Abstract
It is well-known that the Kullback–Leibler support condition implies posterior consistency in the weak topology, but is not sufficient for consistency in the total variation distance. There is a counter–example. Since then many authors have proposed sufficient conditions for strong consistency; and the aim of the present paper is to introduce new conditions with specific application to nonparametric mixture models with heavy–tailed components, such as the Student-$t$. The key is a more focused result on sets of densities where if strong consistency fails then it fails on such densities. This allows us to move away from the traditional types of sieves currently employed.
Highlights
In this paper we consider a novel approach to Bayesian consistency in nonparametric problems, concentrating on mixture models, which are the usual type of nonparametric model used in practice
The first formulation is given by Doob [9]; but this approach has a drawback in infinite dimensional models, see [7, 8]
Instead it is commonly assumed that observations are i.i.d. from some fixed but unknown density function, and a general sufficient condition for weak consistency is given in Schwartz [25]
Summary
In this paper we consider a novel approach to Bayesian consistency in nonparametric problems, concentrating on mixture models, which are the usual type of nonparametric model used in practice. Instead it is commonly assumed that observations are i.i.d. from some fixed but unknown density function, and a general sufficient condition for weak consistency is given in Schwartz [25]. Let the model L be the space of all Lebesgue densities on (R, R) equipped with the total variation metric, and Π be a prior on (L , L), where R and L are Borel σ-algebras. When d is the total variation (Levy–Prokhorov, resp.) metric, it is often called strongly (weakly, resp.) consistent. Along with the KL support condition (1.1), various sufficient conditions for strong consistency have been studied in infinite–dimensional models, see [3, 29, 28, 6] for general conditions. In this paper we present a new sufficient condition for strong consistency and apply it to nonparametric mixture models. The inequality represents “less than up to a constant multiplication,” where the constant is universal (such as 2, π, e) unless specified explicitly
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