A martingale law of large numbers and convergence rates of Bayesian posterior distributions

Details A martingale law of large numbers and convergence rates of Bayesian posterior distributions A martingale law of large numbers and convergence rates of Bayesian posterior distributions

2012-01-15

arxiv.org/abs/1201.3102v1

Mathematics

Statistics Theory

11 pages

Scientific paper

Frequentist-style large-sample properties of Bayesian posterior distributions, such as consistency and convergence rates, are important considerations in nonparametric problems. Martingale methods, in particular, martingale laws of large numbers, have been found to be useful for studying posterior consistency, but these same tools have yet to be applied in the more challenging convergence rates problem. This paper first establishes general sufficient conditions for convergence in probability of a suitably normalized martingale sequence. This general martingale result is then used to prove theorems on convergence rates for predictive densities and posterior distributions under suitable conditions on the prior. In particular, the optimal, nearly parametric posterior convergence rate can be achieved through this martingale-driven analysis.

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