For many classic structured prediction problems, probability distributions over the dependent variables can be efficiently computed using widely-known algorithms and data structures (such as forward-backward, and its corresponding trellis for exact probability distributions in Markov models). However, we know of no previous work studying efficient representations of exact distributions over clusterings. This paper presents definitions and proofs for a dynamic-programming inference procedure that computes the partition function, the marginal probability of a cluster, and the MAP clustering---all exactly. Rather than the Nth Bell number, these exact solutions take time and space proportional to the substantially smaller powerset of N. Indeed, we improve upon the time complexity of the algorithm introduced by Kohonen and Corander (2016) for this problem by a factor of N. While still large, this previously unknown result is intellectually interesting in its own right, makes feasible exact inference for important real-world small data applications (such as medicine), and provides a natural stepping stone towards sparse-trellis approximations that enable further scalability (which we also explore). In experiments, we demonstrate the superiority of our approach over approximate methods in analyzing real-world gene expression data used in cancer treatment.

We develop deep Poisson-gamma dynamical systems (DPGDS) to model sequentially observed multivariate count data, improving previously proposed models by not only mining deep hierarchical latent structure from the data, but also capturing both first-order and long-range temporal dependencies. Using sophisticated but simple-to-implement data augmentation techniques, we derived closed-form Gibbs sampling update equations by first backward and upward propagating auxiliary latent counts, and then forward and downward sampling latent variables. Moreover, we develop stochastic gradient MCMC inference that is scalable to very long multivariate count time series. Experiments on both synthetic and a variety of real-world data demonstrate that the proposed model not only has excellent predictive performance, but also provides highly interpretable multilayer latent structure to represent hierarchical and temporal information propagation.

The Probably Approximately Correct (PAC) Bayes framework (McAllester, 1999) can incorporate knowledge about the learning algorithm and (data) distribution through the use of distribution-dependent priors, yielding tighter generalization bounds on data-dependent posteriors. Using this flexibility, however, is difficult, especially when the data distribution is presumed to be unknown. We show how a differentially private data-dependent prior yields a valid PAC-Bayes bound, and then show how non-private mechanisms for choosing priors can also yield generalization bounds. As an application of this result, we show that a Gaussian prior mean chosen via stochastic gradient Langevin dynamics (SGLD; Welling and Teh, 2011) leads to a valid PAC-Bayes bound due to control of the 2-Wasserstein distance to a differentially private stationary distribution. We study our data-dependent bounds empirically, and show that they can be nonvacuous even when other distribution-dependent bounds are vacuous.

In many structured prediction problems, complex relationships between variables are compactly defined using graphical structures. The most prevalent graphical prediction methods---probabilistic graphical models and large margin methods---have their own distinct strengths but also possess significant drawbacks. Conditional random fields (CRFs) are Fisher consistent, but they do not permit integration of customized loss metrics into their learning process. Large-margin models, such as structured support vector machines (SSVMs), have the flexibility to incorporate customized loss metrics, but lack Fisher consistency guarantees. We present adversarial graphical models (AGM), a distributionally robust approach for constructing a predictor that performs robustly for a class of data distributions defined using a graphical structure. Our approach enjoys both the flexibility of incorporating customized loss metrics into its design as well as the statistical guarantee of Fisher consistency. We present exact learning and prediction algorithms for AGM with time complexity similar to existing graphical models and show the practical benefits of our approach with experiments.

Many probabilistic models of interest in scientific computing and machine learning have expensive, black-box likelihoods that prevent the application of standard techniques for Bayesian inference, such as MCMC, which would require access to the gradient or a large number of likelihood evaluations. We introduce here a novel sample-efficient inference framework, Variational Bayesian Monte Carlo (VBMC). VBMC combines variational inference with Gaussian-process based, active-sampling Bayesian quadrature, using the latter to efficiently approximate the intractable integral in the variational objective. Our method produces both a nonparametric approximation of the posterior distribution and an approximate lower bound of the model evidence, useful for model selection. We demonstrate VBMC both on several synthetic likelihoods and on a neuronal model with data from real neurons. Across all tested problems and dimensions (up to D = 10), VBMC performs consistently well in reconstructing the posterior and the model evidence with a limited budget of likelihood evaluations, unlike other methods that work only in very low dimensions. Our framework shows great promise as a novel tool for posterior and model inference with expensive, black-box likelihoods.

Approximate inference in probabilistic graphical models (PGMs) can be grouped into deterministic methods and Monte-Carlo-based methods. The former can often provide accurate and rapid inferences, but are typically associated with biases that are hard to quantify. The latter enjoy asymptotic consistency, but can suffer from high computational costs. In this paper we present a way of bridging the gap between deterministic and stochastic inference. Specifically, we suggest an efficient sequential Monte Carlo (SMC) algorithm for PGMs which can leverage the output from deterministic inference methods. While generally applicable, we show explicitly how this can be done with loopy belief propagation, expectation propagation, and Laplace approximations. The resulting algorithm can be viewed as a post-correction of the biases associated with these methods and, indeed, numerical results show clear improvements over the baseline deterministic methods as well as over "plain" SMC.

Event datasets include events that occur irregularly over the timeline and are prevalent in numerous domains. We introduce proximal graphical event models (PGEM) as a representation of such datasets. PGEMs belong to a broader family of models that characterize relationships between various types of events, where the rate of occurrence of an event type depends only on whether or not its parents have occurred in the most recent history. The main advantage over the state of the art models is that they are entirely data driven and do not require additional inputs from the user, which can require knowledge of the domain such as choice of basis functions or hyperparameters in graphical event models. We theoretically justify our learning of optimal windows for parental history and the choices of parental sets, and the algorithm are sound and complete in terms of parent structure learning. We present additional efficient heuristics for learning PGEMs from data, demonstrating their effectiveness on synthetic and real datasets.

We introduce collapsed compilation, a novel approximate inference algorithm for discrete probabilistic graphical models. It is a collapsed sampling algorithm that incrementally selects which variable to sample next based on the partial compila- tion obtained so far. This online collapsing, together with knowledge compilation inference on the remaining variables, naturally exploits local structure and context- specific independence in the distribution. These properties are used implicitly in exact inference, but are difficult to harness for approximate inference. More- over, by having a partially compiled circuit available during sampling, collapsed compilation has access to a highly effective proposal distribution for importance sampling. Our experimental evaluation shows that collapsed compilation performs well on standard benchmarks. In particular, when the amount of exact inference is equally limited, collapsed compilation is competitive with the state of the art, and outperforms it on several benchmarks.

Expectation-Maximization (EM) is a popular tool for learning latent variable models, but the vanilla batch EM does not scale to large data sets because the whole data set is needed at every E-step. Stochastic Expectation Maximization (sEM) reduces the cost of E-step by stochastic approximation. However, sEM has a slower asymptotic convergence rate than batch EM, and requires a decreasing sequence of step sizes, which is difficult to tune. In this paper, we propose a variance reduced stochastic EM (sEM-vr) algorithm inspired by variance reduced stochastic gradient descent algorithms. We show that sEM-vr has the same exponential asymptotic convergence rate as batch EM. Moreover, sEM-vr only requires a constant step size to achieve this rate, which alleviates the burden of parameter tuning. We compare sEM-vr with batch EM, sEM and other algorithms on Gaussian mixture models and probabilistic latent semantic analysis, and sEM-vr converges significantly faster than these baselines.

Continuous-time Bayesian networks (CTBNs) constitute a general and powerful framework for modeling continuous-time stochastic processes on networks. This makes them particularly attractive for learning the directed structures among interacting entities. However, if the available data is incomplete, one needs to simulate the prohibitively complex CTBN dynamics. Existing approximation techniques, such as sampling and low-order variational methods, either scale unfavorably in system size, or are unsatisfactory in terms of accuracy. Inspired by recent advances in statistical physics, we present a new approximation scheme based on cluster-variational methods that significantly improves upon existing variational approximations. We can analytically marginalize the parameters of the approximate CTBN, as these are of secondary importance for structure learning. This recovers a scalable scheme for direct structure learning from incomplete and noisy time-series data. Our approach outperforms existing methods in terms of scalability.