Hierarchical probabilistic models, such as Gaussian mixture models, are widely used for unsupervised learning tasks. These models consist of observable and latent variables, which represent the observable data and the underlying data-generation process, respectively. Unsupervised learning tasks, such as cluster analysis, are regarded as estimations of latent variables based on the observable ones. The estimation of latent variables in semi-supervised learning, where some labels are observed, will be more precise than that in unsupervised, and one of the concerns is to clarify the effect of the labeled data. However, there has not been sufficient theoretical analysis of the accuracy of the estimation of latent variables. In a previous study, a distribution-based error function was formulated, and its asymptotic form was calculated for unsupervised learning with generative models. It has been shown that, for the estimation of latent variables, the Bayes method is more accurate than the maximum-likelihood method. The present paper reveals the asymptotic forms of the error function in Bayesian semi-supervised learning for both discriminative and generative models. The results show that the generative model, which uses all of the given data, performs better when the model is well specified.

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.

Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for the sparse spectrum approximation to avoid both issues. We model the covariance function with a finite Fourier series approximation and treat it as a random variable. The random covariance function has a posterior, on which a variational distribution is placed. The variational distribution transforms the random covariance function to fit the data. We study the properties of our approximate inference, compare it to alternative ones, and extend it to the distributed and stochastic domains. Our approximation captures complex functions better than standard approaches and avoids over-fitting.

Existing MAP inference algorithms for determinantal point processes (DPPs) need to calculate determinants or conduct eigenvalue decomposition generally at the scale of the full kernel, which presents a great challenge for real-world applications. In this paper, we introduce a class of DPPs, called BwDPPs, that are characterized by an almost block diagonal kernel matrix and thus can allow efficient block-wise MAP inference. Furthermore, BwDPPs are successfully applied to address the difficulty of selecting change-points in the problem of change-point detection (CPD), which results in a new BwDPP-based CPD method, named BwDppCpd. In BwDppCpd, a preliminary set of change-point candidates is first created based on existing well-studied metrics. Then, these change-point candidates are treated as DPP items, and DPP-based subset selection is conducted to give the final estimate of the change-points that favours both quality and diversity. The effectiveness of BwDppCpd is demonstrated through extensive experiments on five real-world datasets.

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 relational databases, relations between objects, represented by binary matrices or tensors, may be arbitrarily complex. In practice however, there are recurring relational patterns such as transitive, permutation and sequential relationships, that seem to have a regular structure not captured by the classical notion of matrix rank or tensor rank. In this paper, we show that factorizing the relational tensor using a logistic or hinge loss instead of the more standard squared loss is more appropriate because it can accurately model many common relations with a fixed-size embedding that depends sub-linearly on the number of entities in the knowledge base. We illustrate this fact empirically by being able to efficiently predict missing links in several synthetic and real-world experiments. Further, we provide theoretical justification for logistic loss by studying its connection to a complexity measure from the field of information complexity called the sign rank. Sign rank is a more appropriate complexity measure as it has a low value for transitive, permutation, or sequential relationships, while being large for uniformly sampled binary matrices/tensors with a high probability.

High-level representations of uncertainty, such as probabilistic logics and programs, have been around for decades. Lifted inference was initially motivated by the need to make reasoning algorithms high-level as well. While the lifted inference community focused on machine learning applications, the high-level reasoning goal has received less attention recently. We revisit the idea and look at the capabilities of the latest techniques in lifted inference. This lets us conclude that lifted inference is strictly more powerful than propositional inference on high-level reasoning tasks.