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

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

State-space models for bio-loggers: A methodological road map

Ecologists have an unprecedented array of bio-logging technologies available to conduct in situ studies of horizontal and vertical movement patterns of marine animals. These tracking data provide key information about foraging, migratory, and other behaviours that can be linked with bio-physical datasets to understand physiological and ecological influences on habitat selection. In most cases, however, the behavioural context is not directly observable and therefore, must be inferred. Animal movement data are complex in structure, entailing a need for stochastic analysis methods. The recent development of state-space modelling approaches for animal movement data provides statistical rigor for inferring hidden behavioural states, relating these states to bio-physical data, and ultimately for predicting the potential impacts of climate change. Despite the widespread utility, and current popularity, of state-space models for analysis of animal tracking data, these tools are not simple and require considerable care in their use. Here we develop a methodological “road map” for ecologists by reviewing currently available state-space implementations. We discuss appropriate use of state-space methods for location and/or behavioural state estimation from different tracking data types. Finally, we outline key areas where the methodology is advancing, and where it needs further developmen