Editing Markov chain Monte Carlo (section)

=== Central Limit Theorem for MCMC ===

There are several conditions under which the [[Central Limit Theorem]] (CLT) holds for Markov chain Monte Carlo (MCMC) methods. One of the most commonly used is the condition of '''reversibility'''.

;Definition (Reversibility)
A stationary Markov chain <math>(X_n)</math> is said to be '''reversible''' if the distribution of <math>X_{n+1}</math> given <math>X_{n+2} = x</math> is the same as the distribution of <math>X_{n+1}</math> given <math>X_n = x</math>.

This is equivalent to the ''detailed balance condition'', which is defined as follows:

;Definition (Detailed balance)
A Markov chain with transition kernel <math>K</math> satisfies the '''detailed balance condition''' if there exists a function <math>f</math> such that:
:<math>K(y, x) f(y) = K(x, y) f(x)</math>  
for every pair <math>(x, y)</math> in the state space.

;Theorem (CLT under reversibility)
If <math>(X_n)</math> is aperiodic, irreducible, and reversible with invariant distribution <math>\pi</math>, then:

:<math>
\frac{1}{\sqrt{N}} \left( \sum_{n=1}^N \left( h(X_n) - \mathbb{E}^\pi[h] \right) \right)
\overset{\mathcal{L}}{\longrightarrow} \mathcal{N}(0, \gamma_h^2)
</math>

where

:<math>
0 < \gamma_h^2 = \mathbb{E}_\pi \left[ \bar{h}^2(X_0) \right] + 2 \sum_{k=1}^{\infty} \mathbb{E}_\pi \left[ \bar{h}(X_0) \bar{h}(X_k) \right] < +\infty
</math>

and 
:<math>
\bar{h}(\cdot) = h(\cdot) - E[h(\cdot)]
</math>.

Even though reversibility is a restrictive assumption in theory, it is often easily satisfied in practical MCMC algorithms by introducing auxiliary variables or using symmetric proposal mechanisms. There are many other conditions that can be used to establish CLT for MCMC such as geometirc ergodicity  and the discrete state space.