  We propose a flexible nonparametric Bayesian modelling framework for
multivariate time series of count data based on tensor factorisations. Our
models can be viewed as infinite state space Markov chains of known maximal
order with non-linear serial dependence through the introduction of appropriate
latent variables. Alternatively, our models can be viewed as Bayesian
hierarchical models with conditionally independent Poisson distributed
observations. Inference about the important lags and their complex interactions
is achieved via MCMC. When the observed counts are large, we deal with the
resulting computational complexity of Bayesian inference via a two-step
inferential strategy based on an initial analysis of a training set of the
data. Our methodology is illustrated using simulation experiments and analysis
of real-world data.
