  A number of distributions that arise in statistical applications can be
expressed in the form of a weighted density: the product of a base density and
a nonnegative weight function. Generating variates from such a distribution may
be nontrivial and can involve an intractable normalizing constant. Rejection
sampling may be used to generate exact draws, but requires formulation of a
suitable proposal distribution. To be practically useful, the proposal must
both be convenient to sample from and not reject candidate draws too
frequently. A well-known approach to design a proposal involves decomposing the
target density into a finite mixture, whose components may correspond to a
partition of the support. This work considers such a construction that focuses
on majorization of the weight function. This approach may be applicable when
assumptions for adaptive rejection sampling and related algorithms are not met.
An upper bound for the rejection probability based on this construction can be
expressed to evaluate the efficiency of the proposal before sampling. A method
to partition the support is considered where regions are bifurcated based on
their contribution to the bound. Examples based on the von Mises Fisher
distribution and Gaussian Process regression are provided to illustrate the
method.
