The Poisson transform for unnormalised statistical models
This provides a foundational method for simplifying inference in unnormalised models, which are common in fields like spatial statistics and graph modeling, though it builds on existing noise-contrastive divergence ideas.
The paper tackles the difficulty of inference in unnormalised statistical models by mapping it to estimating the intensity of a Poisson point process, effectively turning the normalisation constant into an inferable parameter without information loss, and demonstrates this by fitting spatial Markov chain models for eye movements.
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 & 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.