THE POISSON TRANSFORM FOR UNNORMALISED STATISTICAL MODELS

ABSTRACT. Contrary to standard statistical models, unnormalised statistical models only specify the likelihood function up to a constant. While such models are natural and popular, the lack of normalisation makes inference much more difficult. Here we show that inferring the parameters of a unnormalised model on a space Ω can be mapped onto an equivalent problem of estimating the intensity of a Poisson point process on Ω. The unnormalised statistical model now specifies an intensity function that does not need to be normalised. Effectively, the normalisation constant may now be inferred as just another parameter, at no loss of information. The result can be extended to cover non-IID models, which includes for example unnormalised models for sequences of graphs (dynamical graphs), or for sequences of binary vectors. As a consequence, we prove that unnormalised parameteric inference in non-IID models can be turned into a semi-parametric estimation problem. Moreover, we show that the noise-contrastive divergence of Gutmann and Hyvärinen (2012) can be understood as an approximation of the Poisson transform, and extended to non-IID settings. We use our results to fit spatial Markov chain models of eye movements, where the Poisson transform allows us to turn a highly non-standard model into vanilla semi-parametric logistic regression. Unnormalised statistical models, and especially those based on exponential families, are popular throughout machine learning, statistics and computer vision (Wainwright and Jordan, 2008). The most iconic of these models is probably the Ising model (Ising, 1925), which applies to binary vectors y ∈ {0, 1}d: p(y|a,Q) ∝ exp (aty + ytQy). The Ising model is useful in that it can describe correlated binary data thanks to the couplings induced by Q, but computing th