  We present the fundamentals of a measure transport approach to sampling. The
idea is to construct a deterministic coupling---i.e., a transport map---between
a complex "target" probability measure of interest and a simpler reference
measure. Given a transport map, one can generate arbitrarily many independent
and unweighted samples from the target simply by pushing forward reference
samples through the map. We consider two different and complementary scenarios:
first, when only evaluations of the unnormalized target density are available,
and second, when the target distribution is known only through a finite
collection of samples. We show that in both settings the desired transports can
be characterized as the solutions of variational problems. We then address
practical issues associated with the optimization--based construction of
transports: choosing finite-dimensional parameterizations of the map, enforcing
monotonicity, quantifying the error of approximate transports, and refining
approximate transports by enriching the corresponding approximation spaces.
Approximate transports can also be used to "Gaussianize" complex distributions
and thus precondition conventional asymptotically exact sampling schemes. We
place the measure transport approach in broader context, describing connections
with other optimization--based samplers, with inference and density estimation
schemes using optimal transport, and with alternative transformation--based
approaches to simulation. We also sketch current work aimed at the construction
of transport maps in high dimensions, exploiting essential features of the
target distribution (e.g., conditional independence, low-rank structure). The
approaches and algorithms presented here have direct applications to Bayesian
computation and to broader problems of stochastic simulation.
