Fundamentally, MCMC is just a way to build a pseudo-random number generator; the algorithm is a Big Deal because it lets you build one for complicated, multivariate distributions in an apparently straightforward fashion. (I say “apparently straightforward” because there’s lots to worry about in the actual implementation for complex problems, even though the Metropolis-Hastings algorithm most commonly used for MCMC is incredibly simple.) It just so happens that Bayesian calculations often require integrals of complicated, multivariate distributions, so MCMC has become closely associated with Bayes.

One of my first introductions to MCMC was a review paper by D. Toussaint, “Introduction to algorithms for Monte Carlo simulations and their application to QCD” (Computer Physics Communications, v56, 69-92 (1989)). The first few pages have a physics-flavored intro to the basic ideas of MCMC, motivated by problems in statistical physics and lattice QCD (not a prior or likelihood in sight!). I still send astronomer and physicist colleagues to this paper (among others), to see MCMC ideas in our language. As with Finkebeiner’s lecture, a key element in Toussaint’s presentation is detailed balance. It’s a nice paper, taking the reader from basic Monte Carlo to the Metropolis algorithm, Langevin methods, and ultimately hybrid Monte Carlo, in not very many pages.

(It’s perhaps worth noting that reversibility (detailed balance) is a sufficient but not necessary condition for a Markov chain to have a desired target distribution be its stationary distribution. There has been very little exploration of non-reversible algorithms; Diaconis, Holmes and Neal had a paper on this not long ago.)

]]>The reason for telling my experience is that there is quite statistics about your post, although the equations are purely based on (atomic) physics.

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