We introduce the Gamma-Exponential Process (GEP), a prior over a large family ofcontinuous time stochastic processes. A hierarchical version of this prior (HGEP; the Hierarchical GEP) yields a useful model for analyzing complex time series. Models based on HGEPs display many attractive properties: conjugacy, exchangeability and closed-form predictive distribution for the waiting times, and exact Gibbs updates for the time scale parameters. After establishing these properties, weshow how posterior inference can be carried efficiently using Particle MCMC methods . This yields a MCMC algorithm that can resample entire sequences atomicallywhile avoiding the complications of introducing slice and stick auxiliary variables of the beam sampler . We applied our model to the problem of estimating the disease progression in multiple sclerosis , and to RNA evolutionary modeling. In both domains, we found that our model outperformed the standard rate matrix estimation approach.
Bayesian networks are a popular representation of asymmetric (for example causal) relationships between random variables. Markov random fields (MRFs) are a complementary model of symmetric relationships used in computer vision, spatial modeling, and social and gene expression networks. A chain graph model under the Lauritzen-Wermuth-Frydenberg interpretation (hereafter a chain graph model) generalizes both Bayesian networks and MRFs, and can represent asymmetric and symmetric relationships together.As in other graphical models, the set of marginals from distributions in a chain graph model induced by the presence of hidden variables forms a complex model. One recent approach to the study of marginal graphical models is to consider a well-behaved supermodel. Such a supermodel of marginals of Bayesian networks, defined only by conditional independences, and termed the ordinary Markov model, was studied at length in (Evans and Richardson, 2014).In this paper, we show that special mixed graphs which we call segregated graphs can be associated, via a Markov property, with supermodels of a marginal of chain graphs defined only by conditional independences. Special features of segregated graphs imply the existence of a very natural factorization for these supermodels, and imply many existing results on the chain graph model, and ordinary Markov model carry over. Our results suggest that segregated graphs define an analogue of the ordinary Markov model for marginals of chain graph models.
Most real-world networks are too large to be measured or studied directly and there is substantial interest in estimating global network properties from smaller sub-samples. One of the most important global properties is the number of vertices/nodes in the network. Estimating the number of vertices in a large network is a major challenge in computer science, epidemiology, demography, and intelligence analysis. In this paper we consider a population random graph G = (V;E) from the stochastic block model (SBM) with K communities/blocks. A sample is obtained by randomly choosing a subset W and letting G(W) be the induced subgraph in G of the vertices in W. In addition to G(W), we observe the total degree of each sampled vertex and its block membership. Given this partial information, we propose an efficient PopULation Size Estimation algorithm, called PULSE, that accurately estimates the size of the whole population as well as the size of each community. To support our theoretical analysis, we perform an exhaustive set of experiments to study the effects of sample size, K, and SBM model parameters on the accuracy of the estimates. The experimental results also demonstrate that PULSE significantly outperforms a widely-used method called the network scale-up estimator in a wide variety of scenarios.
Entropy rate quantifies the amount of disorder in a stochastic process. For spiking neurons, the entropy rate places an upper bound on the rate at which the spike train can convey stimulus information, and a large literature has focused on the problem of estimating entropy rate from spike train data. Here we present Bayes Least Squares and Empirical Bayesian entropy rate estimators for binary spike trains using Hierarchical Dirichlet Process (HDP) priors. Our estimator leverages the fact that the entropy rate of an ergodic Markov Chain with known transition probabilities can be calculated analytically, and many stochastic processes that are non-Markovian can still be well approximated by Markov processes of sufficient depth. Choosing an appropriate depth of Markov model presents challenges due to possibly long time dependencies and short data sequences: a deeper model can better account for long time-dependencies, but is more difficult to infer from limited data. Our approach mitigates this difficulty by using a hierarchical prior to share statistical power across Markov chains of different depths. We present both a fully Bayesian and empirical Bayes entropy rate estimator based on this model, and demonstrate their performance on simulated and real neural spike train data.