A Bayesian Approach to Sparse Network Discovery in Spiking Data

In order to reason about the interacting processes giving rise to a set of data, we must understand the causal relationships between those latent processes. One way to uncover these relationships is to discover a directed network structure from the data. Often, these data have the form of discrete events -- for example, neural spikes -- which can be modeled effectively via interacting point processes; the Hawkes process is the classical example of such a joint process.   I will describe a fully-Bayesian treatment of the Hawkes process, introducing convenient conjugate priors for the model parameters, along with a sparsity-promoting spike-and-slab prior on the elements of the interaction matrix.  I will discuss how it is possible to perform efficient inference in this model with Markov chain Monte Carlo.  This makes it possible to both recover posterior samples of the network structure and infer the characteristics of the underlying temporal dynamics.  I will report on explorations with this technique of neural spike train data and financial tick streams.  In both cases, it was possible to uncover sparse structure with meaningful parameters, suggesting that this method could be widely applicable for network discovery.