Rejection Sampling for Weighted Densities by Majorization

A number of density functions of interest can be written in the form of a weighted density: the product of a base density and a nonnegative weight function that provides an adjustment. Generation of random 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 determination of a proposal distribution. To be practical for an intended application, the proposal must both be convenient to sample from and accept draws with large enough probability. A well-known approach to obtain a proposal involves decomposing the target density into a finite mixture where components may correspond to a partition of the support. This work considers focusing such a construction on an envelope for the weight function. This may be applicable when assumptions for adaptive rejection sampling and related algorithms are not met. An upper bound on rejection probability from this proposal construction can be expressed and potentially reduced to a desired tolerance by making suitable refinements. Several example applications will be considered to illustrate the method.

Random variate generation

Proposal distribution

Finite mixture

Truncation

CDF method

Rejection probability 

Mid-Level

Computational Statistics

Symposium on Data Science and Statistics (SDSS) 2023