Austerity in MCMC Land: Cutting the Metropolis-Hastings Budget

A. Distribution of the test statistic In the sequential test, we first compute the test statistic from a mini-batch of size m. If a decision cannot be made with this statistic, we keep increasing the mini-batch size by m datapoints until we reach a decision. This procedure is guaranteed to terminate as explained in Section 4. The parameter ✏ controls the probability of making an error in a single test and not the complete sequential test. As the statistics across multiple tests are correlated with each other, we should first obtain the joint distribution of these statistics in order to estimate the error of the complete sequential test. Let ̄l j and s l,j be the sample mean and standard deviation respectively, computed using the first j mini-batches. Notice that when the size of a mini-batch is large enough, e.g. n > 100, the central limit theorem applies, and also s l,j is an accurate estimate of the population standard deviation. Additionally, since the degrees of freedom is high, the t-statistic in Eqn. 5 reduces to a z-statistic. Therefore, it is reasonable to make the following assumptions: Assumption 1. The joint distribution of the sequence