statisticalrethinkin..

8 Markov Chain Monte Carlo EstimationFor most of human history, chance has been a villain. In classic Roman civilization,chance was personified by Fortuna, goddess of cruel fate, with her ever spinning wheel ofluck. Opposed to her sat Minerva, goddess of wisdom and understanding. Only the desperatewould pray to Fortuna, while everyone implored Minerva for aid. Certainly science wasthe domain of Minerva, a realm with no useful role for Fortuna to play.But by the beginning of the 20th century, the opposition between Fortuna and Minervahad changed to a collaboration. Scientists, servants of Minerva, began publishing booksof random numbers, instruments of chance to be used for learning about the world. Now,chance and wisdom share a cooperative relationship, and few of us are any longer bewilderedby the notion that an understanding of chance could help us acquire wisdom. Everythingfrom weather forecasting to finance to evolutionary biology is dominated by the study ofstochastic processes.is chapter introduces one of the more marvelous examples of how Fortuna and Minervacooperate: the estimation of posterior probability densities using a stochastic processknown as Markov chain Monte Carlo (MCMC) estimation. Unlike in every earlier chapter inthis book, here we’ll produce samples from the joint posterior of a model without maximizinganything. Instead of having to lean on quadratic and other approximations of the shapeof the posterior, now we’ll be able to sample directly from the posterior without assumingany nice Gaussian, or any other, shape for it.e cost of this power is that it may take much longer for our estimates to finish, andusually more work is required to specify the model as well. But the benefit is escaping theawkwardness of assuming multivariate normality. Equally important is the ability to directlyestimate models, such as the multilevel models of later chapters. Such models are not onlyfrequently intractable with the methods we’ve used so far in this book, but also their posteriordistributions are oen non-Gaussian in dramatic ways.e good news is that tools for building and inspecting MCMC estimates are gettingbetter all the time. In this chapter you’ll meet a convenient way to convert the map formulasyou’ve used so far into Markov chains. e engine that makes this possible is STAN (mcstan.org).Stan’s creators describe it as “a probabilistic programming language implementingstatistical inference.” You won’t be working directly in Stan to begin with—the rethinkingpackage provides tools that hide it from you for now. But as you move on to more advancedtechniques, you’ll be able to quickly generate models you already understand in Stan’s language.en you can tinker with them and gain access to Stan’s full power.243