Q:

MCMC-Draws from different MCMC- chains

I have questions about the usage of draws from a MCMC. 
I estimate a hierarchical bayesian Multinomial Logit model (using bayesm in R). I am interested in the ratio of two coefficients 1 and 2, say b1/b2. 
1. Is it better to calculate the mean of draws for each coefficient separately and then the ratio (mean(b1)/mean(b2)) or to calculate the ratio b1/b2 for each draw and then the mean of the ratios (mean(b1/b2 in draw 1; b1/b2 in draw 2 ...))? I assume, that the second idea is better?
2. An equivalent of "significance test" can be obtained for a coefficient by calculating mean and sd of the draws and then (mean(b1-Draws)/sd(b1-Draws). What can I do to test, if the ratio b1/b2 > 0? Can I simply calculate the ratio for each draw and take a look at the relative number larger than 0 (e.g. 97% of the ratios b1/b2 of my draws are >0 is equivalent to a p-value of 0,03)?
3. I also have to compare the ratio of the coefficients in two different experimental groups A and B. The Hypotheses is, that b1A/b2A > b1B/b2B. One idea is to estimate only one model and use group membership as a concomitant variable in the model. 
However, can I also estimate separate models for both groups and compare the draws (Calculate b1A/b2A for each draw in group 1; calculate b1B/b2B for each draw in group 2; look for each draw, if b1A/b2A > b2A/b2B)?  
Thank you for any help!
Stefan

A:

The ratio of the means is not the mean of the ratios. 

$$ \dfrac{\sum_j x_j}{\sum_j y_j} = \sum_j w_j \dfrac{x_j}{y_j},$$
where $w_j = \dfrac{y_j}{\sum_j y_j}$. The correct way in this case is to calculate the ratio of the means of each coefficient, which converges to the ratio of the coefficients by the law of large numbers.

Calculate the ratios $r_j=b_1^{(j)}/b_2^{(j)}$. Using the sample of ratios, approximate the probability $P(b_1/b_2>0)$. This is the posterior probability of your hypothesis. This can be done by approximating the distribution of the ratios using a kernel density estimator and the by integrating this over $(0,\infty)$. A second method consists of dividing the number of positive ratios by the total number of ratios, as you did. THIS IS NOT A P-VALUE, p-values have nothing to do with Bayesian statistics. You have to understand what posterior probabilities mean. Try to read a book on the fundamentals of this theory.
For your last problem, calculate the sample of ratios for each group, say $r_A$ and $r_B$. Then, approximate $P(r_A>r_B) = P(r_B<r_A)$ as indicated in the following answer:

https://stats.stackexchange.com/a/30148

