Expectation Maximization and Complex Duration Distributions for Continuous Time Bayesian Networks

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 the parameters and structure of a CTBN from partially observed data. We show how to apply expectation maximization (EM) and structural expectation maximization (SEM) to CTBNs. The availability of the EM algorithm allows us to extend the representation of CTBNs to allow a much richer class of transition durations distributions, known as phase distributions. This class is a highly expressive semi-parametric representation, which can approximate any duration distribution arbitrarily closely. This extension to the CTBN framework addresses one of the main limitations of both CTBNs and DBNs - the restriction to exponentially / geometrically distributed duration. We present experimental results on a real data set of people's life spans, showing that our algorithm learns reasonable models - structure and parameters - from partially observed data, and, with the use of phase distributions, achieves better performance than DBNs.Comment: Appears in Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (UAI2005

Inference from Randomized Transmissions by Many Backscatter Sensors

Attaining the vision of Smart Cities requires the deployment of an enormous number of sensors for monitoring various conditions of the environment. Backscatter-sensors have emerged to be a promising solution due to the uninterruptible energy supply and relative simple hardwares. On the other hand, backscatter-sensors with limited signal-processing capabilities are unable to support conventional algorithms for multiple-access and channel-training. Thus, the key challenge in designing backscatter-sensor networks is to enable readers to accurately detect sensing-values given simple ALOHA random access, primitive transmission schemes, and no knowledge of channel-states. We tackle this challenge by proposing the novel framework of backscatter sensing featuring random-encoding at sensors and statistical-inference at readers. Specifically, assuming the on/off keying for backscatter transmissions, the practical random-encoding scheme causes the on/off transmission of a sensor to follow a distribution parameterized by the sensing values. Facilitated by the scheme, statistical-inference algorithms are designed to enable a reader to infer sensing-values from randomized transmissions by multiple sensors. The specific design procedure involves the construction of Bayesian networks, namely deriving conditional distributions for relating unknown parameters and variables to signals observed by the reader. Then based on the Bayesian networks and the well-known expectation-maximization principle, inference algorithms are derived to recover sensing-values.Comment: An extended version of a shorter conference submissio

Expectation Propagation for Continuous Time Bayesian Networks

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. As shown previously, exact inference in CTBNs is intractable. We address the problem of approximate inference, allowing for general queries conditioned on evidence over continuous time intervals and at discrete time points. We show how CTBNs can be parameterized within the exponential family, and use that insight to develop a message passing scheme in cluster graphs and allows us to apply expectation propagation to CTBNs. The clusters in our cluster graph do not contain distributions over the cluster variables at individual time points, but distributions over trajectories of the variables throughout a duration. Thus, unlike discrete time temporal models such as dynamic Bayesian networks, we can adapt the time granularity at which we reason for different variables and in different conditions.Comment: Appears in Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence (UAI2005

Learning Continuous-Time Social Network Dynamics

We demonstrate that a number of sociology models for social network dynamics can be viewed as continuous time Bayesian networks (CTBNs). A sampling-based approximate inference method for CTBNs can be used as the basis of an expectation-maximization procedure that achieves better accuracy in estimating the parameters of the model than the standard method of moments algorithmfromthe sociology literature. We extend the existing social network models to allow for indirect and asynchronous observations of the links. A Markov chain Monte Carlo sampling algorithm for this new model permits estimation and inference. We provide results on both a synthetic network (for verification) and real social network data.Comment: Appears in Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence (UAI2009

Factored Filtering of Continuous-Time Systems

We consider filtering for a continuous-time, or asynchronous, stochastic system where the full distribution over states is too large to be stored or calculated. We assume that the rate matrix of the system can be compactly represented and that the belief distribution is to be approximated as a product of marginals. The essential computation is the matrix exponential. We look at two different methods for its computation: ODE integration and uniformization of the Taylor expansion. For both we consider approximations in which only a factored belief state is maintained. For factored uniformization we demonstrate that the KL-divergence of the filtering is bounded. Our experimental results confirm our factored uniformization performs better than previously suggested uniformization methods and the mean field algorithm

Deep Online Learning via Meta-Learning: Continual Adaptation for Model-Based RL

Humans and animals can learn complex predictive models that allow them to accurately and reliably reason about real-world phenomena, and they can adapt such models extremely quickly in the face of unexpected changes. Deep neural network models allow us to represent very complex functions, but lack this capacity for rapid online adaptation. The goal in this paper is to develop a method for continual online learning from an incoming stream of data, using deep neural network models. We formulate an online learning procedure that uses stochastic gradient descent to update model parameters, and an expectation maximization algorithm with a Chinese restaurant process prior to develop and maintain a mixture of models to handle non-stationary task distributions. This allows for all models to be adapted as necessary, with new models instantiated for task changes and old models recalled when previously seen tasks are encountered again. Furthermore, we observe that meta-learning can be used to meta-train a model such that this direct online adaptation with SGD is effective, which is otherwise not the case for large function approximators. In this work, we apply our meta-learning for online learning (MOLe) approach to model-based reinforcement learning, where adapting the predictive model is critical for control; we demonstrate that MOLe outperforms alternative prior methods, and enables effective continuous adaptation in non-stationary task distributions such as varying terrains, motor failures, and unexpected disturbances.Comment: Project website: https://sites.google.com/berkeley.edu/onlineviamet

Continuous Time Markov Networks

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.Comment: Appears in Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence (UAI2006

Variational Bayesian Inference for Audio-Visual Tracking of Multiple Speakers

In this paper we address the problem of tracking multiple speakers via the fusion of visual and auditory information. We propose to exploit the complementary nature of these two modalities in order to accurately estimate smooth trajectories of the tracked persons, to deal with the partial or total absence of one of the modalities over short periods of time, and to estimate the acoustic status -- either speaking or silent -- of each tracked person along time. We propose to cast the problem at hand into a generative audio-visual fusion (or association) model formulated as a latent-variable temporal graphical model. This may well be viewed as the problem of maximizing the posterior joint distribution of a set of continuous and discrete latent variables given the past and current observations, which is intractable. We propose a variational inference model which amounts to approximate the joint distribution with a factorized distribution. The solution takes the form of a closed-form expectation maximization procedure. We describe in detail the inference algorithm, we evaluate its performance and we compare it with several baseline methods. These experiments show that the proposed audio-visual tracker performs well in informal meetings involving a time-varying number of people

Large Scale Estimation in Cyberphysical Systems using Streaming Data: a Case Study with Smartphone Traces

Controlling and analyzing cyberphysical and robotics systems is increasingly becoming a Big Data challenge. Pushing this data to, and processing in the cloud is more efficient than on-board processing. However, current cloud-based solutions are not suitable for the latency requirements of these applications. We present a new concept, Discretized Streams or D-Streams, that enables massively scalable computations on streaming data with latencies as short as a second. We experiment with an implementation of D-Streams on top of the Spark computing framework. We demonstrate the usefulness of this concept with a novel algorithm to estimate vehicular traffic in urban networks. Our online EM algorithm can estimate traffic on a very large city network (the San Francisco Bay Area) by processing tens of thousands of observations per second, with a latency of a few seconds

Compositional Stochastic Modeling and Probabilistic Programming

Probabilistic programming is related to a compositional approach to stochastic modeling by switching from discrete to continuous time dynamics. In continuous time, an operator-algebra semantics is available in which processes proceeding in parallel (and possibly interacting) have summed time-evolution operators. From this foundation, algorithms for simulation, inference and model reduction may be systematically derived. The useful consequences are potentially far-reaching in computational science, machine learning and beyond. Hybrid compositional stochastic modeling/probabilistic programming approaches may also be possible.Comment: Extended Abstract for the Neural Information Processing Systems (NIPS) Workshop on Probabilistic Programming, 201