Edit: ah never mind, variational inference, got it! I was thinking stochastic HMC! --- Ok but that will get an unbiased estimate of the log-likelihood. MCMC o

Edit: ah never mind, variational inference, got it! I was thinking stochastic HMC! --- Ok but that will get an unbiased estimate of the log-likelihood. MCMC or HMC do work with noisy estimators, but they require unbiased estimates of the likelihood. At the very least, you need to do a convexity adjustment by measuring the variance inside your mini batch. Or you can use the Poisson technique which will get you unbiased estimates of exp(x) from unbiased estimates of x (albeit at the cost of introducing a lot of variance).