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Positivity of the prior probability of Kullback-Leibler neighborhood aroundthe true density, commonly known as the Kullback-Leibler property, plays afundamental role in posterior consistency. A popular prior for Bayesianestimation is given by a Dirichlet mixture, where the kernels are chosendepending on the sample space and the class of densities to be estimated. TheKullback-Leibler property of the Dirichlet mixture prior has been shown forsome special kernels like the normal density or Bernstein polynomial, underappropriate conditions. In this paper, we obtain easily verifiable sufficientconditions, under which a prior obtained by mixing a general kernel possessesthe Kullback-Leibler property. We study a wide variety of kernel used inpractice, including the normal, $t$, histogram, gamma, Weibull densities and soon, and show that the Kullback-Leibler property holds if some easily verifiableconditions are satisfied at the true density. This gives a catalog ofconditions required for the Kullback-Leibler property, which can be readilyused in applications.

Kullback Leibler property of kernel mixture priors in Bayesian density estimation