The systematic biases seen in people's probability judgments are typically taken as evidence that people do not reason about probability using the rules of probability theory, but instead use heuristics which sometimes yield reasonable judgments and sometimes systematic biases. This view has had a major impact in economics, law, medicine, and other fields; indeed, the idea that people cannot reason with probabilities has become a widespread truism. We present a simple alternative to this view, where people reason about probability according to probability theory but are subject to random variation or noise in the reasoning process. In this account the effect of noise is cancelled for some probabilistic expressions: analysing data from two experiments we find that, for these expressions, people's probability judgments are strikingly close to those required by probability theory. For other expressions this account produces systematic deviations in probability estimates. These deviations explain four reliable biases in human probabilistic reasoning (conservatism, subadditivity, conjunction and disjunction fallacies). These results suggest that people's probability judgments embody the rules of probability theory, and that biases in those judgments are due to the effects of random noise.

We establish optimal convergence rates for a decomposition-based scalable approach to kernel ridge regression. The method is simple to describe: it randomly partitions a dataset of size N into m subsets of equal size, computes an independent kernel ridge regression estimator for each subset, then averages the local solutions into a global predictor. This partitioning leads to a substantial reduction in computation time versus the standard approach of performing kernel ridge regression on all N samples. Our two main theorems establish that despite the computational speed-up, statistical optimality is retained: as long as m is not too large, the partition-based estimator achieves the statistical minimax rate over all estimators using the set of N samples. As concrete examples, our theory guarantees that the number of processors m may grow nearly linearly for finite-rank kernels and Gaussian kernels and polynomially in N for Sobolev spaces, which in turn allows for substantial reductions in computational cost. We conclude with experiments on both simulated data and a music-prediction task that complement our theoretical results, exhibiting the computational and statistical benefits of our approach.

Graphical models are commonly used tools for modeling multivariate random variables. While there exist many convenient multivariate distributions such as Gaussian distribution for continuous data, mixed data with the presence of discrete variables or a combination of both continuous and discrete variables poses new challenges in statistical modeling. In this paper, we propose a semiparametric model named latent Gaussian copula model for binary and mixed data. The observed binary data are assumed to be obtained by dichotomizing a latent variable satisfying the Gaussian copula distribution or the nonparanormal distribution. The latent Gaussian model with the assumption that the latent variables are multivariate Gaussian is a special case of the proposed model. A novel rank-based approach is proposed for both latent graph estimation and latent principal component analysis. Theoretically, the proposed methods achieve the same rates of convergence for both precision matrix estimation and eigenvector estimation, as if the latent variables were observed. Under similar conditions, the consistency of graph structure recovery and feature selection for leading eigenvectors is established. The performance of the proposed methods is numerically assessed through simulation studies, and the usage of our methods is illustrated by a genetic dataset.

Motivation: Analysis of relationships of drug structure to biological response is key to understanding off-target and unexpected drug effects, and for developing hypotheses on how to tailor drug thera-pies. New methods are required for integrated analyses of a large number of chemical features of drugs against the corresponding genome-wide responses of multiple cell models. Results: In this paper, we present the first comprehensive multi-set analysis on how the chemical structure of drugs impacts on ge-nome-wide gene expression across several cancer cell lines (CMap database). The task is formulated as searching for drug response components across multiple cancers to reveal shared effects of drugs and the chemical features that may be responsible. The com-ponents can be computed with an extension of a very recent ap-proach called Group Factor Analysis (GFA). We identify 11 compo-nents that link the structural descriptors of drugs with specific gene expression responses observed in the three cell lines, and identify structural groups that may be responsible for the responses. Our method quantitatively outperforms the limited earlier studies on CMap and identifies both the previously reported associations and several interesting novel findings, by taking into account multiple cell lines and advanced 3D structural descriptors. The novel observations include: previously unknown similarities in the effects induced by 15-delta prostaglandin J2 and HSP90 inhibitors, which are linked to the 3D descriptors of the drugs; and the induction by simvastatin of leukemia-specific anti-inflammatory response, resem-bling the effects of corticosteroids.

Information geometry applies concepts in differential geometry to probability and statistics and is especially useful for parameter estimation in exponential families where parameters are known to lie on a Riemannian manifold. Connections between the geometric properties of the induced manifold and statistical properties of the estimation problem are well-established. However developing first-order methods that scale to larger problems has been less of a focus in the information geometry community. The best known algorithm that incorporates manifold structure is the second-order natural gradient descent algorithm introduced by Amari. On the other hand, stochastic approximation methods have led to the development of first-order methods for optimizing noisy objective functions. A recent generalization of the Robbins-Monro algorithm known as mirror descent, developed by Nemirovski and Yudin is a first order method that induces non-Euclidean geometries. However current analysis of mirror descent does not precisely characterize the induced non-Euclidean geometry nor does it consider performance in terms of statistical relative efficiency. In this paper, we prove that mirror descent induced by Bregman divergences is equivalent to the natural gradient descent algorithm on the dual Riemannian manifold. Using this equivalence, it follows that (1) mirror descent is the steepest descent direction along the Riemannian manifold of the exponential family; (2) mirror descent with log-likelihood loss applied to parameter estimation in exponential families asymptotically achieves the classical Cram\'er-Rao lower bound and (3) natural gradient descent for manifolds corresponding to exponential families can be implemented as a first-order method through mirror descent.

This paper proposes a simple adaptive sensing and group testing algorithm for sparse signal recovery. The algorithm, termed Compressive Adaptive Sense and Search (CASS), is shown to be near-optimal in that it succeeds at the lowest possible signal-to-noise-ratio (SNR) levels, improving on previous work in adaptive compressed sensing. Like traditional compressed sensing based on random non-adaptive design matrices, the CASS algorithm requires only k log n measurements to recover a k-sparse signal of dimension n. However, CASS succeeds at SNR levels that are a factor log n less than required by standard compressed sensing. From the point of view of constructing and implementing the sensing operation as well as computing the reconstruction, the proposed algorithm is substantially less computationally intensive than standard compressed sensing. CASS is also demonstrated to perform considerably better in practice through simulation. To the best of our knowledge, this is the first demonstration of an adaptive compressed sensing algorithm with near-optimal theoretical guarantees and excellent practical performance. This paper also shows that methods like compressed sensing, group testing, and pooling have an advantage beyond simply reducing the number of measurements or tests -- adaptive versions of such methods can also improve detection and estimation performance when compared to non-adaptive direct (uncompressed) sensing.

As surrogate functions of $L_0$-norm, many nonconvex penalty functions have been proposed to enhance the sparse vector recovery. It is easy to extend these nonconvex penalty functions on singular values of a matrix to enhance low-rank matrix recovery. However, different from convex optimization, solving the nonconvex low-rank minimization problem is much more challenging than the nonconvex sparse minimization problem. We observe that all the existing nonconvex penalty functions are concave and monotonically increasing on $[0,\infty)$. Thus their gradients are decreasing functions. Based on this property, we propose an Iteratively Reweighted Nuclear Norm (IRNN) algorithm to solve the nonconvex nonsmooth low-rank minimization problem. IRNN iteratively solves a Weighted Singular Value Thresholding (WSVT) problem. By setting the weight vector as the gradient of the concave penalty function, the WSVT problem has a closed form solution. In theory, we prove that IRNN decreases the objective function value monotonically, and any limit point is a stationary point. Extensive experiments on both synthetic data and real images demonstrate that IRNN enhances the low-rank matrix recovery compared with state-of-the-art convex algorithms.

Game Theory studies situations in which multiple agents having conflicting objectives have to reach a collective decision. The question of a compact representation language for agents utility function is of crucial importance since the classical representation of a $n$-players game is given by a $n$-dimensional matrix of exponential size for each player. In this paper we use the framework of Constraint Games in which CSP are used to represent utilities. Constraint Programming --including global constraints-- allows to easily give a compact and elegant model to many useful games. Constraint Games come in two flavors: Constraint Satisfaction Games and Constraint Optimization Games, the first one using satisfaction to define boolean utilities. In addition to multimatrix games, it is also possible to model more complex games where hard constraints forbid certain situations. In this paper we study complete search techniques and show that our solver using the compact representation of Constraint Games is faster than the classical game solver Gambit by one to two orders of magnitude.

Almost any large dataset has annotation errors, especially those complex, nuanced datasets commonly used in natural language processing. Low-quality annotations have become even more common in recent years with the rise of Amazon Mechanical Turk, as well as methods like distant supervision and co-training that involve automatically generating training data. Although small amounts of noise may not be detrimental, in some applications the level can be high: upon manually inspecting a relation extraction corpus commonly used in distant supervision, Riedel et al. (2010) report a 31% false positive rate. In cases like these, annotation errors have frequently been observed to hurt performance. Dingare et al. (2005), for example, conduct error analysis on a system to extract relations from biomedical text, and observe that over half of the system's errors could be attributed to inconsistencies in how the data was annotated. Similarly, in a case study on co-training for natural language tasks, Pierce and Cardie (2001) find that the degradation in data quality from automatic labelling prevents these systems from performing comparably to their fully-supervised counterparts.

Regression aims at estimating the conditional mean of output given input. However, regression is not informative enough if the conditional density is multimodal, heteroscedastic, and asymmetric. In such a case, estimating the conditional density itself is preferable, but conditional density estimation (CDE) is challenging in high-dimensional space. A naive approach to coping with high-dimensionality is to first perform dimensionality reduction (DR) and then execute CDE. However, such a two-step process does not perform well in practice because the error incurred in the first DR step can be magnified in the second CDE step. In this paper, we propose a novel single-shot procedure that performs CDE and DR simultaneously in an integrated way. Our key idea is to formulate DR as the problem of minimizing a squared-loss variant of conditional entropy, and this is solved via CDE. Thus, an additional CDE step is not needed after DR. We demonstrate the usefulness of the proposed method through extensive experiments on various datasets including humanoid robot transition and computer art.