Index: Point Processes, Spatial Fields and Networks

Point Processes, Spatial Fields and Networks

Routing Summary

This cluster covers dependence in space, time and graphs: self-exciting (Hawkes/ETAS) and log-Gaussian Cox processes, Gaussian random fields and kriging, BYM2 areal models, Bayesian graph and DAG learning (graphical horseshoe, order MCMC, BGe), latent-space and block network models, phylogenetic regression. It has 9 sub-folders and 37 notes, written to cover the methods used in the PyMC worked examples E07, E18-E21, E23, E65, E79, E86, E89 (examples page). Start from a source overview:

• Friedman 2000: Friedman 2000 - Being Bayesian about Network Structure - Overview
• Hoff 2002: Hoff 2002 - Latent Space Models Overview
• Li 2019: Li 2019 - Graphical Horseshoe Estimator - Overview
• Reinhart 2018: Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes - Overview
• Riebler 2016: Riebler 2016 - BYM2 Intuitive Spatial Model - Overview
• Simpson 2016: Simpson 2016 - Going Off Grid - Overview

Sub-topics

• Friedman 2000 - Being Bayesian about Network Structure (5 notes) — COVERS: Friedman and Koller (UAI 2000; the journal version is Machine Learning 2003) compute the Bayesian posterior probability of structural features of a Bayesian network (edges, Markov blankets) by (i) a closed-form sum over all DAGs with at most $k$ parents consistent with a fixed node ordering, and (ii) MCMC over orderings rather than over structures. Ordering space is much smaller and smoother,…
• Hoff 2002 - Latent Space Approaches to Social Network Analysis (5 notes) — COVERS: Hoff, Raftery and Handcock (JASA 2002) model a network $Y$ by giving each actor a latent position $z_i$ in a low-dimensional Euclidean “social space” and letting tie probability depend on distance, conditionally independent given positions: $\text{logit}P(y_{ij}=1)=\alpha +{\beta }^{\top }{x}_{ij}-|z_i-z_j|$. The latent geometry automatically yields reciprocity and transitivity, supports MLE or MCMC…
• Li 2019 - Graphical Horseshoe Estimator (4 notes) — COVERS: Li, Craig and Bhadra (2019) place horseshoe priors on the off-diagonal entries of a Gaussian precision matrix $\Omega$ (diagonal flat, $\Omega$ positive definite) and give a full block Gibbs sampler. Theory: a smaller Kullback-Leibler risk upper bound than priors with bounded density at zero when ${\Omega }_0$ is sparse (Thm 3.2), and a posterior mean within $O(1/|\hat{\omega }|)$ of least squares for…
• Network Outcome Models Concept Notes (1 notes) — COVERS: Concept notes (no raw source): Network Outcome Models - Stochastic Block Models and Network Checks
• Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes (7 notes) — COVERS: Reinhart (2018, Statistical Science) reviews self-exciting spatio-temporal point processes (Hawkes processes and ETAS-type models): event rates that depend on the history of earlier events, so that events trigger further events. The paper unifies estimation (maximum likelihood, EM with branching structure), stochastic declustering, simulation, inference, Bayesian methods and diagnostics, and…
• Riebler 2016 - BYM2 Intuitive Spatial Model (5 notes) — COVERS: Riebler, Sorbye, Simpson and Rue (2016, Statistical Methods in Medical Research; arXiv:1601.01180) explain and test the BYM2 parameterisation of the Besag-York-Mollie disease-mapping model: $b={\tau }_b^{-1/2}(\sqrt{1-\varphi }v+\sqrt{\varphi }{u}^{\star })$ with a scaled structured component $u^{\star }$. Then ${\tau }_b$ is a marginal precision independent of the graph, $\varphi$ is the share of variance…
• Simpson 2016 - Going Off Grid LGCP Inference (6 notes) — COVERS: Simpson, Illian, Lindgren, Sorbye and Rue (Biometrika 2016) show how to fit log-Gaussian Cox processes (LGCPs) without binning points to a fine lattice: represent the latent field with a finite basis $Z(s)={\sum }_i{z}_i{\varphi }_i(s)$ (SPDE/Matern on a triangulation), approximate only the intensity integral by a quadrature rule, and obtain an exact-in-the-points Poisson-form likelihood. A lattice of…
• Spatial Fields Concept Notes (1 notes) — COVERS: Concept notes (no raw source): Spatial Fields - Gaussian Random Fields, Kriging and Field-Level Inference
• Structure Learning and Phylogenetics Concept Notes (3 notes) — COVERS: Concept notes (no raw source): Bayesian Causal Discovery - BGe Score, Interventions and Sachs Data, Bayesian Gaussian Graphical Models - G-Wishart and Neighbourhood Selection, Phylogenetic Regression - Brownian Motion, OU and Pagel Lambda

Motivating PyMC examples

E07, E18-E21, E23, E65, E79, E86, E89 — see the worked examples page; each source Overview lists the examples it supports.

Sources

• Friedman 2000 - Being Bayesian about Network Structure
• Hoff 2002 - Latent Space Approaches to Social Network Analysis
• Li 2019 - The Graphical Horseshoe Estimator
• Reinhart 2018 - Self-Exciting Spatio-Temporal Point Processes Review
• Riebler 2016 - BYM2 An Intuitive Bayesian Spatial Model for Disease Mapping
• Simpson 2016 - Going Off Grid LGCP Inference

See Also

• Applied Bayesian Model Families — sibling clusters