  We consider estimating the marginal likelihood in settings with independent
and identically distributed (i.i.d.) data. We propose estimating the predictive
distributions in a sequential factorization of the marginal likelihood in such
settings by using stochastic gradient Markov Chain Monte Carlo techniques. This
approach is far more efficient than traditional marginal likelihood estimation
techniques such as nested sampling and annealed importance sampling due to its
use of mini-batches to approximate the likelihood. Stability of the estimates
is provided by an adaptive annealing schedule. The resulting stochastic
gradient annealed importance sampling (SGAIS) technique, which is the key
contribution of our paper, enables us to estimate the marginal likelihood of a
number of models considerably faster than traditional approaches, with no
noticeable loss of accuracy. An important benefit of our approach is that the
marginal likelihood is calculated in an online fashion as data becomes
available, allowing the estimates to be used for applications such as online
weighted model combination.
