mcmc

http://chi-feng.github.io/mcmc-demo/
$ E_\pi[T(X)] = \int T(x)\pi(x) dx. $

In Bayesian inference, we are interested in posterior mean $E(\theta|y)$ or posterior variance $Var(\theta|y)$.

One solution is to draw independent samples $ ( X^{(1)}, X^{(2)}, \cdots, X^{(N)} )$ from $\pi(x)$, then we can approximate
$ E_\pi[T(X)] \approx \frac{1}{N} \sum_{t=1}^N T( X^{(t) }) $

Law of large numbers → 위 근사는 adoptable

it is known that above approximation is still possible if we sample using a Markov chain. This is the main idea of MCMC method.

code