A promising class of methods for large-scale population genomic inference use the conditional sampling distribution (CSD), which approximates the probability of sampling an individual with a particular DNA sequence, given that a collection of sequences from the population has already been observed. The CSD has a wide range of applications, including imputing missing sequence data, estimating recombination rates, inferring human colonization history and identifying tracts of distinct ancestry in admixed populations. Most well-used CSDs are based on hidden Markov models (HMMs). Although computationally efficient in principle, methods resulting from the common implementation of the relevant HMM techniques remain intractable for large genomic datasets.