# SRN – Scalable Bayesian Inference for Excitatory Point Process Networks by Linderman and Adams

The topic of this week’s SRN is latent network discovery. The paper ‘Scalable bayesian inference for excitatory point process networks‘ by Scott Linderman and Ryan Adams is suggested to me by Gonzalo Mena.

Hawkes process v.s. Poisson process, by Steven Mores. Source: https://stmorse.github.io/journal/Hawkes-python.html.

In a multivariate Hawkes process, there are $K$ individual point processes and events on one constituent process are more likely to happen right after an event from a neighboring process. To put it in other words, events are ‘induce’ subsequent events according to a network describing their interactions. If we imagine a network of neurons, if neuron A and B are connected and when neuron A fires it can excite neuron B and neuron B has a higher chance of firing. When we want to infer connectivity, if neuron B consistently after neuron A fires, we might  infer that there is likely an excitatory connection between them.

In the work by Linderman and Adams, they extend previous works on continuous time Hawkes process and study the discrete time network Hawkes model. This discretization is motivated by computational concerns. In Simma and Jordan (2010), the authors invoke auxiliary variable $z_{k,n}$ that which denotes the origin of the $n$-th event on process $k$. So the number of auxiliary variables needed is large for highly active networks, which could be a computational bottleneck. But working in the discrete model assuming a time scale $\Delta t$, we can ‘bin’ events and ignore interactions within the same bin. This reduction in number of auxiliary variables improves the run time of inference algorithms such as Gibbs sampler.

Comparison of run time per Gibbs sweep. Source: Fig 2 of Linderman and Adams (2015) https://arxiv.org/pdf/1507.03228.pdf

Besides Gibbs sampler, Linderman and Adams also derived stochastic variational inference algorithm for the discrete time network Hawkes model. This work focuses on scaling to long duration $T$ and the problem of scaling with size of network $K$ remains open.

References:

• Linderman, S. W., & Adams, R. P. (2015). Scalable bayesian inference for excitatory point process networks. arXiv preprint arXiv:1507.03228.
• Linderman, S., & Adams, R. (2014, January). Discovering latent network structure in point process data. In International Conference on Machine Learning (pp. 1413-1421).
• Simma, A., & Jordan, M. I. (2012). Modeling events with cascades of Poisson processes. arXiv preprint arXiv:1203.3516.
• Steven Morse (2017, June). Python class for Hawkes Process. [Blog post] https://stmorse.github.io/journal/Hawkes-python.html.