Convergence rates of posterior distributions

Subhashis Ghosal, Jayanta K. Ghosh, Aad W. van der Vaart
2000 Annals of Statistics  
We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinite-dimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, log-spline models, Dirichlet processes and interval censoring. Therefore, we can verify (2.7) for n = P θ θ ∞ ≤ M and every ε n such that J n log 1 + J −α n ε n nε 2 n Next, we have, with vol J the volume of the J − 1
more » ... lume of the J − 1 -dimensional unit ball, By Lemma 4.3 and the assumption that p 0 is bounded, the norms p 0 /p θ ∞ are uniformly bounded over θ ranging over a set of bounded θ ∞ . Therefore, By assumption, the first term is of the order c J . Thus condition (2.9) is satisfied if, for all sufficiently large j, J log j nε 2 n j 2 and ε n J −α This gives ε n of the order 1/n α/ 2α+1 for J n of the order ε −1/α n
doi:10.1214/aos/1016218228 fatcat:pcqu3s22qjaspeh65mudxxsdo4