# Physical meaning behind the choice of the proposal distribution in Markov Chain Monte Carlo (MCMC) methods

Let us consider the conventions on names used in the theoretical derivation of Metropolis-Hastings Monte Carlo as outlined here, for the sake of common nomenclature.

What we are building is a step-by-step Markov Chain Monte Carlo (MCMC) algorithm to describe the evolution of a system in an initial state towards a final state distributed according to a desired probability distribution $P(x)$. This final sentence is to be read in the sense that repeated iterations of the algorithm, on distinct initial states, lead to an ensemble of states distributed according to a desired distribution $P(x)$.

For each of these iterations, during each step, given an initial state $x$ for the system and a final state $x'$, the probability of the system moving from $x$ to $x'$ is factorized into the proposal probability $g(x'|x)$ and the acceptance probability $A(x'|x)$ -- i.e. $P(x'|x) = g(x'|x) A(x'|x)$.

The meaning of the proposal probability is that of being the probability associated to proposing the next state to be $x'$ if we start from the state $x$. That of the acceptance probability is the probability of accepting the state $x'$ if we start from the initial state $x$ and derives from the final desired distribution of the states $P(x)$ -- aside from physically justified fluctuations, the most probable states are accepted and the least ones rejected.

All that is stated above is valid for any MCMC method.

In a common scenario, the specific case of the Metropolis process -- which is a particular MCMC method -- we choose $g(x'|x)$ to be symmetrical. But beside it being so, we generally pose no further constraints on this choice.

It left me with some open questions: how does the choice of the form of the proposal distribution influence the MCMC algorithm? Does it depend on the system in analysis? Specifically, is there any physical meaning behind the choice of the proposal distribution $g(x'|x)$?

[Cross posted from physics.stackexchange]

• What do you mean by "physical meaning"? – Tim Jun 27 '17 at 20:28
• As for the acceptance rate. Take for example the situation in which you consider the P(x) to be that of the classical thermal state. In this case the acceptance rate can be read as: "If $x'$ is lower in energy compared to $x$, then accept it (physical principium of minimal energy), if not consider thermal fluctuations to determine if it will be accepted." – user213575 Jun 28 '17 at 10:32
• In the case of the proposal distribution, I imagine it could be something like: if we take for example an hypothetical random equally distributed sampling of the configuration space, "each final state is equally probable to be considered as any other". In another case, if I consider a certain order in which to propose new states, maybe it can be read as that being caused by some characteristic of the system such as certain transitions to be more probable to happen before some other. But I can think of a million more possible cases, without necessarily figuring out a physical explaination. – user213575 Jun 28 '17 at 10:38