Editing Markov chain Monte Carlo (section)

===Effective sample size (ESS)===
The effective sample size <math>N_{\text{eff}}</math> is a useful diagnostic that translates the autocorrelation in a chain into an equivalent number of independent samples. It is defined by the formula: 
 
:<math>
N_{\text{eff}} = \frac{N}{1 + 2 \sum_{k=1}^{\infty} \rho_k}
</math>

so that <math>N_{\text{eff}}</math> is the number of independent draws that would yield the same estimation precision as the <math>N</math> dependent draws from the Markov chain. For example, if <math>1 + 2\sum_{k=1}^{\infty} \rho_k = 5</math>, then <math>N_{\text{eff}} = N/5</math>, meaning the chain of length <math>N</math> carries information equivalent to <math>N/5</math> independent samples. In an ideal scenario with no correlation, <math>\rho_k=0</math> and thus <math>N_{\text{eff}}\approx N</math>. But in a poorly mixing chain with strong autocorrelation, <math>N_{\text{eff}}</math> can be much smaller than <math>N</math>. In practice, monitoring the ESS for each parameter is a way to gauge how much correlation is present: a low ESS indicates that many more iterations may be needed to achieve a desired effective sample of independent draws.