  Markov chain Monte Carlo (MCMC) is a popular and successful general-purpose
tool for Bayesian inference. However, MCMC cannot be practically applied to
large data sets because of the prohibitive cost of evaluating every likelihood
term at every iteration. Here we present Firefly Monte Carlo (FlyMC) an
auxiliary variable MCMC algorithm that only queries the likelihoods of a
potentially small subset of the data at each iteration yet simulates from the
exact posterior distribution, in contrast to recent proposals that are
approximate even in the asymptotic limit. FlyMC is compatible with a wide
variety of modern MCMC algorithms, and only requires a lower bound on the
per-datum likelihood factors. In experiments, we find that FlyMC generates
samples from the posterior more than an order of magnitude faster than regular
MCMC, opening up MCMC methods to larger datasets than were previously
considered feasible.
