Methods of Data Analysis Metropolis Monte Carlo and Entropic Sampling

Many problems in statistical physics, machine learning and statistical inference require us to draw samples from (potentially very) high-dimensional distributions, P (~x). Often, one does not have an explicit expression for the probability distribution but (as we will see) can evaluate a function f(~x) ∝ P (~x). Markov Chain Monte Carlo is a way of sequentially generating samples (in a “chain”) using only knowledge of f , such that after some “burn-in” time the samples will be drawn from the desired distribution P . Today, MCMC is often a method of choice: given the processor speeds, it is possible to sample quickly, and the method is both conceptually clear, straightforward to implement, theoretically well-understood, and correct in the limit as the number of samples tends to infinity. There is a large body of literature, but also “an art” to efficient MCMC sampling. Many versions of MCMC exist that differ mostly in elementary “moves” that generate a new sample from the old one; Metropolis MCMC is one such approach often used to sample from Boltzmann distributions describing physical systems at equilibrium. A successful showcase use of MCMC has been to obtain a posterior distribution over ∼ 10 cosmological parameters given the likelihood of the experimental data (e.g., from cosmic microwave background and large scale surveys), using sampling runs on large computer clusters. A strength of the MCMC approach for inference is that it can provide confidence bounds and correlations among the parameters specifying the origin and fate of our universe.