\[ \newcommand{\Exg}{\operatorname{\mathbb{E}}} \newcommand{\Ex}{\mathbb{E}} \newcommand{\Ind}{\mathbb{I}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Cov}{\operatorname{Cov}} \newcommand{\Corr}{\operatorname{Corr}} \newcommand{\ee}{\mathrm{e}} \]

22  MCMC error

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Summary:

• We can analyse the error in Monte Carlo estimation with the output of a Markov chain by assuming it begins “in equilibrium”.

• MCMC is approximately unbiased and with MSE \({\displaystyle \approx \frac{\sigma^2}{n} \left(1 + 2 \sum_{k = 1}^{\infty} \rho(k) \right)}\).

• We want the autocorrelation \(\rho(k)\) to decay as quickly as possible.

Read more: Voss, An Introduction to Statistical Computing, Subsection ?.?.?.