  Annealed importance sampling (AIS) and related algorithms are highly
effective tools for marginal likelihood estimation, but are not fully
differentiable due to the use of Metropolis-Hastings correction steps.
Differentiability is a desirable property as it would admit the possibility of
optimizing marginal likelihood as an objective using gradient-based methods. To
this end, we propose Differentiable AIS (DAIS), a variant of AIS which ensures
differentiability by abandoning the Metropolis-Hastings corrections. As a
further advantage, DAIS allows for mini-batch gradients. We provide a detailed
convergence analysis for Bayesian linear regression which goes beyond previous
analyses by explicitly accounting for the sampler not having reached
equilibrium. Using this analysis, we prove that DAIS is consistent in the
full-batch setting and provide a sublinear convergence rate. Furthermore,
motivated by the problem of learning from large-scale datasets, we study a
stochastic variant of DAIS that uses mini-batch gradients. Surprisingly,
stochastic DAIS can be arbitrarily bad due to a fundamental incompatibility
between the goals of last-iterate convergence to the posterior and elimination
of the accumulated stochastic error. This is in stark contrast with other
settings such as gradient-based optimization and Langevin dynamics, where the
effect of gradient noise can be washed out by taking smaller steps. This
indicates that annealing-based marginal likelihood estimation with stochastic
gradients may require new ideas.
