We tackle the fundamental problem of Bayesian active learning with noise, where we need to adaptively select from a number of expensive tests in order to identify an unknown hypothesis sampled from a known prior distribution. In the case of noise-free observations, a greedy algorithm called generalized binary search (GBS) is known to perform near-optimally. We show that if the observations are noisy, perhaps surprisingly, GBS can perform very poorly. We develop EC2, a novel, greedy active learning algorithm and prove that it is competitive with the optimal policy, thus obtaining the first competitiveness guarantees for Bayesian active learning with noisy observations. Our bounds rely on a recently discovered diminishing returns property called adaptive submodularity, generalizing the classical notion of submodular set functions to adaptive policies.

Optimal coding provides a guiding principle for understanding the representation of sensory variables in neural populations. Here we consider the influence of a prior probability distribution over sensory variables on the optimal allocation of cells and spikes in a neural population. We model the spikes of each cell as samples from an independent Poisson process with rate governed by an associated tuning curve. For this response model, we approximate the Fisher information in terms of the density and amplitude of the tuning curves, under the assumption that tuning width varies inversely with cell density. We consider a family of objective functions based on the expected value, over the sensory prior, of a functional of the Fisher information.

Figure/ground assignment, in which the visual image is divided into nearer (figural) and farther (ground) surfaces, is an essential step in visual processing, but its underlying computational mechanisms are poorly understood. Figural assignment (often referred to as border ownership) can vary along a contour, suggesting a spatially distributed process whereby local and global cues are combined to yield local estimates of border ownership. In this paper we model figure/ground estimation in a Bayesian belief network, attempting to capture the propagation of border ownership across the image as local cues (contour curvature and T-junctions) interact with more global cues to yield a figure/ground assignment. Our network includes as a nonlocal factor skeletal (medial axis) structure, under the hypothesis that medial structure draws'' border ownership so that borders are owned by their interiors. We also briefly present a psychophysical experiment in which we measured local border ownership along a contour at various distances from an inducing cue (a T-junction).

Gaussian graphical models with sparsity in the inverse covariance matrix are of significant interest in many modern applications. For the problem of recovering the graphical structure, information criteria provide useful optimization objectives for algorithms searching through sets of graphs or for selection of tuning parameters of other methods such as the graphical lasso, which is a likelihood penalization technique. In this paper we establish the asymptotic consistency of an extended Bayesian information criterion for Gaussian graphical models in a scenario where both the number of variables p and the sample size n grow. Compared to earlier work on the regression case, our treatment allows for growth in the number of non-zero parameters in the true model, which is necessary in order to cover connected graphs. We demonstrate the performance of this criterion on simulated data when used in conjuction with the graphical lasso, and verify that the criterion indeed performs better than either cross-validation or the ordinary Bayesian information criterion when p and the number of non-zero parameters q both scale with n.

We present the Copula Bayesian Network model for representing multivariate continuous distributions. Our approach builds on a novel copula-based parameterization of a conditional density that, joined with a graph that encodes independencies, offers great flexibility in modeling high-dimensional densities, while maintaining control over the form of the univariate marginals. We demonstrate the advantage of our framework for generalization over standard Bayesian networks as well as tree structured copula models for varied real-life domains that are of substantially higher dimension than those typically considered in the copula literature. Papers published at the Neural Information Processing Systems Conference.

To cope with concept drift, we placed a probability distribution over the location of the most-recent drift point. We used Bayesian model comparison to update this distribution from the predictions of models trained on blocks of consecutive observations and pruned potential drift points with low probability. We compare our approach to a non-probabilistic method for drift and a probabilistic method for change-point detection. In our experiments, our approach generally yielded improved accuracy and/or speed over these other methods. Papers published at the Neural Information Processing Systems Conference.

In this paper, we consider Bayesian reinforcement learning (BRL) where actions incur costs in addition to rewards, and thus exploration has to be constrained in terms of the expected total cost while learning to maximize the expected long-term total reward. In order to formalize cost-sensitive exploration, we use the constrained Markov decision process (CMDP) as the model of the environment, in which we can naturally encode exploration requirements using the cost function. We extend BEETLE, a model-based BRL method, for learning in the environment with cost constraints. We demonstrate the cost-sensitive exploration behaviour in a number of simulated problems. Papers published at the Neural Information Processing Systems Conference.

In many hierarchical inverse problems, not only do we want to estimate high- or infinite-dimensional model parameters in the parameter-to-observable maps, but we also have to estimate hyperparameters that represent critical assumptions in the statistical and mathematical modeling processes. As a joint effect of high-dimensionality, nonlinear dependence, and non-concave structures in the joint posterior posterior distribution over model parameters and hyperparameters, solving inverse problems in the hierarchical Bayesian setting poses a significant computational challenge. In this work, we aim to develop scalable optimization-based Markov chain Monte Carlo (MCMC) methods for solving hierarchical Bayesian inverse problems with nonlinear parameter-to-observable maps and a broader class of hyperparameters. Our algorithmic development is based on the recently developed scalable randomize-then-optimize (RTO) method [4] for exploring the high- or infinite-dimensional model parameter space. By using RTO either as a proposal distribution in a Metropolis-within-Gibbs update or as a biasing distribution in the pseudo-marginal MCMC [2], we are able to design efficient sampling tools for hierarchical Bayesian inversion. In particular, the integration of RTO and the pseudo-marginal MCMC has sampling performance robust to model parameter dimensions. We also extend our methods to nonlinear inverse problems with Poisson-distributed measurements. Numerical examples in PDE-constrained inverse problems and positron emission tomography (PET) are used to demonstrate the performance of our methods.

Comparing the underlying distributions of two given datasets has been an important task in machine learning community and has a wide range of applications. For example, change detection algorithms Kawahara and Sugiyama ((2012)) compare datasets collected at different time points and report how the underlying distribution has shifted over time; Transfer learning algorithms Quionero-Candela et al. ((2009)) utilize the estimated differences between two datasets to efficiently share information between different tasks. Generative Adversarial Net (GAN) Goodfellow et al. ((2014)) learns an implicit generative model whose output minimizes the differences between an artificial dataset and a real dataset. Various computational methods have been proposed for comparing underlying distributions given two sets of observations. For example, Maximum Mean Discrepancy (MMD) Gretton et al. ((2012)) computes the distance between the kernel mean embeddings of two datasets in Reproducing Kernel Hilbert Space (RKHS).

A challenging problem in hierarchical classification is to leverage the hierarchical relations among classes for improving classification performance. An even greater challenge is to do so in a manner that is computationally feasible for the large scale problems usually encountered in practice. This paper proposes a set of Bayesian methods to model hierarchical dependencies among class labels using multivari- ate logistic regression. Specifically, the parent-child relationships are modeled by placing a hierarchical prior over the children nodes centered around the parame- ters of their parents; thereby encouraging classes nearby in the hierarchy to share similar model parameters. We present new, efficient variational algorithms for tractable posterior inference in these models, and provide a parallel implementa- tion that can comfortably handle large-scale problems with hundreds of thousands of dimensions and tens of thousands of classes.