<SYSTEM_TASK:>
Solve the following problem using Python, implementing the functions described below, one line at a time
<END_TASK>
<USER_TASK:>
Description:
 def gelman_rubin(x, return_var=False):
 """ Returns estimate of R for a set of traces. The Gelman-Rubin diagnostic tests for lack of convergence by comparing the variance between multiple chains to the variance within each chain. If convergence has been achieved, the between-chain and within-chain variances should be identical. To be most effective in detecting evidence for nonconvergence, each chain should have been initialized to starting values that are dispersed relative to the target distribution. Parameters x : array-like An array containing the 2 or more traces of a stochastic parameter. That is, an array of dimension m x n x k, where m is the number of traces, n the number of samples, and k the dimension of the stochastic. return_var : bool Flag for returning the marginal posterior variance instead of R-hat (defaults of False). Returns ------- Rhat : float Return the potential scale reduction factor, :math:`\hat{R}` Notes ----- The diagnostic is computed by: .. math:: \hat{R} = \sqrt{\frac{\hat{V}}{W}} where :math:`W` is the within-chain variance and :math:`\hat{V}` is the posterior variance estimate for the pooled traces. This is the potential scale reduction factor, which converges to unity when each of the traces is a sample from the target posterior. Values greater than one indicate that one or more chains have not yet converged. References Brooks and Gelman (1998) Gelman and Rubin (1992)"""


    if np.shape(x) < (2,):
        raise ValueError(
            'Gelman-Rubin diagnostic requires multiple chains of the same length.')

    try:
        m, n = np.shape(x)
    except ValueError:
        return [gelman_rubin(np.transpose(y)) for y in np.transpose(x)]

    # Calculate between-chain variance
    B_over_n = np.sum((np.mean(x, 1) - np.mean(x)) ** 2) / (m - 1)

    # Calculate within-chain variances
    W = np.sum(
        [(x[i] - xbar) ** 2 for i,
         xbar in enumerate(np.mean(x,
                                   1))]) / (m * (n - 1))

    # (over) estimate of variance
    s2 = W * (n - 1) / n + B_over_n
    
    if return_var:
        return s2

    # Pooled posterior variance estimate
    V = s2 + B_over_n / m

    # Calculate PSRF
    R = V / W

    return np.sqrt(R)