The Effective Sample Size (ESS) and Monte Carlo Standard Error (MCSE) are described in Kai Xu’s thesis at page 16:

**Effective Sample Size** ESS is a measure of how well a continuous chain is mixing. For a give chain \{x_i\}_{1:n}, ESS is defined by

\text{ESS} = \frac{n}{1 + \Sigma_{k=1}^\infty \rho_k},

where n is the total number of samples in the chain and \rho_k is the autocorrelation factor at lag k of the chain [11].

An ESS measures how many samples are effective in the chain, the larger this value is, the higher the sampling efficiency. Different MCMC samplers can be evaluated by generating the same number of samples and comparing the ESS for each sampling results.

**Monte Carlo Standard Error** MCSE is an estimate of the inaccuracy of MC samples. There are multiple ways to estimate MCSE, among which the batch mean method proposed in [12] is believed to be the most popular one.

[…]

As MCSE measures the inaccuracy of MC samples, a smaller value of MCSE is an indicator of better sampling performance.

However, it has been argued that MCSE is generally unimportant when the goal of inference is parameters themselves rather than the expectation of parameters, in which case the ESS is would be a more important measure [13].

### References

[11] Dr. Orlaith Burke. *Statistical Methods Autocorrelation: MCMC Output Analysis.* Department of Statistics, University of Oxford, 2012

[12] James M Flegal, Murali Haran, and Galin L Jones. Markov chain monte carlo: Can we trust the third significant figure? Statistical Science, pages250–260, 2008.

[13] Andrew Gelman, John B Carlin, Hal S Stern, and Donald B Rubin. Bayesian data analysis. texts in statistical science series, 2004