Abstract
Positivity of the prior probability of Kullback-Leibler neighborhood around the true density, commonly known as the Kullback-Leibler property, plays a fundamental role in posterior consistency. A popular prior for Bayesian estimation is given by a Dirichlet mixture, where the kernels are chosen depending on the sample space and the class of densities to be estimated. The Kullback-Leibler property of the Dirichlet mixture prior has been shown for some special kernels like the normal density or Bernstein polynomial, under appropriate conditions. In this paper, we obtain easily verifiable sufficient conditions, under which a prior obtained by mixing a general kernel possesses the Kullback-Leibler property. We study a wide variety of kernel used in practice, including the normal, t, histogram, gamma, Weibull densities and so on, and show that the Kullback-Leibler property holds if some easily verifiable conditions are satisfied at the true density. This gives a catalog of conditions required for the Kullback-Leibler property, which can be readily used in applications.
Highlights
Density estimation, which is relevant in various applications such as cluster analysis and robust estimation, is a fundamental nonparametric inference problem
In Bayesian approach to density estimation, a prior such as a Gaussian process, a Polya tree process, or a Dirichlet mixture is constructed on the space of probability densities
Note that the variable x and the parameters θ, φ and ξ mentioned above are not necessarily one-dimensional. Asymptotic properties, such as consistency, and rate of convergence of the posterior distribution based on kernel mixture priors were established by Ghosal, Ghosh and Ramamoorthi [11], Tokdar [29], and Ghosal and van der Vaart [13; 14], when the kernel is chosen to be a normal probability density
Summary
Density estimation, which is relevant in various applications such as cluster analysis and robust estimation, is a fundamental nonparametric inference problem. Let Φ be the space of φ and supp(μ) denote the support of μ With such a random hyper parameter in the chosen kernel, the prior on densities is induced by μ × Π via the map (φ, P ) → fP,φ(x) := K(x; θ, φ)dP (θ). Note that the variable x and the parameters θ, φ and ξ mentioned above are not necessarily one-dimensional Asymptotic properties, such as consistency, and rate of convergence of the posterior distribution based on kernel mixture priors were established by Ghosal, Ghosh and Ramamoorthi [11], Tokdar [29], and Ghosal and van der Vaart [13; 14], when the kernel is chosen to be a normal probability density (and the prior distribution of the mixing distribution is DP).
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