21079951947681205 0.15898177262116883 0.3161777806841421


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1647.77draws/s]


    2.1021782620182323 0.5870022447487627 0.326257315960336 0.7559248168115869 0.9942310711216205


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1567.83draws/s]


    1.6001361174289848 1.321683029631699 0.8728441841664698 1.456352116637333 1.257841820275396


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1632.47draws/s]


    0.43831594931306755 0.40958836121552106 0.3486031262748968 0.43493091125173233 0.3543467856997513


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1857.40draws/s]


    0.33600593890611713 0.3450525857650336 0.27818691288716163 0.344466646395206 0.3684143892725536


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1566.95draws/s]


    0.17961185056174106 0.17505496512722876 0.1631532811342217 0.18146187347782566 0.19923655373017538


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1591.69draws/s]


    2.2703785119145157 1.2108548716559027 0.6494296230311847 1.917872897933939 1.3193089494512982


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1872.03draws/s]


    0.8696892397377729 0.5019893685061404 0.35672033825079724 0.43822092606892754 0.7170072490184245


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1312.57draws/s]


    1.089206104733469 0.7173554916004529 0.671044505101281 0.4860828496932009 0.8240179352955006


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1739.05draws/s]


    0.3313411013329047 0.2928679662086464 0.31020400153257743 0.3067299808692629 0.42671814265470404


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1096.10draws/s]


    0.19768073450603335 0.2143172801541301 0.1941070570281819 0.18897209921811947 0.23203717906274135


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1678.43draws/s]


    5.285004890356163 0.8899904570887279 0.48789852350298835 1.2935534587679598 1.4489833498788622


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1771.32draws/s]


    0.4790813596905396 0.37084203198015125 0.38920622094435897 0.45673133499108565 0.9311252209490102


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1647.31draws/s]


    0.9258060959221833 0.718703059318442 0.5674790579699922 0.8129717906640104 1.0808774086310213


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1756.75draws/s]


    0.21176182198336385 0.23229011400018618 0.22205504659057337 0.2636849315359272 0.3505312671469008


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1398.69draws/s]


    0.15819079145538198 0.1498931583561081 0.15031124387102374 0.15503284442898718 0.2397578287008638


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1687.02draws/s]


    406.7962724119899 0.7453604096410855 0.35510838956075885 1.001370173392419 1.7569481527554296


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1665.57draws/s]


    0.5679382966159887 0.22285414515270024 0.2070276808185162 0.3123076791272322 0.3507225378685394


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1529.88draws/s]


    0.837300439798787 0.7174913223222205 0.6924773503999782 0.6508804610959812 0.9678003661532233


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1669.12draws/s]


    0.21510454504722218 0.2571320435214034 0.19332050764615155 0.28538792871000396 0.3842353168346526


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1439.58draws/s]


    0.2609396854954789 0.23992887360908932 0.2518548205813766 0.1667431908065955 0.17401119530877499


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1844.80draws/s]


    181.43196974778704 0.2684076029472243 0.21846335108650847 0.24150237605389158 0.6310754111525655


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1787.70draws/s]


    0.8621949098809278 0.7806073473285307 0.6034215552284181 0.9723299230461233 1.1460100038530991


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1833.15draws/s]


    0.41182027121464454 0.462394039270163 0.2891384013203954 0.6302110097719732 1.0585204917788495


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1814.55draws/s]


    0.3429352318409238 0.33325688778389073 0.27170968656693245 0.30790156576370176 0.543580761753026


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1475.16draws/s]


    0.21811320285295538 0.21779971851671737 0.21097385596450496 0.18340763312449662 0.26266896989590016


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1723.97draws/s]


    208.59317928886435 0.44508790982808094 0.2842666351270646 0.5591264987018241 0.988136358071886


    Sampling 2 chains, 1 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1649.76draws/s]
    There was 1 divergence after tuning. Increase `target_accept` or reparameterize.


    0.9183344379695458 0.8279492991018096 0.6480581717641996 1.3476267023394046 1.0963173418764234


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1869.37draws/s]


    0.6113797876004651 0.4237594686325306 0.36146767857040685 0.4508887357163076 0.6946310364973154


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1299.93draws/s]


    0.3401262591909968 0.3335897448530124 0.2937770735484223 0.35892672929837477 0.5551818444229368


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1453.03draws/s]


    0.18583826812161544 0.1794145317294417 0.17717261559980688 0.18758514572527377 0.23798424084712727


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1922.32draws/s]


    0.533341177628416 0.41008187880883906 0.3228516329139283 0.3740976226402518 0.5560352406209351


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1742.85draws/s]


    1.5309878457570707 0.8903332565687582 0.6190773031146896 1.0780610865065992 1.104144184587196


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1827.10draws/s]


    0.36954328396076114 0.38186567165770946 0.3581207794755128 0.4295144231979688 0.8276310845501997


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1658.55draws/s]


    0.32970715971161 0.32177051289152697 0.29838931347332176 0.35202579299731623 0.6103788544954237


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1459.70draws/s]


    0.17662668984631968 0.1895141459098156 0.15776544737300863 0.1780607875114451 0.2243870018018995


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1720.38draws/s]


    0.9008564570955322 0.8529251011654182 0.5175575991675287 1.4685788503046342 1.391843820528999


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1784.76draws/s]


    0.992136663626995 0.48359530319651195 0.4067141914741041 0.5836031662050816 0.7654607396502624


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1800.54draws/s]


    0.5382374871084613 0.44179331369500346 0.30040473888055363 0.41169928584250237 0.7399282920127688


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:03<00:00, 522.91draws/s] 


    0.4568849557449593 0.4237597887813614 0.4061919549826041 0.3544023982789657 0.5611581276819557


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1540.46draws/s]


    0.26437605118521523 0.2297411476913707 0.2617917153688173 0.20934044208520702 0.3553327468521476


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1802.79draws/s]


    124.68539384198391 0.4508690105050574 0.3203387362420064 0.6668724732527589 0.8958365820624148


    Sampling 2 chains, 2 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1616.27draws/s]
    There were 2 divergences after tuning. Increase `target_accept` or reparameterize.


    0.5440249893660887 0.5873615874679128 0.5027970438623678 0.7295908732421553 0.8896924586789572


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1694.84draws/s]


    0.7086130295772649 0.6665542219784168 0.6616705689048243 0.755298142041265 0.8986162977396515


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1683.96draws/s]


    0.21950190508888845 0.256424293776392 0.19281069128727762 0.27894413974203147 0.42198434147228525


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1576.67draws/s]


    0.2639595760603601 0.2664748308811448 0.2527971917408978 0.18929851192149405 0.193954295147609


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1771.68draws/s]


    0.49394443694044615 0.5405039482677032 0.3609482526901883 0.6144904687639705 1.0086654623790066


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1795.93draws/s]


    0.6675148521345301 0.5686137311890141 0.39944316555728976 0.7961302908047047 0.9395204115100682


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1662.97draws/s]


    0.6891038832666871 0.6420904441679607 0.4579526054157484 0.7840180881290669 0.959157216763375


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1683.80draws/s]


    0.3896780143566499 0.3565028729224615 0.3777992287478743 0.3381535296152587 0.33431393622464095


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1439.34draws/s]


    0.1720854691041488 0.16606482731542724 0.16210451647628996 0.17185261296587456 0.2102642120831462


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1544.17draws/s]


    1.3041999211381339 1.5191063200092108 0.9919211169765892 3.2789635219302493 1.5622774295441233


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1695.96draws/s]


    187.99420522932996 0.8031022406932623 0.7442061869818465 0.9504570635016762 1.2333992478888698


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1870.55draws/s]


    0.4244196430386301 0.42390572752688466 0.33291192302591127 0.4500828605047958 0.48962983406474386


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1740.63draws/s]


    0.2278144343656455 0.24593893848656667 0.20317911327467447 0.2682302637417595 0.2843069995923044


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1540.99draws/s]


    0.18162340387335116 0.1710430529283377 0.16354763479239182 0.17941054402530057 0.344124681357966


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1673.03draws/s]


    224.10617231387923 0.6967832957084618 0.4551135267382522 0.7602627281690398 1.0094196779417812


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1839.79draws/s]


    0.797148141195788 0.5575177060022106 0.42152702387356167 0.6749567028373172 0.9672839890370876


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1813.07draws/s]


    0.6315005820870631 0.49236662641298 0.42706190211826056 0.5179775278561781 0.7362999273539158


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1769.34draws/s]


    0.2410357943462761 0.22826330085106675 0.23361832748202285 0.23131274854140957 0.49196011500448067


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1408.40draws/s]


    0.14095845732003706 0.1534749974075031 0.12880548037239134 0.1752893464748288 0.22358456202836646


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1828.81draws/s]


    265.5056086070083 0.5766903926213124 0.35772141345945113 0.6461958853076624 0.8493563718360327


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1449.30draws/s]


    2.1342939356314563 1.3382164265110992 0.8728316469201538 1.2045375581176903 1.1404157138865838


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1696.64draws/s]


    0.8257531831908275 0.8733804228889362 0.6895875566009321 0.7757431984119468 0.9069823647301314


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1773.62draws/s]


    0.3381164593398363 0.3245942069217864 0.2883636734988951 0.3620231206418328 0.6127872408610288


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1563.10draws/s]


    0.13995854625972234 0.14648039461283227 0.136902048783037 0.1393064363395709 0.16852579479589586


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1739.77draws/s]


    0.4944078280304507 0.408186110629975 0.17235194676822502 0.5742748002205162 0.9768876347762432


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1806.18draws/s]


    1.0343366549530093 0.6892949231091033 0.42946375823080607 1.0140698107311255 1.2336859114467462


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1845.79draws/s]


    0.5068010645798032 0.592979127283338 0.5559537449876853 0.3776052063813933 0.6791903606652224


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1839.88draws/s]


    0.34504318658596816 0.33358556802792627 0.288008600529548 0.35905455678819387 0.6435741314698924


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1510.70draws/s]


    0.1970899811333953 0.17200319751336818 0.19708444995406552 0.17912670067894285 0.2489713917158121


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1800.81draws/s]


    0.8813409049885076 0.8858510695436388 0.579974309499823 1.0730097193592834 1.1771690487458906


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1574.32draws/s]


    1.6749608187677327 1.4711631331172863 1.003937256835019 1.5035041554770476 1.2022539545224413


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1873.88draws/s]


    0.563059281536693 0.44796554926920223 0.36041910149208606 0.5405897940418037 0.782116923745564


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1698.03draws/s]


    0.3329660417230761 0.32549183843879875 0.26417024514556237 0.3148164298338066 0.3774548872627484


    Sampling 2 chains, 0 divergences: 100%|██████████| 2000/2000 [00:01<00:00, 1569.55draws/s]


    0.2027403615356476 0.20273835263676462 0.19009271253778784 0.17976101612861098 0.257406129853108
    



```python
for res in [bavg_results, chodera_results, sokal_results,
            correction_results, bayes_results]:
    ci_95 = 1.96*res[:135]
    print(((ci_95-np.abs(estimated_means[:135]))>0).sum(0)/100)

```

    [0.86 0.88 0.92 0.92 0.95]
    [0.71 0.82 0.88 0.91 0.96]
    [0.49 0.75 0.85 0.87 0.95]
    [0.79 0.87 0.92 0.95 0.94]
    [0.95 0.95 0.96 0.99 0.95]



```python
my_results = list()
for res in [bavg_results, chodera_results, sokal_results,
            correction_results, bayes_results]:
    ci_95 = 1.96*res
    print(((ci_95-np.abs(estimated_means))>0).sum(0))
    my_results.append(((ci_95-np.abs(estimated_means))>0).sum(0))
    
    
z=1.96
for col, datasize, label in zip(range(6), [30, 100, 300,1000,3000], ['Block avg', 'Chodera' ,'Sokal', 'Correction', 'Bayes']):
    for color, item in enumerate(np.array(my_results)[:,col]):
        it = item/100
        confint = z* np.sqrt((it*(1-it))/100)
        plt.errorbar(datasize+np.random.uniform(-datasize*0.1, datasize*0.1), 
                     it, mfc='white' , elinewidth=2, capsize=2,yerr=confint,c='C'+str(color), label=label)
        
plt.xscale('log')
plt.axhline(0.95)
plt.legend()
```


```python
np.random.uniform(-0.1, 0.1, len(datasizes))
```




    array([ 0.00987476, -0.03704028, -0.09737911, -0.06560206, -0.06717733])




```python
items = np.array(my_results)/100
confint = z*((items*(1-items))/100)**0.5
datasizes = [30, 100, 300,1000,3000]
for b, lab, ci in zip(items, ['Block avg', 'Chodera' ,'Sokal', 'Correction', 'Bayes'], confint):
    plt.errorbar(np.log10(datasizes)+np.random.uniform(-0.025, 0.025, len(datasizes)),  
                 b, yerr=ci, label=lab)
    
#plt.xscale('log')
plt.axhline(0.95, c='k')
plt.legend()

```


```python
ci
```




    array([0.000931  , 0.000931  , 0.00075264, 0.00019404, 0.000931  ])




```python
rpts = 1000
bavg_results2 = np.zeros([rpts,6])
chodera_results2 = np.zeros([rpts,6])
sokal_results2 = np.zeros([rpts,6])
correction_results2 = np.zeros([rpts,6])
estimated_means2 = np.zeros([rpts,6])
```


```python
rpts = 1000
bavg_results2 = np.zeros([rpts,7])
chodera_results2 = np.zeros([rpts,7])
sokal_results2 = np.zeros([rpts,7])
correction_results2 = np.zeros([rpts,7])
estimated_means2 = np.zeros([rpts,7])
for row in tqdm.tqdm_notebook(range(rpts)):
    for col, datasize in enumerate([30, 100, 300,1000,3000]):
        cor = gen_correlated_curve(0.9, datasize)
        estimated_means2[row,col]=cor.mean()
    
        #block avg:
        s = do_block_averaging(cor)

        popt, pcov = optimize.curve_fit(block_avg_fit, np.arange(1, int(datasize**(2/3))+1), np.array(s), maxfev=25000)

            
        block_avg_sem = popt[0] *np.pi/2
        bavg_results2[row,col]=block_avg_sem

    
        #chodera, sokal:
        chodera_autoc_time = statineff(cor)
        sokal_autoc_time = sokal_autocorr_time(cor)
        n=len(cor)
        chodera_sem = np.std(cor) / np.sqrt(n/chodera_autoc_time)
        chodera_results2[row,col] = chodera_sem
        sokal_sem = np.std(cor) / np.sqrt(n/sokal_autoc_time)
        sokal_results2[row,col] = sokal_sem
    
        #correction factor:
        result = AutoReg(cor-cor.mean(), lags = [1]).fit()
        estimated_rho = result.params[1]
        correction = np.sqrt((1+estimated_rho)/(1-estimated_rho))
        naive_sem = sem(cor)
        corrected_sem = naive_sem*correction
        correction_results2[row,col]=corrected_sem
        
        
```


```python
cor = gen_correlated_curve(0.6, 30)
print(np.mean(cor))
#block avg:
s = do_block_averaging(cor)
popt, pcov = optimize.curve_fit(block_avg_fit, np.arange(1, int(datasize**(2/3))+1), np.array(s), maxfev=25000)
#plt.plot(s)
```


```python
my_results = list()
for res in [bavg_results2, chodera_results2, sokal_results2,
            correction_results2]:
    ci_95 = 1.96*res
    print(((ci_95-np.abs(estimated_means2))>0).sum(0)/10)
    my_results.append(((ci_95-np.abs(estimated_means2))>0).sum(0)/10)
    
    
z=1.96
for col, datasize in zip(range(6), [30, 100, 300,1000,3000,10000]):
    for color, item in enumerate(np.array(my_results)[:,col]):
        it = item/100
        confint = z* np.sqrt((it*(1-it))/1000)
        plt.errorbar(datasize+np.random.uniform(-datasize*0.1, datasize*0.1), 
                     it, mfc='white' , elinewidth=2, capsize=2,yerr=confint,c='C'+str(color))
        
plt.xscale('log')
plt.axhline(0.95)
```

    [78.  87.  91.4 95.1 94.2 94.8  0. ]
    [58.2 80.9 89.4 94.8 94.5 95.1  0. ]
    [39.7 70.2 83.1 93.1 93.5 94.3  0. ]
    [70.6 84.9 90.4 95.3 94.5 94.7  0. ]



```python
z=1.96
for col, datasize in zip(range(6), [30, 100, 300,1000,3000,10000]):
    for color, item in enumerate(np.array(my_results)[:,col]):
        it = item/100
        confint = z* np.sqrt((it*(1-it))/1000)
        plt.errorbar(datasize+np.random.uniform(-datasize*0.1, datasize*0.1), 
                     it, mfc='white' , elinewidth=2, capsize=2,yerr=confint,c='C'+str(color))
        
plt.xscale('log')
plt.axhline(0.95)
```


```python
confint
```




    0.04391056200961219




```python
690000+750000-65000
```




    1375000



See: 
*Confidence intervals in stationary autocorrelated time series*. Halkos, George and Kevork, Ilias,
at: https://mpra.ub.uni-muenchen.de/31840/1/MPRA_paper_31840.pdf , who reference:
`Fishman, G., 1978. Principles of Discrete Event Simulation. Wiley, New York.`

the below is from a cooky approach - fitting the np.log(periodogram) by linear regression to estimate the value at 0. 
`A Spectral Method for Confidence Interval Generation and Run Length Control in Simulations`

I found it in Thomson:
`A Comparison of Methods for Computing Autocorrelation Time`


```python
from scipy.signal import periodogram
#out = periodogram(correlated)
#out= welch(correlated)
peri=np.abs(np.fft.fft(correlated)**2)/len(correlated)
out = peri[:peri.shape[0]//2]
from statsmodels.tools import add_constant
from statsmodels.regression.linear_model import OLS
x_ = np.arange(len(out[1:200]))
y_ = np.log(out[1:200])

res = OLS(y_, add_constant(x_)).fit()

I = np.e**res.params[0]
print(I, I/np.var(correlated))
```

When you don't have enough data, the above can be a bad approximation. The full version is a bit messier:

\begin{align}
k = \sqrt{\frac{1+2\delta / n}{1 - 2\delta/(n(n-1))}}
\end{align}

where

\begin{align}
\delta = \frac{(n-1)\rho - n\rho^2 + \rho^{n+1}}{(1-\rho)^2}
\end{align}


```python

```

a093a41154089582b339ec10c120462af89c8690
ipynb
Jupyter Notebook
.ipynb_checkpoints/autocorrelation-checkpoint.ipynb
ljmartin/estimating_sem_timeseries
e2824edb924405a3140e01420304368d8e942dd3
MIT
2020-04-29T03:34:54.000Z
2021-05-09T12:14:16.000Z
.ipynb_checkpoints/autocorrelation-checkpoint.ipynb
ljmartin/estimating_sem_timeseries
e2824edb924405a3140e01420304368d8e942dd3
MIT
2020-05-02T02:39:43.000Z
2021-01-05T04:12:13.000Z
.ipynb_checkpoints/autocorrelation-checkpoint.ipynb
ljmartin/estimating_sem_timeseries
e2824edb924405a3140e01420304368d8e942dd3
MIT
Qwen/Qwen-72B
1. YES
2. YES