Efficient data augmentation for fitting stochastic epidemic models to prevalence data

Stochastic epidemic models describe the dynamics of an epidemic as a disease spreads through a population. Typically, only a fraction of cases are observed at a set of discrete times. The absence of complete information about the time evolution of an epidemic gives rise to a complicated latent variable problem in which the state space size of the epidemic grows large as the population size increases. This makes analytically integrating over the missing data infeasible for populations of even moderate size. We present a data augmentation Markov chain Monte Carlo (MCMC) framework for Bayesian estimation of stochastic epidemic model parameters, in which measurements are augmented with subject-level disease histories. In our MCMC algorithm, we propose each new subject-level path, conditional on the data, using a time-inhomogeneous continuous-time Markov process with rates determined by the infection histories of other individuals. The method is general, and may be applied, with minimal modifications, to a broad class of stochastic epidemic models. We present our algorithm in the context of multiple stochastic epidemic models in which the data are binomially sampled prevalence counts, and apply our method to data from an outbreak of influenza in a British boarding school

Maximum likelihood estimation for randomized shortest paths with trajectory data

Randomized shortest paths (RSPs) are tool developed in recent years for different graph and network analysis applications, such as modelling movement or flow in networks. In essence, the RSP framework considers the temperature-dependent Gibbs–Boltzmann distribution over paths in the network. At low temperatures, the distribution focuses solely on the shortest or least-cost paths, while with increasing temperature, the distribution spreads over random walks on the network. Many relevant quantities can be computed conveniently from this distribution, and these often generalize traditional network measures in a sensible way. However, when modelling real phenomena with RSPs, one needs a principled way of estimating the parameters from data. In this work, we develop methods for computing the maximum likelihood estimate of the model parameters, with focus on the temperature parameter, when modelling phenomena based on movement, flow or spreading processes. We test the validity of the derived methods with trajectories generated on artificial networks as well as with real data on the movement of wild reindeer in a geographic landscape, used for estimating the degree of randomness in the movement of the animals. These examples demonstrate the attractiveness of the RSP framework as a generic model to be used in diverse applications. randomized shortest paths; random walk; shortest path; parameter estimation; maximum likelihood; animal movement modellingpublishedVersio

Maximum likelihood estimation for randomized shortest paths with trajectory data

Randomized shortest paths (RSPs) are tool developed in recent years for different graph and network analysis applications, such as modelling movement or flow in networks. In essence, the RSP framework considers the temperature-dependent Gibbs–Boltzmann distribution over paths in the network. At low temperatures, the distribution focuses solely on the shortest or least-cost paths, while with increasing temperature, the distribution spreads over random walks on the network. Many relevant quantities can be computed conveniently from this distribution, and these often generalize traditional network measures in a sensible way. However, when modelling real phenomena with RSPs, one needs a principled way of estimating the parameters from data. In this work, we develop methods for computing the maximum likelihood estimate of the model parameters, with focus on the temperature parameter, when modelling phenomena based on movement, flow or spreading processes. We test the validity of the derived methods with trajectories generated on artificial networks as well as with real data on the movement of wild reindeer in a geographic landscape, used for estimating the degree of randomness in the movement of the animals. These examples demonstrate the attractiveness of the RSP framework as a generic model to be used in diverse applications. randomized shortest paths; random walk; shortest path; parameter estimation; maximum likelihood; animal movement modellingpublishedVersio

Neural Probabilistic Methods for Event Sequence Modeling

This thesis focuses on modeling event sequences, namely, sequences of discrete events in continuous time. We build a family of generative probabilistic models that is able to reason about what events will happen in the future and when, given the history of previous events. Under our models, each event—as it happens—is allowed to update the future intensities of multiple event types, and the intensity of each event type—as nothing happens—is allowed to evolve with time along a trajectory. We use neural networks to allow the “updates” and “trajectories” to be complex and realistic. In the purely neural version of our model, all future event intensities are conditioned on the hidden state of a continuous-time LSTM, which has consumed every past event as it happened. To exploit domain-specific knowledge of how an event might only affect a few—but not all—future event intensities, we propose to introduce domain-specific structure into the model. We design a modeling language, by which a domain expert can write down the rules of a temporal deductive database. The database tracks facts over time; the rules deduce facts from other facts and from past events. Each fact has a time-varying state, computed by a neural network whose topology is determined by the fact’s provenance, including its experience of the past events that have contributed to deducing it. The possible event types at any time are given by special facts, whose intensities are neurally modeled alongside their states. We develop efficient methods for training our models, and doing inference with them. Applying the general principle of noise-contrastive estimation, we work out a stochastic training objective that is less expensive to optimize than the log-likelihood, which people typically maximize for parameter estimation. As in the discrete-time case that inspired us, the parameters that maximize our objective will provably maximize the log-likelihood as well. For the scenarios where we are given incomplete sequences, we propose particle smoothing—a form of sequential importance sampling—to impute the missing events. This thesis includes extensive experiments, demonstrating the effectiveness of our models and algorithms. On many synthetic and real-world datasets, on held-out sequences, we show empirically: (1) our purely neural model achieves competitive likelihood and predictive accuracy; (2) our neural-symbolic model improves prediction by encoding appropriate domain knowledge in the architecture; (3) for models to achieve the same level of log-likelihood, our noise-contrastive estimation needs considerably fewer function evaluations and less wall-clock time than maximum likelihood estimation; (4) our particle smoothing method is effective at inferring the ground-truth unobserved events. In this thesis, I will also discuss a few future research directions, including embedding our models within a reinforcement learner to discover causal structure and learn an intervention policy

Tutorial on structured continuous-time markov processes

A continuous-time Markov process (CTMP) is a collection of variables indexed by a continuous quantity, time. It obeys the Markov property that the distribution over a future variable is independent of past variables given the state at the present time. We introduce continuous-time Markov process representations and algorithms for filtering, smoothing, expected sufficient statistics calculations, and model estimation, assuming no prior knowledge of continuous-time processes but some basic knowledge of probability and statistics. We begin by describing "flat" or unstructured Markov processes and then move to structured Markov processes (those arising from state spaces consisting of assignments to variables) including Kronecker, decision-diagram, and continuous-time Bayesian network representations. We provide the first connection between decision-diagrams and continuous-time Bayesian networks