Dynamic Bayesian networks have been well explored in the literature as discrete-time models; however, their continuous-time extensions have seen comparatively little attention. In this paper, we propose the first constraint-based algorithm for learning the structure of continuous-time Bayesian networks. We discuss the different statistical tests and the underlying hypotheses used by our proposal to establish conditional independence. Finally, we validate its performance using synthetic data, and discuss its strengths and limitations. We find that score-based is more accurate in learning networks with binary variables, while our constraint-based approach is more accurate with variables assuming more than two values. However, more experiments are needed for confirmation.
Continuous time Bayesian networks (CTBNs) describe structured stochastic processes with finitely many states that evolve over continuous time. A CTBN is a directed (possibly cyclic) dependency graph over a set of variables, each of which represents a finite state continuous time Markov process whose transition model is a function of its parents. We address the problem of learning parameters and structure of a CTBN from fully observed data. We define a conjugate prior for CTBNs, and show how it can be used both for Bayesian parameter estimation and as the basis of a Bayesian score for structure learning. Because acyclicity is not a constraint in CTBNs, we can show that the structure learning problem is significantly easier, both in theory and in practice, than structure learning for dynamic Bayesian networks (DBNs). Furthermore, as CTBNs can tailor the parameters and dependency structure to the different time granularities of the evolution of different variables, they can provide a better fit to continuous-time processes than DBNs with a fixed time granularity.
Learning temporal dependencies between variables over continuous time is an important and challenging task. Continuous-time Bayesian networks effectively model such processes but are limited by the number of conditional intensity matrices, which grows exponentially in the number of parents per variable. We develop a partition-based representation using regression trees and forests whose parameter spaces grow linearly in the number of node splits. Using a multiplicative assumption we show how to update the forest likelihood in closed form, producing efficient model updates. Our results show multiplicative forests can be learned from few temporal trajectories with large gains in performance and scalability.
A central task in many applications is reasoning about processes that change in a continuous time. The mathematical framework of Continuous Time Markov Processes provides the basic foundations for modeling such systems. Recently, Nodelman et al introduced continuous time Bayesian networks (CTBNs), which allow a compact representation of continuous-time processes over a factored state space. In this paper, we introduce continuous time Markov networks (CTMNs), an alternative representation language that represents a different type of continuous-time dynamics. In many real life processes, such as biological and chemical systems, the dynamics of the process can be naturally described as an interplay between two forces - the tendency of each entity to change its state, and the overall fitness or energy function of the entire system. In our model, the first force is described by a continuous-time proposal process that suggests possible local changes to the state of the system at different rates. The second force is represented by a Markov network that encodes the fitness, or desirability, of different states; a proposed local change is then accepted with a probability that is a function of the change in the fitness distribution. We show that the fitness distribution is also the stationary distribution of the Markov process, so that this representation provides a characterization of a temporal process whose stationary distribution has a compact graphical representation. This allows us to naturally capture a different type of structure in complex dynamical processes, such as evolving biological sequences. We describe the semantics of the representation, its basic properties, and how it compares to CTBNs. We also provide algorithms for learning such models from data, and discuss its applicability to biological sequence evolution.
A principled mechanism for identifying conditional dependencies in time-series data is provided through structure learning of dynamic Bayesian networks (DBNs). An important assumption of DBN structure learning is that the data are generated by a stationary processâan assumption that is not true in many important settings. In this paper, we introduce a new class of graphical models called non-stationary dynamic Bayesian networks, in which the conditional dependence structure of the underlying data-generation process is permitted to change over time. Non-stationary dynamic Bayesian networks represent a new framework for studying problems in which the structure of a network is evolving over time. We define the non-stationary DBN model, present an MCMC sampling algorithm for learning the structure of the model from time-series data under different assumptions, and demonstrate the effectiveness of the algorithm on both simulated and biological data.