# 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 $g(x'|x)$?

• It's a good question but I think the answer is "With enough iterations and instances $g(x'|x)$ doesn't matter." because it is rarely discussed (the preferred direction in $A(x'|x)$ probably plays a role in this). Perhaps a more interesting questions is "What makes a good choice of $g(x'|x)$?" because "enough iterations" may depend on that choice. Jun 27, 2017 at 17:14
• I don't know if either Computational Science or Cross Validated would have more experts in a position to comment on the details. The above comment represents the limit of my current understanding—I often use some kind of simulated annealing procedure on $g$, reducing the 'range' as the simulation proceeds. Jun 27, 2017 at 17:17
• I am expecially interested in the physical meaningm though also finding out the convenience in terms of the algorithm is a good thing. Thanks for the suggestion, I'll definitely try. Jun 27, 2017 at 18:49

$g(x'\vert x)$ does not have a physical meaning, in the sense that it has no influence on the ensemble that is sampled and thus no influence on the physical observables (as long as detailed balance is preserved and the full phase space is accessible). However, the choice of $g(x'\vert x)$ is an implementation detail that can and should be tuned to a specific problem. Thus it will probably contain prior knowledge of the system. For example, when simulating a molecule you want to suggest small changes in the bond lengths, but larger changes in the dihedral angles, as the dihedral potential is typically much softer than the bond length potential.
In general, the choice of $g(x'\vert x)$ has to balance two aspects: Are the steps too large, an overwhelmingly amount of them will be rejected. Are the steps too small, you need a huge amount of steps to move through phase-space.