Markov Chain, Monte Carlo, and Markov Chain Monte Carlo

Question

kindly explain to this to me (like im 5). what they are, the difference between Monte Carlo and MCMC simulations, how they’re used in Physics and Astronomy. I’ve been wondering since I heard about this from my seniors.

In: 5

Answer 1

Markov chains are a kind of process where the inputs for one step are solely determined by the outputs of the step directly before it. Anything that exhibits that behavior is a Markov chain.

Monte Carlo sampling is a way of describing a parameter space by repeatedly trying random points in that space and calculating their value. Think of it as trying to find the altitude of a hill by going to a bunch of random points on it and just measuring the altitude there. The more points you check, the more confident you can be in your answer.

A Markov Chain Monte Carlo then, is a kind of Markov chain that is focused toward describing a parameter space. And there are a number of different algorithms made for that purpose, to the point that I don’t really understand them all.

Why would you want to do this rather than just doing the regular Monte Carlo sample? Well sometimes you have a lot of different parameters, so randomly trying points would be silly, as you could never get enough points to be confident that your answer is correct.

So you get a bit clever, and find ways of localizing where your answer might be.

In the hill example, if your current altitude is lower than your previous one, why would you try looking close to that lower guess? It would probably be more efficient to guess close to places that you know to be higher.

But because there could be two peaks of the hill, you don’t just go close to prior high values, you randomize the process and sometimes jump to unrelated parts of the hill, and see if they’re taller.

As for where they’re used in astrophysics and cosmology, well there are a bunch of parameters we can measure that interact with each other. So it becomes more efficient to describe that parameter space with the Markov chain Monte Carlo simulation than by just completely guessing at what each parameter should be.