  In this paper, we study the effects of different prior and likelihood choices
for Bayesian matrix factorisation, focusing on small datasets. These choices
can greatly influence the predictive performance of the methods. We identify
four groups of approaches: Gaussian-likelihood with real-valued priors,
nonnegative priors, semi-nonnegative models, and finally Poisson-likelihood
approaches. For each group we review several models from the literature,
considering sixteen in total, and discuss the relations between different
priors and matrix norms. We extensively compare these methods on eight
real-world datasets across three application areas, giving both inter- and
intra-group comparisons. We measure convergence runtime speed, cross-validation
performance, sparse and noisy prediction performance, and model selection
robustness. We offer several insights into the trade-offs between prior and
likelihood choices for Bayesian matrix factorisation on small datasets - such
as that Poisson models give poor predictions, and that nonnegative models are
more constrained than real-valued ones.
