  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.
