This paper introduces constrained mixtures for continuous distributions, characterized by a mixture of distributions where each distribution has a shape similar to the base distribution and disjoint domains. This new concept is used to create generalized asymmetric versions of the Laplace and normal distributions, which are shown to define exponential families, with known conjugate priors, and to have maximum likelihood estimates for the original parameters, with known closed-form expressions. The asymmetric and symmetric normal distributions are compared in a linear regression example, showing that the asymmetric version performs at least as well as the symmetric one, and in a real world time-series problem, where a hidden Markov model is used to fit a stock index, indicating that the asymmetric version provides higher likelihood and may learn distribution models over states and transition distributions with considerably less entropy.

Structural damage due to excessive loading or environmental degradation typically occurs in localized areas in the absence of collapse. This prior information about the spatial sparseness of structural damage is exploited here by a hierarchical sparse Bayesian learning framework with the goal of reducing the source of ill-conditioning in the stiffness loss inversion problem for damage detection. Sparse Bayesian learning methodologies automatically prune away irrelevant or inactive features from a set of potential candidates, and so they are effective probabilistic tools for producing sparse explanatory subsets. We have previously proposed such an approach to establish the probability of localized stiffness reductions that serve as a proxy for damage by using noisy incomplete modal data from before and after possible damage. The core idea centers on a specific hierarchical Bayesian model that promotes spatial sparseness in the inferred stiffness reductions in a way that is consistent with the Bayesian Ockham razor. In this paper, we improve the theory of our previously proposed sparse Bayesian learning approach by eliminating an approximation and, more importantly, incorporating a constraint on stiffness increases. Our approach has many appealing features that are summarized at the end of the paper. We validate the approach by applying it to the Phase II simulated and experimental benchmark studies sponsored by the IASC-ASCE Task Group on Structural Health Monitoring. The results show that it can reliably detect, locate and assess damage by inferring substructure stiffness losses from the identified modal parameters. The occurrence of missed and false damage alerts is effectively suppressed.

Estimation of response functions is an important task in dynamic medical imaging. This task arises for example in dynamic renal scintigraphy, where impulse response or retention functions are estimated, or in functional magnetic resonance imaging where hemodynamic response functions are required. These functions can not be observed directly and their estimation is complicated because the recorded images are subject to superposition of underlying signals. Therefore, the response functions are estimated via blind source separation and deconvolution. Performance of this algorithm heavily depends on the used models of the response functions. Response functions in real image sequences are rather complicated and finding a suitable parametric form is problematic. In this paper, we study estimation of the response functions using non-parametric Bayesian priors. These priors were designed to favor desirable properties of the functions, such as sparsity or smoothness. These assumptions are used within hierarchical priors of the blind source separation and deconvolution algorithm. Comparison of the resulting algorithms with these priors is performed on synthetic dataset as well as on real datasets from dynamic renal scintigraphy. It is shown that flexible non-parametric priors improve estimation of response functions in both cases. MATLAB implementation of the resulting algorithms is freely available for download.

Recently, there has been focus on penalized log-likelihood covariance estimation for sparse inverse covariance (precision) matrices. The penalty is responsible for inducing sparsity, and a very common choice is the convex $l_1$ norm. However, the best estimator performance is not always achieved with this penalty. The most natural sparsity promoting "norm" is the non-convex $l_0$ penalty but its lack of convexity has deterred its use in sparse maximum likelihood estimation. In this paper we consider non-convex $l_0$ penalized log-likelihood inverse covariance estimation and present a novel cyclic descent algorithm for its optimization. Convergence to a local minimizer is proved, which is highly non-trivial, and we demonstrate via simulations the reduced bias and superior quality of the $l_0$ penalty as compared to the $l_1$ penalty.

We propose a sequential learning policy for noisy discrete global optimization and ranking and selection (R\&S) problems with high dimensional sparse belief functions, where there are hundreds or even thousands of features, but only a small portion of these features contain explanatory power. We aim to identify the sparsity pattern and select the best alternative before the finite budget is exhausted. We derive a knowledge gradient policy for sparse linear models (KGSpLin) with group Lasso penalty. This policy is a unique and novel hybrid of Bayesian R\&S with frequentist learning. Particularly, our method naturally combines B-spline basis expansion and generalizes to the nonparametric additive model (KGSpAM) and functional ANOVA model. Theoretically, we provide the estimation error bounds of the posterior mean estimate and the functional estimate. Controlled experiments show that the algorithm efficiently learns the correct set of nonzero parameters even when the model is imbedded with hundreds of dummy parameters. Also it outperforms the knowledge gradient for a linear model.

In the recent years, the desire and need to understand sequential data has been increasing, with particular interest in sequential contexts such as patient monitoring, understanding daily activities, video surveillance, stock market and the like. Along with the constant flow of data, it is critical to classify and segment the observations on-the-fly, without being limited to a rigid number of classes. In addition, the model needs to be capable of updating its parameters to comply with possible evolutions. This interesting problem, however, is not adequately addressed in the literature since many studies focus on offline classification over a pre-defined class set. In this paper, we propose a principled solution to this gap by introducing an adaptive online system based on Markov switching models with hierarchical Dirichlet process priors. This infinite adaptive online approach is capable of segmenting and classifying the sequential data over unlimited number of classes, while meeting the memory and delay constraints of streaming contexts. The model is further enhanced by introducing a learning rate, responsible for balancing the extent to which the model sustains its previous learning (parameters) or adapts to the new streaming observations. Experimental results on several variants of stationary and evolving synthetic data and two video datasets, TUM Assistive Kitchen and collatedWeizmann, show remarkable performance in segmentation and classification, particularly for evolutionary sequences with changing distributions and/or containing new, unseen classes.

A network of agents attempt to learn some unknown state of the world drawn by nature from a finite set. Agents observe private signals conditioned on the true state, and form beliefs about the unknown state accordingly. Each agent may face an identification problem in the sense that she cannot distinguish the truth in isolation. However, by communicating with each other, agents are able to benefit from side observations to learn the truth collectively. Unlike many distributed algorithms which rely on all-time communication protocols, we propose an efficient method by switching between Bayesian and non-Bayesian regimes. In this model, agents exchange information only when their private signals are not informative enough; thence, by switching between the two regimes, agents efficiently learn the truth using only a few rounds of communications. The proposed algorithm preserves learnability while incurring a lower communication cost. We also verify our theoretical findings by simulation examples.

Probabilistic programming languages can simplify the development of machine learning techniques, but only if inference is sufficiently scalable. Unfortunately, Bayesian parameter estimation for highly coupled models such as regressions and state-space models still scales poorly; each MCMC transition takes linear time in the number of observations. This paper describes a sublinear-time algorithm for making Metropolis-Hastings (MH) updates to latent variables in probabilistic programs. The approach generalizes recently introduced approximate MH techniques: instead of subsampling data items assumed to be independent, it subsamples edges in a dynamically constructed graphical model. It thus applies to a broader class of problems and interoperates with other general-purpose inference techniques. Empirical results, including confirmation of sublinear per-transition scaling, are presented for Bayesian logistic regression, nonlinear classification via joint Dirichlet process mixtures, and parameter estimation for stochastic volatility models (with state estimation via particle MCMC). All three applications use the same implementation, and each requires under 20 lines of probabilistic code.

Multivariate categorical data occur in many applications of machine learning. One of the main difficulties with these vectors of categorical variables is sparsity. The number of possible observations grows exponentially with vector length, but dataset diversity might be poor in comparison. Recent models have gained significant improvement in supervised tasks with this data. These models embed observations in a continuous space to capture similarities between them. Building on these ideas we propose a Bayesian model for the unsupervised task of distribution estimation of multivariate categorical data. We model vectors of categorical variables as generated from a non-linear transformation of a continuous latent space. Non-linearity captures multi-modality in the distribution. The continuous representation addresses sparsity. Our model ties together many existing models, linking the linear categorical latent Gaussian model, the Gaussian process latent variable model, and Gaussian process classification. We derive inference for our model based on recent developments in sampling based variational inference. We show empirically that the model outperforms its linear and discrete counterparts in imputation tasks of sparse data.

Computer vision is hard because of a large variability in lighting, shape, and texture; in addition the image signal is non-additive due to occlusion. Generative models promised to account for this variability by accurately modelling the image formation process as a function of latent variables with prior beliefs. Bayesian posterior inference could then, in principle, explain the observation. While intuitively appealing, generative models for computer vision have largely failed to deliver on that promise due to the difficulty of posterior inference. As a result the community has favoured efficient discriminative approaches. We still believe in the usefulness of generative models in computer vision, but argue that we need to leverage existing discriminative or even heuristic computer vision methods. We implement this idea in a principled way with an "informed sampler" and in careful experiments demonstrate it on challenging generative models which contain renderer programs as their components. We concentrate on the problem of inverting an existing graphics rendering engine, an approach that can be understood as "Inverse Graphics". The informed sampler, using simple discriminative proposals based on existing computer vision technology, achieves significant improvements of inference.