Posterior Consistency for some Semi-parametric Problems

The Bayesian approach to analyzing semi-parametric models are gaining popularity in practice. For the Cox proportional hazard model, it has been shown recently that the posterior is consistent and leads to asymptotically accurate confidence intervals under a Lévy process prior on the cumulative hazard rate. The explicit expression of the posterior distribution together with independent increment structure of Lévy process play a key role in the development. However, except for one-dimensional linear regression with an unknown error distribution and binary response regression with unknown link function, even consistency of Bayesian procedures has not been studied for a general prior distribution. We consider consistency of Bayesian inference for several semi-parametric models including multiple linear regression with an unknown error distribution, exponential frailty model, generalized linear model with unknown link function, Cox proportional hazard model where the baseline hazard function is unknown, accelerated failure time models and partial linear regression model. We give sufficient conditions under which the posterior distribution of the parametric part is consistent in the Euclidean distance while the non-parametric part is consistent with respect to some topology such as the weak topology. Our results are obtained by verifying the conditions of an appropriate modification of a celebrated result of Schwartz. Our general consistency result applies also to the case of independent, non-identically distributed observations. Application of our theorem requires showing the existence of exponentially consistent tests for the complements of the neighborhoods of the “true” value of the parameter and the prior positivity of a Kullback-Leibler type of neighborhood of the true distribution of the observations. We construct the required tests and give sufficient conditions for positivity of prior probabilities of Kullback-Leibler neighborhoods in all the examples we consider in this paper. AMS (2000) subject classification. Primary 62C10,62F15,60G09; secondary 62F03, 62H15.