Editing Markov chain Monte Carlo (section)

===Autocorrelation and efficiency===
The effect of correlation on estimation can be quantified through the [[Markov chain central limit theorem]]. For a chain targeting a distribution with variance <math>\sigma^2</math>, the variance of the sample mean after <math>N</math> steps is approximately <math>{\sigma^2}\big/N_{\text{eff}}</math>, where <math>N_{\text{eff}}</math> is an effective sample size smaller than <math>N</math>. Equivalently, one can express this as: 

:<math>
\mathrm{Var}(\bar{X}_N) \approx \frac{\sigma^2}{N} \left(1 + 2 \sum_{k=1}^{\infty} \rho_k \right)
</math>

where <math>\bar{X}_N</math> is the sample mean and <math>\rho_k</math> is the autocorrelation of the chain at lag <math>k</math>, defined as <math>\rho_k = \frac{\mathrm{Cov}(X_0, X_k)}{\sqrt{\mathrm{Var}(X_0)\mathrm{Var}(X_k)}}</math>. The term in parentheses, <math>1 + 2\sum_{k= 1}^\infty\rho_k</math>, is often called the integrated autocorrelation. When the chain has no autocorrelation (<math>\rho_k=0</math> for all <math>k\ge1</math>), this factor equals 1, and one recovers the usual <math>\sigma^2/N</math> variance for independent samples. If the chain's samples are highly correlated, the sum of autocorrelations is large, leading to a much bigger variance for <math>\bar{X}_N</math> than in the independent case.