We consider Markov decision processes where the values of the parameters are uncertain. This uncertainty is described by a sequence of nested sets (that is, each set contains the previous one), each of which corresponds to a probabilistic guarantee for a different confidence level so that a set of admissible probability distributions of the unknown parameters is specified. This formulation models the case where the decision maker is aware of and wants to exploit some (yet imprecise) a-priori information of the distribution of parameters, and arises naturally in practice where methods to estimate the confidence region of parameters abound. We propose a decision criterion based on distributional robustness: the optimal policy maximizes the expected total reward under the most adversarial probability distribution over realizations of the uncertain parameters that is admissible (i.e., it agrees with the a-priori information). We show that finding the optimal distributionally robust policy can be reduced to a standard robust MDP where the parameters belong to a single uncertainty set, hence it can be computed in polynomial time under mild technical conditions.

The performance of Markov chain Monte Carlo methods is often sensitive to the scaling and correlations between the random variables of interest. An important source of information about the local correlation and scale is given by the Hessian matrix of the target distribution, but this is often either computationally expensive or infeasible. In this paper we propose MCMC samplers that make use of quasi-Newton approximations from the optimization literature, that approximate the Hessian of the target distribution from previous samples and gradients generated by the sampler. A key issue is that MCMC samplers that depend on the history of previous states are in general not valid. We address this problem by using limited memory quasi-Newton methods, which depend only on a fixed window of previous samples.

Factored Decentralized Partially Observable Markov Decision Processes (Dec-POMDPs) form a powerful framework for multiagent planning under uncertainty, but optimal solutions require a rigid history-based policy representation. In this paper we allow inter-agent communication which turns the problem in a centralized Multiagent POMDP (MPOMDP). We map belief distributions over state factors to an agent's local actions by exploiting structure in the joint MPOMDP policy. The key point is that when sparse dependencies between the agents' decisions exist, often the belief over its local state factors is sufficient for an agent to unequivocally identify the optimal action, and communication can be avoided. We formalize these notions by casting the problem into convex optimization form, and present experimental results illustrating the savings in communication that we can obtain.

Focusing on short term trend prediction in a financial context, we consider the problem of selective prediction whereby the predictor can abstain from prediction in order to improve performance. We examine two types of selective mechanisms for HMM predictors. The first is a rejection in the spirit of Chow's well-known ambiguity principle. The second is a specialized mechanism for HMMs that identifies low quality HMM states and abstain from prediction in those states. We call this model selective HMM (sHMM).

We present a type of Temporal Restricted Boltzmann Machine that defines a probability distribution over an output sequence conditional on an input sequence. It shares the desirable properties of RBMs: efficient exact inference, an exponentially more expressive latent state than HMMs, and the ability to model nonlinear structure and dynamics. We apply our model to a challenging real-world graphics problem: facial expression transfer. Our results demonstrate improved performance over several baselines modeling high-dimensional 2D and 3D data.

We present an asymptotic analysis of Viterbi Training (VT) and contrast it with a more conventional Maximum Likelihood (ML) approach to parameter estimation in Hidden Markov Models. While ML estimator works by (locally) maximizing the likelihood of the observed data, VT seeks to maximize the probability of the most likely hidden state sequence. We develop an analytical framework based on a generating function formalism and illustrate it on an exactly solvable model of HMM with one unambiguous symbol. For this particular model the ML objective function is continuously degenerate. VT objective, in contrast, is shown to have only finite degeneracy.

We present explicit classes of probability distributions that can be learned by Restricted Boltzmann Machines (RBMs) depending on the number of units that they contain, and which are representative for the expressive power of the model. We use this to show that the maximal Kullback-Leibler divergence to the RBM model with n visible and m hidden units is bounded from above by (n-1)-log(m 1). In this way we can specify the number of hidden units that guarantees a sufficiently rich model containing different classes of distributions and respecting a given error tolerance.

We propose a novel Adaptive Markov Chain Monte Carlo algorithm to compute the partition function. In particular, we show how to accelerate a flat histogram sampling technique by significantly reducing the number of null moves'' in the chain, while maintaining asymptotic convergence properties. Our experiments show that our method converges quickly to highly accurate solutions on a range of benchmark instances, outperforming other state-of-the-art methods such as IJGP, TRW, and Gibbs sampling both in run-time and accuracy. We also show how obtaining a so-called density of states distribution allows for efficient weight learning in Markov Logic theories.

We propose a novel class of Bayesian nonparametric models for sequential data called fragmentation-coagulation processes (FCPs). An FCP is exchangeable, projective, stationary and reversible, and its equilibrium distributions are given by the Chinese restaurant process. As opposed to hidden Markov models, FCPs allow for flexible modelling of the number of clusters, and they avoid label switching non-identifiability problems. We develop an efficient Gibbs sampler for FCPs which uses uniformization and the forward-backward algorithm. Our development of FCPs is motivated by applications in population genetics, and we demonstrate the utility of FCPs on problems of genotype imputation with phased and unphased SNP data.

This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linear-Gaussian models, hidden Markov models, Gaussian mixture models, and Markov evolutionary trees. The setting is one where we only have samples from certain observed variables in the tree, and our goal is to estimate the tree structure (i.e., the graph of how the underlying hidden variables are connected to each other and to the observed variables). We propose the Spectral Recursive Grouping algorithm, an efficient and simple bottom-up procedure for recovering the tree structure from independent samples of the observed variables. Our finite sample size bounds for exact recovery of the tree structure reveal certain natural dependencies on underlying statistical and structural properties of the underlying joint distribution. Furthermore, our sample complexity guarantees have no explicit dependence on the dimensionality of the observed variables, making the algorithm applicable to many high-dimensional settings.