We consider the estimation of sparse graphical models that characterize the dependency structure of high-dimensional tensor-valued data. To facilitate the estimation of the precision matrix corresponding to each way of the tensor, we assume the data follow a tensor normal distribution whose covariance has a Kronecker product structure. The penalized maximum likelihood estimation of this model involves minimizing a non-convex objective function. In spite of the non-convexity of this estimation problem, we prove that an alternating minimization algorithm, which iteratively estimates each sparse precision matrix while fixing the others, attains an estimator with the optimal statistical rate of convergence as well as consistent graph recovery. Notably, such an estimator achieves estimation consistency with only one tensor sample, which is unobserved in previous work. Our theoretical results are backed by thorough numerical studies.

We consider the problem of binary classification when the covariates conditioned on the each of the response values follow multivariate Gaussian distributions. We focus on the setting where the covariance matrices for the two conditional distributions are the same. The corresponding generative model classifier, derived via the Bayes rule, also called Linear Discriminant Analysis, has been shown to behave poorly in high-dimensional settings. We present a novel analysis of the classification error of any linear discriminant approach given conditional Gaussian models. This allows us to compare the generative model classifier, other recently proposed discriminative approaches that directly learn the discriminant function, and then finally logistic regression which is another classical discriminative model classifier. As we show, under a natural sparsity assumption, and letting $s$ denote the sparsity of the Bayes classifier, $p$ the number of covariates, and $n$ the number of samples, the simple ($\ell_1$-regularized) logistic regression classifier achieves the fast misclassification error rates of $O\left(\frac{s \log p}{n}\right)$, which is much better than the other approaches, which are either inconsistent under high-dimensional settings, or achieve a slower rate of $O\left(\sqrt{\frac{s \log p}{n}}\right)$.

In simple perceptual decisions the brain has to identify a stimulus based on noisy sensory samples from the stimulus. Basic statistical considerations state that the reliability of the stimulus information, i.e., the amount of noise in the samples, should be taken into account when the decision is made. However, for perceptual decision making experiments it has been questioned whether the brain indeed uses the reliability for making decisions when confronted with unpredictable changes in stimulus reliability. We here show that even the basic drift diffusion model, which has frequently been used to explain experimental findings in perceptual decision making, implicitly relies on estimates of stimulus reliability. We then show that only those variants of the drift diffusion model which allow stimulus-specific reliabilities are consistent with neurophysiological findings. Our analysis suggests that the brain estimates the reliability of the stimulus on a short time scale of at most a few hundred milliseconds.

Inference is typically intractable in high-treewidth undirected graphical models, making maximum likelihood learning a challenge. One way to overcome this is to restrict parameters to a tractable set, most typically the set of tree-structured parameters. This paper explores an alternative notion of a tractable set, namely a set of “fast-mixing parameters” where Markov chain Monte Carlo (MCMC) inference can be guaranteed to quickly converge to the stationary distribution. While it is common in practice to approximate the likelihood gradient using samples obtained from MCMC, such procedures lack theoretical guarantees. This paper proves that for any exponential family with bounded sufficient statistics, (not just graphical models) when parameters are constrained to a fast-mixing set, gradient descent with gradients approximated by sampling will approximate the maximum likelihood solution inside the set with high-probability. When unregularized, to find a solution epsilon-accurate in log-likelihood requires a total amount of effort cubic in 1/epsilon, disregarding logarithmic factors. When ridge-regularized, strong convexity allows a solution epsilon-accurate in parameter distance with an effort quadratic in 1/epsilon. Both of these provide of a fully-polynomial time randomized approximation scheme.

We propose an exploratory approach to statistical model criticism using maximum mean discrepancy (MMD) two sample tests. Typical approaches to model criticism require a practitioner to select a statistic by which to measure discrepancies between data and a statistical model. MMD two sample tests are instead constructed as an analytic maximisation over a large space of possible statistics and therefore automatically select the statistic which most shows any discrepancy. We demonstrate on synthetic data that the selected statistic, called the witness function, can be used to identify where a statistical model most misrepresents the data it was trained on. We then apply the procedure to real data where the models being assessed are restricted Boltzmann machines, deep belief networks and Gaussian process regression and demonstrate the ways in which these models fail to capture the properties of the data they are trained on.

In this paper, we propose a novel parameter estimator for probabilistic models on discrete space. The proposed estimator is derived from minimization of homogeneous divergenceand can be constructed without calculation of the normalization constant, which is frequently infeasible for models in the discrete space. We investigate statisticalproperties of the proposed estimator such as consistency and asymptotic normality, and reveal a relationship with the information geometry. Some experiments show that the proposed estimator attains comparable performance tothe maximum likelihood estimator with drastically lower computational cost.

For many complex diseases, there is a wide variety of ways in which an individual can manifest the disease. The challenge of personalized medicine is to develop tools that can accurately predict the trajectory of an individual's disease, which can in turn enable clinicians to optimize treatments. We represent an individual's disease trajectory as a continuous-valued continuous-time function describing the severity of the disease over time. We propose a hierarchical latent variable model that individualizes predictions of disease trajectories. This model shares statistical strength across observations at different resolutions--the population, subpopulation and the individual level. We describe an algorithm for learning population and subpopulation parameters offline, and an online procedure for dynamically learning individual-specific parameters. Finally, we validate our model on the task of predicting the course of interstitial lung disease, a leading cause of death among patients with the autoimmune disease scleroderma. We compare our approach against state-of-the-art and demonstrate significant improvements in predictive accuracy.

Restricted Boltzmann machines are undirected neural networks which have been shown tobe effective in many applications, including serving as initializations fortraining deep multi-layer neural networks. One of the main reasons for their success is theexistence of efficient and practical stochastic algorithms, such as contrastive divergence,for unsupervised training. We propose an alternative deterministic iterative procedure based on an improved mean field method from statistical physics known as the Thouless-Anderson-Palmer approach. We demonstrate that our algorithm provides performance equal to, and sometimes superior to, persistent contrastive divergence, while also providing a clear and easy to evaluate objective function. We believe that this strategycan be easily generalized to other models as well as to more accurate higher-order approximations, paving the way for systematic improvements in training Boltzmann machineswith hidden units.

In this paper, we address the question of identifiability and learning algorithms for large-scale Poisson Directed Acyclic Graphical (DAG) models. We define general Poisson DAG models as models where each node is a Poisson random variable with rate parameter depending on the values of the parents in the underlying DAG. First, we prove that Poisson DAG models are identifiable from observational data, and present a polynomial-time algorithm that learns the Poisson DAG model under suitable regularity conditions. The main idea behind our algorithm is based on overdispersion, in that variables that are conditionally Poisson are overdispersed relative to variables that are marginally Poisson. Our algorithms exploits overdispersion along with methods for learning sparse Poisson undirected graphical models for faster computation. We provide both theoretical guarantees and simulation results for both small and large-scale DAGs.

Both learning and inference tasks on Bayesian networks are NP-hard in general. Bounded tree-width Bayesian networks have recently received a lot of attention as a way to circumvent this complexity issue; however, while inference on bounded tree-width networks is tractable, the learning problem remains NP-hard even for tree-width~2. In this paper, we propose bounded vertex cover number Bayesian networks as an alternative to bounded tree-width networks. In particular, we show that both inference and learning can be done in polynomial time for any fixed vertex cover number bound $k$, in contrast to the general and bounded tree-width cases; on the other hand, we also show that learning problem is W[1]-hard in parameter $k$. Furthermore, we give an alternative way to learn bounded vertex cover number Bayesian networks using integer linear programming (ILP), and show this is feasible in practice.