  Many Bayesian inference problems involve target distributions whose density
functions are computationally expensive to evaluate. Replacing the target
density with a local approximation based on a small number of carefully chosen
density evaluations can significantly reduce the computational expense of
Markov chain Monte Carlo (MCMC) sampling. Moreover, continual refinement of the
local approximation can guarantee asymptotically exact sampling. We devise a
new strategy for balancing the decay rate of the bias due to the approximation
with that of the MCMC variance. We prove that the error of the resulting local
approximation MCMC (LA-MCMC) algorithm decays at roughly the expected
$1/\sqrt{T}$ rate, and we demonstrate this rate numerically. We also introduce
an algorithmic parameter that guarantees convergence given very weak tail
bounds, significantly strengthening previous convergence results. Finally, we
apply LA-MCMC to a computationally intensive Bayesian inverse problem arising
in groundwater hydrology.
