  In genetic studies, haplotype data provide more refined information than data
about separate genetic markers. However, large-scale studies that genotype
hundreds to thousands of individuals may only provide results of pooled data,
where only the total allele counts of each marker in each pool are reported.
Methods for inferring haplotype frequencies from pooled genetic data that scale
well with pool size rely on a normal approximation, which we observe to produce
unreliable inference when applied to real data. We illustrate cases where the
approximation breaks down, due to the normal covariance matrix being
near-singular. As an alternative to approximate methods, in this paper we
propose exact methods to infer haplotype frequencies from pooled genetic data
based on a latent multinomial model, where the observed allele counts are
considered integer combinations of latent, unobserved haplotype counts. One of
our methods, latent count sampling via Markov bases, achieves approximately
linear runtime with respect to pool size. Our exact methods produce more
accurate inference over existing approximate methods for synthetic data and for
data based on haplotype information from the 1000 Genomes Project. We also
demonstrate how our methods can be applied to time-series of pooled genetic
data, as a proof of concept of how our methods are relevant to more complex
hierarchical settings, such as spatiotemporal models.
