Metropolis–Hastings algorithm: Difference between revisions

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==Formal derivation==

The purpose of the Metropolis–Hastings algorithm is to generate a collection of states according to a desired distribution <math>P(x)</math>. To accomplish this, the algorithm uses a [[Markov process]], which asymptotically reaches a unique [[Markov chain#Steady-state analysis and limiting distributions|stationary distribution]]{{Broken anchor|date=2024-06-13|bot=User:Cewbot/log/20201008/configuration|target_link=Markov chain#Steady-state analysis and limiting distributions|reason= The anchor (Steady-state analysis and limiting distributions) [[Special:Diff/970694186|has been deleted]].}} <math>\pi(x)</math> such that <math>\pi(x) = P(x)</math>.<ref name=Roberts_Casella/>

A Markov process is uniquely defined by its transition probabilities <math>P(x' \mid x)</math>, the probability of transitioning from any given state <math>x</math> to any other given state <math>x'</math>. It has a unique stationary distribution <math>\pi(x)</math> when the following two conditions are met:<ref name=Roberts_Casella/>