Perspectives in Mathematical Sciences I

The following sections are included:

• Foreword

• Preface

• Contents

The following sections are included:

• Introduction

• Relative Entropy and Gibbs-Boltzmann Measures
  • Entropy Maximization Results
  • Weak Convergence of Gibbs-Boltzmann Distribution
  • Relative Entropy and Conditioning

• Measure Free Martingales, Weak Martingales, Martingales
  • Finite Range Case
  • The General Case

• Equivalent Martingale Measures

• References

Methods for quantile estimation based on massive streaming data are reviewed. Marginal quantiles help in the exploration of massive multivariate data. Asymptotic properties of the joint distribution of marginal sample quantiles of multivariate data are also reviewed. The results include weak convergence to Gaussian random elements. Asymptotics for the mean of functions of order statistics are also presented. Application of the latter result to regression analysis under partial or complete loss of association among the multivariate data is described.

This article provides an exposition of recent developments on the analysis of landmark based shapes in which a k-ad, i.e., a set of k points or landmarks on an object or a scene, are observed in 2D or 3D, for purposes of identification, discrimination, or diagnostics. Depending on the way the data are collected or recorded, the appropriate shape of an object is the maximal invariant specified by the space of orbits under a group G of transformations. All these spaces are manifolds, often with natural Riemannian structures. The statistical analysis based on Riemannian structures is said to be intrinsic. In other cases, proper distances are sought via an equivariant embedding of the manifold M in a vector space E, and the corresponding statistical analysis is called extrinsic.

This article advocates the viewpoint that reinforcement learning algorithms, primarily meant for approximate dynamic programming, can also be cast as a technique for estimating stationary averages and stationary distributions of Markov chains. In this role, they lie somewhere between standard deterministic numerical schemes and Markov chain Monte Carlo, and capture a trade-off between the advantages of either – lower per iterate computation than the former and lower variance than the latter. Issues arising from the ‘curse of dimensionality’ and convergence rate are also discussed.

The following sections are included:

• Introduction

• Some Basic Properties of Factors and Roots

• Factor Sets

• Compactness

• Roots

• One-Parameter Semigroups

• References

Higher criticism has been proposed as a tool for highly multiple hypothesis testing or signal detection, initially in cases where the distribution of a test statistic (or the noise in a signal) is known and the component tests are statistically independent. In this paper we explore the extent to which the assumptions of known distribution and independence can be relaxed, and we consider too the application of higher criticism to classification. It is shown that effective distribution approximations can be achieved by using a threshold approach; that is, by disregarding data components unless their significance level exceeds a sufficiently high value. This method exploits the good relative accuracy of approximations to light-tailed distributions. In particular, it can be effective when the true distribution is founded on something like a Studentised mean, or on an average of related type, which is commonly the case in practice. The issue of dependence among vector components is also shown not to be a serious difficulty in many circumstances.

Motivated by the problems in genomics, astronomy and some other emerging fields, multiple hypothesis testing has come to the forefront of statistical research in the recent years. In the context of multiple testing, new error measures such as the false discovery rate (FDR) occupy important roles comparable to the role of type I error in classical hypothesis testing. Assuming that a random mechanism decides the truth of a hypothesis, substantial gain in power is possible by estimating error measures from the data. Nonparametric Bayesian approaches are proven to be particularly suitable for estimation of error measure in multiple testing situation. A Bayesian approach based on a nonparametric mixture model for p-values can utilize special features of the distribution of p-values that significantly improves the quality of estimation. In this paper we describe the nonparametric Bayesian modeling exercise of the distribution of the p-values. We begin with a brief review of Bayesian nonparametric concepts of Dirichlet process and Dirichlet mixtures and classical multiple hypothesis testing. We then review recently proposed nonparametric Bayesian methods for estimating errors based on a Dirichlet mixture of prior for the p-value density. When the test statistics are independent, a mixture of beta kernels can adequately model the p-value density, whereas in the dependent case one can consider a Dirichlet mixture of multivariate skew-normal kernel prior for probit transforms of the p-values. We conclude the paper by illustrating the scope of these methods in some real-life applications.

This survey covers state-of-the-art Bayesian techniques for the estimation of mixtures. It complements the earlier work [31] by studying new types of distributions, the multinomial, latent class and t distributions. It also exhibits closed form solutions for Bayesian inference in some discrete setups. Lastly, it sheds a new light on the computation of Bayes factors via the approximation of [8].

The paper is a review of results on the asymptotic behavior of Markov processes generated by i.i.d. iterates of monotone maps. Of particular importance is the notion of splitting introduced by [12]. Some extensions to more general frameworks are outlined, and, finally, a number of applications are indicated.

The following sections are included:

• Introduction

• Elements of Finite Dimensional Quantum Probability

• Quantum Error-Correcting Codes

• Testing Quantum Hypotheses

• References

The following sections are included:

• Introduction

• Non-Interacting Case

• Simple Exclusion Processes

• Large Deviations

• References

The following sections are included:

• Author Index

• Subject Index

• Contents of Part II