  Bayesian methods and their implementations by means of sophisticated Monte
Carlo techniques, such as Markov chain Monte Carlo (MCMC) and particle filters,
have become very popular in signal processing over the last years. However, in
many problems of practical interest these techniques demand procedures for
sampling from probability distributions with non-standard forms, hence we are
often brought back to the consideration of fundamental simulation algorithms,
such as rejection sampling (RS). Unfortunately, the use of RS techniques
demands the calculation of tight upper bounds for the ratio of the target
probability density function (pdf) over the proposal density from which
candidate samples are drawn. Except for the class of log-concave target pdf's,
for which an efficient algorithm exists, there are no general methods to
analytically determine this bound, which has to be derived from scratch for
each specific case. In this paper, we introduce new schemes for (a) obtaining
upper bounds for likelihood functions and (b) adaptively computing proposal
densities that approximate the target pdf closely. The former class of methods
provides the tools to easily sample from a posteriori probability distributions
(that appear very often in signal processing problems) by drawing candidates
from the prior distribution. However, they are even more useful when they are
exploited to derive the generalized adaptive RS (GARS) algorithm introduced in
the second part of the paper. The proposed GARS method yields a sequence of
proposal densities that converge towards the target pdf and enable a very
efficient sampling of a broad class of probability distributions, possibly with
multiple modes and non-standard forms.
