Overlapping batch means for MCMC standard errors

Inputs: Real outputs $y_1,\dots ,y_n$, an integer $n\ge 2$, and an integer batch size $b$ with $1\le b<n$. For an MCMC run, $y_i=f(X_i)$ for the chosen real-valued observable $f$.

Outputs: The sample mean ${\bar{y}}_n$, the overlapping-batch-means estimate ${\widehat{\sigma }}_{\mathrm{O}\mathrm{B}\mathrm{M}}^2$, and its estimated Monte Carlo standard error.

1. Compute the sample mean ${\bar{y}}_n=\frac{1}{n}{\sum }_{i=1}^n{y}_i\mathrm{.}$

2. For each starting index $j=1,\dots ,n-b+1$, compute the overlapping batch mean ${\bar{y}}_j(b)=\frac{1}{b}{\sum }_{i=j}^{j+b-1}{y}_i\mathrm{.}$

3. Return the overlapping-batch-means variance estimate ${\widehat{\sigma }}_{\mathrm{O}\mathrm{B}\mathrm{M}}^2=\frac{nb}{(n-b)(n-b+1)}{\sum }_{j=1}^{n-b+1}({\bar{y}}_j(b)-{\bar{y}}_n{)}^2\mathrm{.}$

4. Return the estimated Monte Carlo standard error $\stackrel{}{\mathrm{MCSE}}({\bar{y}}_n)=\sqrt{{\widehat{\sigma }}_{\mathrm{O}\mathrm{B}\mathrm{M}}^2\mathrm{/}n}\mathrm{.}$

All sums are finite, the denominator is positive because $b<n$, and the variance estimate is nonnegative. The procedure always returns a number for the stated finite input. It does not by itself assert a CLT or consistency; consistency requires the hypotheses of the overlapping-batch-means theorem.