Foundational principles for large scale inference: Illustrations through correlation mining

When can reliable inference be drawn in the "Big Data" context? This paper presents a framework for answering this fundamental question in the context of correlation mining, with implications for general large scale inference. In large scale data applications like genomics, connectomics, and eco-informatics the dataset is often variable-rich but sample-starved: a regime where the number $n$ of acquired samples (statistical replicates) is far fewer than the number $p$ of observed variables (genes, neurons, voxels, or chemical constituents). Much of recent work has focused on understanding the computational complexity of proposed methods for "Big Data." Sample complexity however has received relatively less attention, especially in the setting when the sample size $n$ is fixed, and the dimension $p$ grows without bound. To address this gap, we develop a unified statistical framework that explicitly quantifies the sample complexity of various inferential tasks. Sampling regimes can be divided into several categories: 1) the classical asymptotic regime where the variable dimension is fixed and the sample size goes to infinity; 2) the mixed asymptotic regime where both variable dimension and sample size go to infinity at comparable rates; 3) the purely high dimensional asymptotic regime where the variable dimension goes to infinity and the sample size is fixed. Each regime has its niche but only the latter regime applies to exa-scale data dimension. We illustrate this high dimensional framework for the problem of correlation mining, where it is the matrix of pairwise and partial correlations among the variables that are of interest. We demonstrate various regimes of correlation mining based on the unifying perspective of high dimensional learning rates and sample complexity for different structured covariance models and different inference tasks

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

The Bayesian Analysis of Complex, High-Dimensional Models: Can It Be CODA?

We consider the Bayesian analysis of a few complex, high-dimensional models and show that intuitive priors, which are not tailored to the fine details of the model and the estimated parameters, produce estimators which perform poorly in situations in which good, simple frequentist estimators exist. The models we consider are: stratified sampling, the partial linear model, linear and quadratic functionals of white noise and estimation with stopping times. We present a strong version of Doob's consistency theorem which demonstrates that the existence of a uniformly $\sqrt{n}$-consistent estimator ensures that the Bayes posterior is $\sqrt{n}$-consistent for values of the parameter in subsets of prior probability 1. We also demonstrate that it is, at least, in principle, possible to construct Bayes priors giving both global and local minimax rates, using a suitable combination of loss functions. We argue that there is no contradiction in these apparently conflicting findings.Comment: Published in at http://dx.doi.org/10.1214/14-STS483 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

A Simple Approach to Maximum Intractable Likelihood Estimation

Approximate Bayesian Computation (ABC) can be viewed as an analytic approximation of an intractable likelihood coupled with an elementary simulation step. Such a view, combined with a suitable instrumental prior distribution permits maximum-likelihood (or maximum-a-posteriori) inference to be conducted, approximately, using essentially the same techniques. An elementary approach to this problem which simply obtains a nonparametric approximation of the likelihood surface which is then used as a smooth proxy for the likelihood in a subsequent maximisation step is developed here and the convergence of this class of algorithms is characterised theoretically. The use of non-sufficient summary statistics in this context is considered. Applying the proposed method to four problems demonstrates good performance. The proposed approach provides an alternative for approximating the maximum likelihood estimator (MLE) in complex scenarios

Bayesian Subset Simulation: a kriging-based subset simulation algorithm for the estimation of small probabilities of failure

The estimation of small probabilities of failure from computer simulations is a classical problem in engineering, and the Subset Simulation algorithm proposed by Au & Beck (Prob. Eng. Mech., 2001) has become one of the most popular method to solve it. Subset simulation has been shown to provide significant savings in the number of simulations to achieve a given accuracy of estimation, with respect to many other Monte Carlo approaches. The number of simulations remains still quite high however, and this method can be impractical for applications where an expensive-to-evaluate computer model is involved. We propose a new algorithm, called Bayesian Subset Simulation, that takes the best from the Subset Simulation algorithm and from sequential Bayesian methods based on kriging (also known as Gaussian process modeling). The performance of this new algorithm is illustrated using a test case from the literature. We are able to report promising results. In addition, we provide a numerical study of the statistical properties of the estimator.Comment: 11th International Probabilistic Assessment and Management Conference (PSAM11) and The Annual European Safety and Reliability Conference (ESREL 2012), Helsinki : Finland (2012

Probabilistic Motion Estimation Based on Temporal Coherence

We develop a theory for the temporal integration of visual motion motivated by psychophysical experiments. The theory proposes that input data are temporally grouped and used to predict and estimate the motion flows in the image sequence. This temporal grouping can be considered a generalization of the data association techniques used by engineers to study motion sequences. Our temporal-grouping theory is expressed in terms of the Bayesian generalization of standard Kalman filtering. To implement the theory we derive a parallel network which shares some properties of cortical networks. Computer simulations of this network demonstrate that our theory qualitatively accounts for psychophysical experiments on motion occlusion and motion outliers.Comment: 40 pages, 7 figure