Monday, July 3, 2017

Bayes, Jeffreys, MCMC, Statistics, and Econometrics

In Ch. 3 of their brilliant book, Efron and Tibshirani (ET) assert that:
Jeffreys’ brand of Bayesianism [i.e., "uninformative" Jeffreys priors] had a dubious reputation among Bayesians in the period 1950-1990, with preference going to subjective analysis of the type advocated by Savage and de Finetti. The introduction of Markov chain Monte Carlo methodology was the kind of technological innovation that changes philosophies. MCMC ... being very well suited to Jeffreys-style analysis of Big Data problems, moved Bayesian statistics out of the textbooks and into the world of computer-age applications.
Interestingly, the situation in econometrics strikes me as rather the opposite.  Pre-MCMC, much of the leading work emphasized Jeffreys priors (RIP Arnold Zellner), whereas post-MCMC I see uniform at best (still hardly uninformative as is well known and as noted by ET), and often Gaussian or Wishart or whatever.  MCMC of course still came to dominate modern Bayesian econometrics, but for a different reason: It facilitates calculation of the marginal posteriors of interest, in contrast to the conditional posteriors of old-style analytical calculations. (In an obvious notation and for an obvious normal-gamma regression problem, for example, one wants posterior(beta), not posterior(beta | sigma).) So MCMC has moved us toward marginal posteriors, but moved us away from uninformative priors.