Networks capture our intuition about relationships in the world. They describe the friendships between Facebook users, interactions in financial markets, and synapses connecting neurons in the brain. These networks are richly structured with cliques of friends, sectors of stocks, and a smorgasbord of cell types that govern how neurons connect. Some networks, like social network friendships, can be directly observed, but in many cases we only have an indirect view of the network through the actions of its constituents and an understanding of how the network mediates that activity. In this work, we focus on the problem of latent network discovery in the case where the observable activity takes the form of a mutually-excitatory point process, also known as a Hawkes process. We build on previous work that has taken a Bayesian approach to this problem, specifying prior distributions over the latent network structure and a likelihood of observed activity given this network. We extend this work by proposing a discretetime formulation and developing a computationally efficient stochastic variational inference (SVI) algorithm that allows us to scale the approach to long sequences of observations. We demonstrate our algorithm on the calcium imaging data used in the Chalearn neural connectomics challenge. 1. Introduction. Networks are abstractions of the relationships and connections between realworld objects, such as people, stocks, or neurons. These connections reflect relationships like "Wilson and Brady are friends," or "When neuron A fires it excites neuron B." Sometimes the networks themselves are observed data, as in the case of social network friendships, but often our view of the network is indirect. We are left to infer the latent connections between objects based on our observations of their behavior. In our neural example, recording techniques can often provide a measure of the neurons' activity but cannot resolve the individual synaptic connections between neurons. Given our knowledge of how synapses work, however, we might infer that if one neuron consistently fires shortly after another then there is likely an excitatory connection between them. This is one example of the latent network discovery problem that this work addresses. We focus on the case where our observations come in the form of a series of discrete events, like a sequence of Twitter messages or the firing pattern of a population of neurons. These events do not happen independently; rather, events induce subsequent events according to an excitatory network of interactions. A connection from one object to another indicates that events by the first object increase the probability of subsequent events by the second. We model these observations with a multivariate Hawkes process, a type of point process tailored to excitatory networks of interaction. Building on the previous work, we combine the Hawkes process observation model with a prior distribution over networks in a unified Bayesian model (Simma and Jordan, 2010; Blundell et al., 2012; Perry and Wolfe, 2013; DuBois et al., 2013; Linderman and Adams, 2014; Guo et al., 2015) . Most real-world networks are not simply random, but have highly structured patterns of interaction. For example, a friendship can often be traced back to some commonality between two people, such as belonging to the same club or attending the same school. A simple model for social networks might assign each person to a group, and then connect people according to whether or not they are
To tackle this problem, Linderman and Adams REF discretize the time, which introduces noise in the model.
Scalable Bayesian Inference for Excitatory Point Process Networks
arXiv: Machine Learning
Mathematics Computer Science