Papers Published
Abstract:
The Markov Chain Monte Carlo method is the dominant paradigm for posterior
computation in Bayesian analysis. It is common to control computation time by
making approximations to the Markov transition kernel. Comparatively little
attention has been paid to computational optimality in these approximating
Markov Chains, or when such approximations are justified relative to obtaining
shorter paths from the exact kernel. We give simple, sharp bounds for uniform
approximations of uniformly mixing Markov chains. We then suggest a notion of
optimality that incorporates computation time and approximation error, and use
our bounds to make generalizations about properties of good approximations in
the uniformly mixing setting. The relevance of these properties is demonstrated
in applications to a minibatching-based approximate MCMC algorithm for large
$n$ logistic regression and low-rank approximations for Gaussian processes.