Bayesian Gaussian process factor analysis with copula for count data

Gaussian process factor analysis models aim to capture the latent trajectories that generated observed multi-variate data by factorizing the observations. These methods assume conditional independence of observations, given the factor scores and loadings. For continuous data this restrictive assumption can be resolved by using a non-diagonal covariance matrix. However, this does not work for count data. We propose a new model which pairs a negative binomial Gaussian process factor analysis with a Gaussian copula to find latent trajectories and also accounts for the residual covariance not captured by these trajectories. We provide a fully Bayesian implementation of the model and use augmented likelihood for inference with Hamiltonian Monte Carlo. We compare the proposed method to other Gaussian process factor analysis models on 12 toy data sets, finding latent qualities of NBA teams, and forecasting disease counts. The results show that the proposed method is useful for latent structure extraction and out-of-sample prediction of multivariate counts.