Monte Carlo standard error ($\mathrm{MCSE}(\bar{h})\simeq \sqrt{{\sigma }_h^2/N_{\mathrm{eff}}}$)

The Monte Carlo standard error estimates the standard deviation of a simulation-based estimate across repeated simulation runs. For an average of $N$ independent random variables with finite variance ${\sigma }_h^2$, it is ${\sigma }_h/\sqrt{N}$. Under a suitable central limit theorem for Markov chain Monte Carlo, correlated draws instead give the approximation ${\sigma }_h/\sqrt{N_{\mathrm{eff}}}$, where $N_{\mathrm{eff}}$ is the effective sample size of a Markov chain for the particular estimated function $h$. This measures numerical uncertainty in an estimate, not uncertainty in the unknown parameter itself.