Permanental point processes

2019-12-04 — 2019-12-04

Wherein a point process is described whose intensity is given by the square of a Gaussian process, and whose likelihood is seen to involve a matrix permanent in its formulation.

linear algebra
Monte Carlo
point processes
Figure 1

Placeholder notes for a type of point process, with which I am unfamiliar, but about which I am incidentally curious.

This is, AFAICT, a point process whose intensity is a squared Gaussian process. The term permanental is because the matrix permanent arises somewhere in the model of this process although I know not where (Walder and Bishop 2017). From some incidental comments at a seminar, I presumed the permanental process was actually a Gibbs point process (i.e. determined by interactions between points, not a latent process) like its determinantal cousin and I am surprised to find otherwise.

1 References

Ben Hough, Krishnapur, Peres, et al. 2006. “Determinantal Processes and Independence.” Probability Surveys.
Eisenbaum, and Kaspi. 2009. “On Permanental Processes.” Stochastic Processes and Their Applications.
Lavancier, Møller, and Rubak. 2015. “Determinantal Point Process Models and Statistical Inference.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
McCullagh, and Møller. 2006. “The Permanental Process.” Advances in Applied Probability.
Møller, and Waagepetersen. 2017. “Some Recent Developments in Statistics for Spatial Point Patterns.” Annual Review of Statistics and Its Application.
Walder, and Bishop. 2017. “Fast Bayesian Intensity Estimation for the Permanental Process.” In International Conference on Machine Learning.