## Batch mean variance estimators for MCMC

Galin Jones and James Flegal posted on arXiv a new version of their batch mean convergence assessment paper. As posted earlier, I think this is a very interesting approach to MCMC convergence assessment and stopping rule, and I have included their method in the Convergence Assessment chapter of EnteR Monte Carlo Statistical Methods. The stopping rule studied in the paper is of the form

$t_* \frac{\hat\sigma_n}{\sqrt{n}}+p(n) \le \epsilon$

where $t_*$ is an appropriate Student’s t quantile and $p(n)$ is a positive function of order $o(n^{-1/2})$. The estimation of the asymptotic variance by $\hat\sigma_n^2$ is a delicate issue for MCMC output, which can be addressed by regenerative sampling when this is feasible, or by batch means, which is a general solution. The batch mean estimate of $\sigma^2$ is given by

$\hat\sigma^2_n=\frac{b_n}{a_n-1} \sum_{k=0}^{a_n-1} (\bar Y_k - \bar Y_n)^2$

where $a_n=b_n=[n^{1/2}]$. The overlapping batch mean estimator allows for the partial averages to use common simulations. Those estimators are consistent, but the overlapping batch mean estimator slightly decreases the variance. The paper also covers the case of the spectral approaches to estimators $\hat\sigma_n^2$. As mentioned earlier, this is an interesting paper (at least) in that it relates theory (which was unknown to me) with practice, in the construction of proper stopping rules that apply in a general setting (modulo mathematical assumptions that may be hard to verify in practice but cannot be weakened).

### 3 Responses to “Batch mean variance estimators for MCMC”

1. […] Jones. (As stressed in Introducing Monte Carlo Methods with R, I particularly like the idea of Flegal and Jones to validate a bootstrap approach to confidence evaluation!) The two next chapters are covering […]

2. FordPrefect Says:

The equations are difficult to see … maybe my eyes need testing. In the meantime … can a different colour be used?