How to estimate posterior variance from dependent MCMC samples? Does the usual sample covariance still work?

Question

Suppose we want to estimate posterior variance of $\alpha$ given x, i.e. var($\alpha|x$). We have MCMC posterior samples $a_1,\dots, a_B$, which are not independent. Does $\hat{\text{var}}(\alpha|x) = \frac{1}{B} \sum (a_i-\bar{a})^2$ still serve as a good estimator for var($\alpha|x$)? Since the samples are not independent, we can't get consistency of $\hat{\text{var}}(\alpha|x)$ from law of large number.

In an extreme case of correlated posterior samples, if $a_i $s are all the same, then $\hat{\text{var}}(\alpha|x)$ is definitely not a good estimator.

So in practice how should we estimate posterior variance from MCMC samples?

Answer 1

The usual sample variance still works because the strong law still holds for ergodic Markov chains. That is, we can apply a strong law of large numbers.

If the Markov chain used in MCMC is Harris ergodic (irreducible, aperiodic, and Harris recurrent), the strong law of large numbers holds for functions with finite expectations. That is for Markov chain samples $X_1, X_2, \dots, X_n$, with stationary distribution $\pi$, and a function $f$ such that $\int f \pi(dx) < \infty$ $$\dfrac{1}{n} \sum_{i = 1}^{n} f(X_i) \overset{a.s.}{\to} \int f \pi(dx) \,. $$

The above is called the Birkhoff ergodic theorem, and essentially why MCMC is used so widely. Most reaonsbale MCMC algorithms usually satisfy the Harris ergodicity conditions.

So essentially, if you can write the sample variance as a function of Monte Carlo averages, strong consistency would follow naturally. The Math post here does exactly that.