We introduce a Gaussian process model of functions which areadditive . An additive function is one which decomposes into a sum of low-dimensional functions, each depending on only a subset of the input variables. Additive GPs generalize both Generalized Additive Models, and the standard GP models which use squared-exponential kernels. Hyperparameter learning in this model can be seen as Bayesian Hierarchical Kernel Learning (HKL). We introduce an expressive but tractable parameterization of the kernel function, which allows efficient evaluation of all input interaction terms, whose number is exponential in the input dimension. The additional structure discoverable by this model results in increased interpretability, as well as state-of-the-art predictive power in regression tasks.

Exact inference in the linear regression model with spike and slab priors is often intractable. Expectation propagation (EP) can be used for approximate inference. However, the regular sequential form of EP (R-EP) may fail to converge in this model when the size of the training set is very small. As an alternative, we propose a provably convergent EP algorithm (PC-EP). PC-EP is proved to minimize an energy function which, under some constraints, is bounded from below and whose stationary points coincide with the solution of R-EP. Experiments with synthetic data indicate that when R-EP does not converge, the approximation generated by PC-EP is often better. By contrast, when R-EP converges, both methods perform similarly.

Inverse inference, or "brain reading", is a recent paradigm for analyzing functional magnetic resonance imaging (fMRI) data, based on pattern recognition and statistical learning. By predicting some cognitive variables related to brain activation maps, this approach aims at decoding brain activity. Inverse inference takes into account the multivariate information between voxels and is currently the only way to assess how precisely some cognitive information is encoded by the activity of neural populations within the whole brain. However, it relies on a prediction function that is plagued by the curse of dimensionality, since there are far more features than samples, i.e., more voxels than fMRI volumes. To address this problem, different methods have been proposed, such as, among others, univariate feature selection, feature agglomeration and regularization techniques. In this paper, we consider a sparse hierarchical structured regularization. Specifically, the penalization we use is constructed from a tree that is obtained by spatially-constrained agglomerative clustering. This approach encodes the spatial structure of the data at different scales into the regularization, which makes the overall prediction procedure more robust to inter-subject variability. The regularization used induces the selection of spatially coherent predictive brain regions simultaneously at different scales. We test our algorithm on real data acquired to study the mental representation of objects, and we show that the proposed algorithm not only delineates meaningful brain regions but yields as well better prediction accuracy than reference methods.

Probabilistic graphical models combine the graph theory and probability theory to give a multivariate statistical modeling. They provide a unified description of uncertainty using probability and complexity using the graphical model. Especially, graphical models provide the following several useful properties: - Graphical models provide a simple and intuitive interpretation of the structures of probabilistic models. On the other hand, they can be used to design and motivate new models. - Graphical models provide additional insights into the properties of the model, including the conditional independence properties. - Complex computations which are required to perform inference and learning in sophisticated models can be expressed in terms of graphical manipulations, in which the underlying mathematical expressions are carried along implicitly. The graphical models have been applied to a large number of fields, including bioinformatics, social science, control theory, image processing, marketing analysis, among others. However, structure learning for graphical models remains an open challenge, since one must cope with a combinatorial search over the space of all possible structures. In this paper, we present a comprehensive survey of the existing structure learning algorithms.

The popular cubic smoothing spline estimate of a regression function arises as the minimizer of the penalized sum of squares $\sum_j(Y_j - {\mu}(t_j))^2 + {\lambda}\int_a^b [{\mu}"(t)]^2 dt$, where the data are $t_j,Y_j$, $j=1,..., n$. The minimization is taken over an infinite-dimensional function space, the space of all functions with square integrable second derivatives. But the calculations can be carried out in a finite-dimensional space. The reduction from minimizing over an infinite dimensional space to minimizing over a finite dimensional space occurs for more general objective functions: the data may be related to the function ${\mu}$ in another way, the sum of squares may be replaced by a more suitable expression, or the penalty, $\int_a^b [{\mu}"(t)]^2 dt$, might take a different form. This paper reviews the Reproducing Kernel Hilbert Space structure that provides a finite-dimensional solution for a general minimization problem. Particular attention is paid to penalties based on linear differential operators. In this case, one can sometimes easily calculate the minimizer explicitly, using Green's functions.

Structure learning in random fields has attracted considerable attention due to its difficulty and importance in areas such as remote sensing, computational biology, natural language processing, protein networks, and social network analysis. We consider the problem of estimating the probabilistic graph structure associated with a Gaussian Markov Random Field (GMRF), the Ising model and the Potts model, by extending previous work on $l_1$ regularized neighborhood estimation to include the elastic net $l_1+l_2$ penalty. Additionally, we show numerical evidence that the edge density plays a role in the graph recovery process. Finally, we introduce a novel method for augmenting neighborhood estimation by leveraging pair-wise neighborhood union estimates.

In mixtures-of-experts (ME) model, where a number of submodels (experts) are combined, there have been two longstanding problems: (i) how many experts should be chosen, given the size of the training data? (ii) given the total number of parameters, is it better to use a few very complex experts, or is it better to combine many simple experts? In this paper, we try to provide some insights to these problems through a theoretic study on a ME structure where $m$ experts are mixed, with each expert being related to a polynomial regression model of order $k$. We study the convergence rate of the maximum likelihood estimator (MLE), in terms of how fast the Kullback-Leibler divergence of the estimated density converges to the true density, when the sample size $n$ increases. The convergence rate is found to be dependent on both $m$ and $k$, and certain choices of $m$ and $k$ are found to produce optimal convergence rates. Therefore, these results shed light on the two aforementioned important problems: on how to choose $m$, and on how $m$ and $k$ should be compromised, for achieving good convergence rates.

Most classification methods provide either a prediction of class membership or an assessment of class membership probability. In the case of two-group classification the predicted probability can be described as "risk" of belonging to a "special" class . When the required output is a set of ordinal-risk groups, a discretization of the continuous risk prediction is achieved by two common methods: by constructing a set of models that describe the conditional risk function at specific points (quantile regression) or by dividing the output of an "optimal" classification model into adjacent intervals that correspond to the desired risk groups. By defining a new error measure for the distribution of risk onto intervals we are able to identify lower bounds on the accuracy of these methods, showing sub-optimality both in their distribution of risk and in the efficiency of their resulting partition into intervals. By adding a new form of constraint to the existing maximum likelihood optimization framework and by introducing a penalty function to avoid degenerate solutions, we show how existing methods can be augmented to solve the ordinal risk-group classification problem. We implement our method for logistic regression (LR) and show a numeric example.

Technology has had an unquestionable impact on the way people watch sports. Along with this technological evolution has come a higher standard to ensure a good viewing experience for the casual sports fan. It can be argued that the pervasion of statistical analysis in sports serves to satiate the fan's desire for detailed sports statistics. The goal of statistical analysis in sports is a simple one: to eliminate subjective analysis. In this paper, we review previous work that attempts to analyze various aspects in sports by using ideas from Markov Chains, Bayesian Inference and Markov Chain Monte Carlo (MCMC) methods. The unifying goal of these works is to achieve an accurate representation of the player's ability, the sport, or the environmental effects on the player's performance. With the prevalence of cheap computation, it is possible that using techniques in Artificial Intelligence could improve the result of statistical analysis in sport. This is best illustrated when evaluating football using Neuro Dynamic Programming, a Control Theory paradigm heavily based on theory in Stochastic processes. The results from this method suggest that statistical analysis in sports may benefit from using ideas from the area of Control Theory or Machine Learning

Recently, Mahoney and Orecchia demonstrated that popular diffusion-based procedures to compute a quick \emph{approximation} to the first nontrivial eigenvector of a data graph Laplacian \emph{exactly} solve certain regularized Semi-Definite Programs (SDPs). In this paper, we extend that result by providing a statistical interpretation of their approximation procedure. Our interpretation will be analogous to the manner in which $\ell_2$-regularized or $\ell_1$-regularized $\ell_2$-regression (often called Ridge regression and Lasso regression, respectively) can be interpreted in terms of a Gaussian prior or a Laplace prior, respectively, on the coefficient vector of the regression problem. Our framework will imply that the solutions to the Mahoney-Orecchia regularized SDP can be interpreted as regularized estimates of the pseudoinverse of the graph Laplacian. Conversely, it will imply that the solution to this regularized estimation problem can be computed very quickly by running, e.g., the fast diffusion-based PageRank procedure for computing an approximation to the first nontrivial eigenvector of the graph Laplacian. Empirical results are also provided to illustrate the manner in which approximate eigenvector computation \emph{implicitly} performs statistical regularization, relative to running the corresponding exact algorithm.