Abstract:
Parallel tempering (PT) is a class of Markov chain
Monte Carlo algorithms that constructs a path of
distributions annealing between a tractable reference
and an intractable target, and then interchanges
states along the path to improve mixing
in the target. The performance of PT depends on
how quickly a sample from the reference distribution
makes its way to the target, which in turn
depends on the particular path of annealing distributions.
However, past work on PT has used
only simple paths constructed from convex combinations
of the reference and target log-densities.
This paper begins by demonstrating that this path
performs poorly in the setting where the reference
and target are nearly mutually singular. To address
this issue, we expand the framework of PT
to general families of paths, formulate the choice
of path as an optimization problem that admits
tractable gradient estimates, and propose a flexible
new family of spline interpolation paths for
use in practice. Theoretical and empirical results
both demonstrate that our proposed methodology
breaks previously-established upper performance
limits for traditional paths.
