Fast likelihood‐based inference for latent count models using the saddlepoint approximation

Latent count models constitute an important modeling class in which a latent vector of counts, z, is summarized or corrupted for reporting, yielding observed data y=Tz where T is a known but non‐invertible matrix. The observed vector y generally follows an unknown multivariate distribution with a complicated dependence structure. Latent count models arise in diverse fields, such as estimation of population size from capture‐recapture studies; inference on multi‐way contingency tables summarized by marginal totals; or analysis of route flows in networks based on traffic counts at a subset of nodes. Currently, inference under these models relies primarily on stochastic algorithms for sampling the latent vector z, typically in a Bayesian data‐augmentation framework. These schemes involve long computation times and can be difficult to implement. Here, we present a novel maximum‐likelihood approach using likelihoods constructed by the saddlepoint approximation. We show how the saddlepoint likelihood may be maximized efficiently, yielding fast inference even for large problems. For the case where z has a multinomial distribution, we validate the approximation by applying it to a specific model for which an exact likelihood is available. We implement the method for several models of interest, and evaluate its performance empirically and by comparison with other estimation approaches. The saddlepoint method consistently gives fast and accurate inference, even when y is dominated by small counts.