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Permanental point processes

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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.

Refs

Ben Hough, J., Manjunath Krishnapur, Yuval Peres, and Bálint Virág. 2006. “Determinantal Processes and Independence.” Probability Surveys 3 (0): 206–29. https://doi.org/10.1214/154957806000000078.

Eisenbaum, Nathalie, and Haya Kaspi. 2009. “On Permanental Processes.” Stochastic Processes and Their Applications 119 (5): 1401–15. https://doi.org/10.1016/j.spa.2008.07.003.

Lavancier, Frédéric, Jesper Møller, and Ege Rubak. 2015. “Determinantal Point Process Models and Statistical Inference.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77 (4): 853–77. https://doi.org/10.1111/rssb.12096.

McCullagh, Peter, and Jesper Møller. 2006. “The Permanental Process.” Advances in Applied Probability 38 (4): 873–88. https://doi.org/10.1017/S0001867800001361.

Møller, Jesper, and Rasmus Waagepetersen. 2017. “Some Recent Developments in Statistics for Spatial Point Patterns.” Annual Review of Statistics and Its Application 4 (1): 317–42. https://doi.org/10.1146/annurev-statistics-060116-054055.

Walder, Christian J., and Adrian N. Bishop. 2017. “Fast Bayesian Intensity Estimation for the Permanental Process.” In International Conference on Machine Learning, 3579–88. http://proceedings.mlr.press/v70/walder17a.html.