We describe an adaptation of the simulated annealing algorithm to nonparametric clustering and related probabilistic models. This new algorithm learns nonparametric latent structure over a growing and constantly churning subsample of training data, where the portion of data subsampled can be interpreted as the inverse temperature beta(t) in an annealing schedule. Gibbs sampling at high temperature (i.e., with a very small subsample) can more quickly explore sketches of the final latent state by (a) making longer jumps around latent space (as in block Gibbs) and (b) lowering energy barriers (as in simulated annealing). We prove subsample annealing speeds up mixing time N^2 -> N in a simple clustering model and exp(N) -> N in another class of models, where N is data size. Empirically subsample-annealing outperforms naive Gibbs sampling in accuracy-per-wallclock time, and can scale to larger datasets and deeper hierarchical models. We demonstrate improved inference on million-row subsamples of US Census data and network log data and a 307-row hospital rating dataset, using a Pitman-Yor generalization of the Cross Categorization model.

Hierarchical statistical models are widely employed in information science and data engineering. The models consist of two types of variables: observable variables that represent the given data and latent variables for the unobservable labels. An asymptotic analysis of the models plays an important role in evaluating the learning process; the result of the analysis is applied not only to theoretical but also to practical situations, such as optimal model selection and active learning. There are many studies of generalization errors, which measure the prediction accuracy of the observable variables. However, the accuracy of estimating the latent variables has not yet been elucidated. For a quantitative evaluation of this, the present paper formulates distribution-based functions for the errors in the estimation of the latent variables. The asymptotic behavior is analyzed for both the maximum likelihood and the Bayes methods.

Determinantal point processes (DPPs) are well-suited for modeling repulsion and have proven useful in many applications where diversity is desired. While DPPs have many appealing properties, such as efficient sampling, learning the parameters of a DPP is still considered a difficult problem due to the non-convex nature of the likelihood function. In this paper, we propose using Bayesian methods to learn the DPP kernel parameters. These methods are applicable in large-scale and continuous DPP settings even when the exact form of the eigendecomposition is unknown. We demonstrate the utility of our DPP learning methods in studying the progression of diabetic neuropathy based on spatial distribution of nerve fibers, and in studying human perception of diversity in images.

CMF is a technique for simultaneously learning low-rank representations based on a collection of matrices with shared entities. A typical example is the joint modeling of user-item, item-property, and user-feature matrices in a recommender system. The key idea in CMF is that the embeddings are shared across the matrices, which enables transferring information between them. The existing solutions, however, break down when the individual matrices have low-rank structure not shared with others. In this work we present a novel CMF solution that allows each of the matrices to have a separate low-rank structure that is independent of the other matrices, as well as structures that are shared only by a subset of them. We compare MAP and variational Bayesian solutions based on alternating optimization algorithms and show that the model automatically infers the nature of each factor using group-wise sparsity. Our approach supports in a principled way continuous, binary and count observations and is efficient for sparse matrices involving missing data. We illustrate the solution on a number of examples, focusing in particular on an interesting use-case of augmented multi-view learning.

In this paper we consider a problem of searching a space of predictive models for a given training data set. We propose an iterative procedure for deriving a sequence of improving models and a corresponding sequence of sets of non-linear features on the original input space. After a finite number of iterations N, the non-linear features become 2^N -degree polynomials on the original space. We show that in a limit of an infinite number of iterations derived non-linear features must form an associative algebra: a product of two features is equal to a linear combination of features from the same feature space for any given input point. Because each iteration consists of solving a series of convex problems that contain all previous solutions, the likelihood of the models in the sequence is increasing with each iteration while the dimension of the model parameter space is set to a limited controlled value.

Partition functions arise in a variety of settings, including conditional random fields, logistic regression, and latent gaussian models. In this paper, we consider semistochastic quadratic bound (SQB) methods for maximum likelihood estimation based on partition function optimization. Batch methods based on the quadratic bound were recently proposed for this class of problems, and performed favorably in comparison to state-of-the-art techniques. Semistochastic methods fall in between batch algorithms, which use all the data, and stochastic gradient type methods, which use small random selections at each iteration. We build semistochastic quadratic bound-based methods, and prove both global convergence (to a stationary point) under very weak assumptions, and linear convergence rate under stronger assumptions on the objective. To make the proposed methods faster and more stable, we consider inexact subproblem minimization and batch-size selection schemes. The efficacy of SQB methods is demonstrated via comparison with several state-of-the-art techniques on commonly used datasets.

We present a formal model of human decision-making in explore-exploit tasks using the context of multi-armed bandit problems, where the decision-maker must choose among multiple options with uncertain rewards. We address the standard multi-armed bandit problem, the multi-armed bandit problem with transition costs, and the multi-armed bandit problem on graphs. We focus on the case of Gaussian rewards in a setting where the decision-maker uses Bayesian inference to estimate the reward values. We model the decision-maker's prior knowledge with the Bayesian prior on the mean reward. We develop the upper credible limit (UCL) algorithm for the standard multi-armed bandit problem and show that this deterministic algorithm achieves logarithmic cumulative expected regret, which is optimal performance for uninformative priors. We show how good priors and good assumptions on the correlation structure among arms can greatly enhance decision-making performance, even over short time horizons. We extend to the stochastic UCL algorithm and draw several connections to human decision-making behavior. We present empirical data from human experiments and show that human performance is efficiently captured by the stochastic UCL algorithm with appropriate parameters. For the multi-armed bandit problem with transition costs and the multi-armed bandit problem on graphs, we generalize the UCL algorithm to the block UCL algorithm and the graphical block UCL algorithm, respectively. We show that these algorithms also achieve logarithmic cumulative expected regret and require a sub-logarithmic expected number of transitions among arms. We further illustrate the performance of these algorithms with numerical examples.

ABSTRACT Score calibration enables automatic speaker recognizers to make cost-effective accept / reject decisions. Traditional calibration requires supervised data, which is an expensive resource. We propose a 2-component GMM for unsupervised calibration and demonstrate good performance relative to a supervised baseline on NIST SRE'10 and SRE'12. A Bayesian analysis demonstrates that the uncertainty associated with the unsupervised calibration parameter estimates is surprisingly small. Index Terms-- calibration, unsupervised learning, Laplace approximation, automatic speaker recognition 1. INTRODUCTION Automatic speaker recognizers map trials to scores.

The law of total probability may be deployed in binary classification exercises to estimate the unconditional class probabilities if the class proportions in the training set are not representative of the population class proportions. We argue that this is not a conceptually sound approach and suggest an alternative based on the new law of total odds. We quantify the bias of the total probability estimator of the unconditional class probabilities and show that the total odds estimator is unbiased. The sample version of the total odds estimator is shown to coincide with a maximum-likelihood estimator known from the literature. The law of total odds can also be used for transforming the conditional class probabilities if independent estimates of the unconditional class probabilities of the population are available. Keywords: Total probability, likelihood ratio, Bayes' formula, binary classification, relative odds, unbiased estimator, supervised learning, dataset shift.

Markov chain Monte Carlo (MCMC) sampling has been the main workhorse of Bayesian computation since the 1990s. A canonical MCMC algorithm proposes samples from a distribution q and then accepts or rejects these proposals with a certain probability given by the Metropolis-Hastings (MH) formula [Metropolis et al., 1953, Hastings, 1970]. For each proposed sample, the MH rule needs to examine the likelihood of all dataitems. When the number of data-cases is large this is an awful lot of computation for one bit of information, namely whether to accept or reject a proposal. In today's Big Data world, we need to rethink our Bayesian inference algorithms. Standard MCMC methods do not meet the Big Data challenge for the reason described above. Researchers have made some progress in terms of making MCMC more efficient, mostly by focusing on parallelization. Very few question the algorithm itself: is the standard MCMC paradigm really optimally efficient in achieving its goals?