Abstract

Will MCMC survive the “Big Data revolution”? Current MCMC methods for posterior inference, compute the likelihood of a model twice for every data-­‐case in order to make a single binary decision: to accept or reject a proposed parameter value. Compare this with stochastic gradient descent that uses O(1) computations per iteration. In this talk I will discuss two MCMC algorithms that cut the computational budget of an MCMC update. The first algorithm, “stochastic gradient Langevin dynamics” (and its successor “stochastic gradient Fisher scoring”) performs updates based on stochastic gradients and ignore the Metropolis-­‐Hastings step altogether. The second algorithm uses an approximate Metropolis-­‐Hastings rule where accept/reject decisions are made with high (but not perfect) confidence based on sequential hypothesis tests. We argue that for any finite sampling window, we can choose hyper-­‐parameters (stepsize, confidence level) such that the extra bias introduced by these algorithms is more than compensated by the reduction in variance due to the fact that we can draw more samples. We anticipate a new framework where bias and variance contributions to the sampling error a optimally traded-­‐off.