Metropolis Algorithm

Creator

Creator

Created

Created

2022 Apr 3 15:50

Editor

Editor

Edited

Edited

2025 Apr 29 1:52

Refs

Refs

Rejection Sampling

Metropolis–Hastings algorithm (MH algorithm)

A flagship MCMC algorithm that uses proposal distribution with Monte Carlo sampling and filters based on Acceptance Criteria.

MH makes local changes to a current state. We use the proposal

q

and the transition

T

to define an ”acceptance ratio”

A

:

A \;=\; \min\!\left(1,\;\frac{T(x' \mid x)\,q(x \mid x')}{T(x \mid x')\,q(x' \mid x)}\right).

Symmetric MH

Note that in the (popular) choice of Gaussian perturbations,

q(x' \mid x) \;=\; \mathcal{N}\bigl(x' \mid x, \sigma^2\bigr)

then the proposal is ”symmetric”

q(x'|x) = q(x|x')

A \;=\; \min\!\left(1,\;\frac{T(x' \mid x)}{T(x \mid x')}\right).

sample a value from proposal $x' \sim q(\cdot|x)$

if $T(x'|x)>T(x|x')$ , then move to $x'$ with certainity $1$

otherwise, move to $x'$ with probability $A$

Metropolis-Hastings algorithm - Wikipedia

In statistics and statistical physics, the Metropolis-Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g. to generate a histogram) or to compute an integral (e.g.

Metropolis-Hastings algorithm - Wikipedia

https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm

Metropolis-Hastings algorithm - Wikipedia

Recommendations

/////////