Ryan Martin

Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference

Posterior probabilistic statistical inference without priors is an important but so far elusive goal. Fisher’s fiducial inference, Dempster–Shafer theory of belief functions, and Bayesian inference with default priors are attempts to achieve this goal but, to date, none has given a completely satisfactory picture. This article presents a new framework for probabilistic inference, based on inferential models (IMs), which not only provides data-dependent probabilistic measures of uncertainty about the unknown parameter, but also does so with an automatic long-run frequency-calibration property. The key to this new approach is the identification of an unobservable auxiliary variable associated with observable data and unknown parameter, and the prediction of this auxiliary variable with a random set before conditioning on data. Here we present a three-step IM construction, and prove a frequency-calibration property of the IM’s belief function under mild conditions. A corresponding optimality theory is developed, which helps to resolve the nonuniqueness issue. Several examples are presented to illustrate this new approach.

Dempster-Shafer Theory and Statistical Inference with Weak Beliefs

The Dempster-Shafer (DS) theory is a powerful tool for probabilistic reasoning based on a formal calculus for combining evi- dence. DS theory has been widely used in computer science and engi- neering applications, but has yet to reach the statistical mainstream, perhaps because the DS belief functions do not satisfy long-run fre- quency properties. Recently, two of the authors proposed an extension of DS, called the weak belief (WB) approach, that can incorporate de- sirable frequency properties into the DS framework by systematically enlarging the focal elements. The present paper reviews and extends this WB approach. We present a general description of WB in the context of inferential models, its interplay with the DS calculus, and the maximal belief solution. New applications of the WB method in two high-dimensional hypothesis testing problems are given. Simula- tions show that the WB procedures, suitably calibrated, perform well compared to popular classical methods. Most importantly, the WB ap- proach combines the probabilistic reasoning of DS with the desirable frequency properties of classical statistics.

CONSISTENCY OF A RECURSIVE ESTIMATE OF MIXING DISTRIBUTIONS

Mixture models have received considerable attention recently and Newton [Sankhyā Ser. A 64 (2002) 306-322] proposed a fast recursive algorithm for estimating a mixing distribution. We prove almost sure consistency of this recursive estimate in the weak topology under mild conditions on the family of densities being mixed. This recursive estimate depends on the data ordering and a permutation-invariant modification is proposed, which is an average of the original over permutations of the data sequence. A Rao-Blackwell argument is used to prove consistency in probability of this alternative estimate. Several simulations are presented, comparing the finite-sample performance of the recursive estimate and a Monte Carlo approximation to the permutation-invariant alternative along with that of the nonparametric maximum likelihood estimate and a nonparametric Bayes estimate.

Marginal Inferential Models: Prior-Free Probabilistic Inference on Interest Parameters

The inferential models (IM) framework provides prior-free, frequency-calibrated, and posterior probabilistic inference. The key is the use of random sets to predict unobservable auxiliary variables connected to the observable data and unknown parameters. When nuisance parameters are present, a marginalization step can reduce the dimension of the auxiliary variable which, in turn, leads to more efficient inference. For regular problems, exact marginalization can be achieved, and we give conditions for marginal IM validity. We show that our approach provides exact and efficient marginal inference in several challenging problems, including a many-normal-means problem. In nonregular problems, we propose a generalized marginalization technique and prove its validity. Details are given for two benchmark examples, namely, the Behrens–Fisher and gamma mean problems.

Stochastic Approximation and Newton’s Estimate of a Mixing Distribution

Many statistical problems involve mixture models and the need for computationally efficient methods to estimate the mixing distribution has increased dramatically in recent years. Newton [Sankhya Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating the mixing distribution, which we study as a special case of stochastic approximation (SA). We begin with a review of SA, some recent statistical applications, and the theory necessary for analysis of a SA algorithm, which includes Lyapunov functions and ODE stability theory. Then standard SA results are used to prove consistency of Newtons estimate in the case of a finite mixture. We also propose a modification of Newtons algorithm that allows for estimation of an additional unknown parameter in the model, and prove its consistency.

Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector

For the important classical problem of inference on a sparse high-dimensional normal mean vector, we propose a novel empirical Bayes model that admits a posterior distribution with desirable properties under mild conditions. In particular, our empirical Bayes posterior distribution concentrates on balls, centered at the true mean vector, with squared radius proportional to the minimax rate, and its posterior mean is an asymptotically minimax estimator. We also show that, asymptotically, the support of our empirical Bayes posterior has roughly the same effective dimension as the true sparse mean vector. Simulation from our empirical Bayes posterior is straightforward, and our numerical results demonstrate the quality of our method compared to others having similar large-sample properties.

Tripartite version of the Corrádi–Hajnal theorem

Let G be a tripartite graph with N vertices in each vertex class. If each vertex is adjacent to at least (2/3)N vertices in each of the other classes, then either G contains a subgraph that consists of N vertex-disjoint triangles or G is a specific graph in which each vertex is adjacent to exactly (2/3)N vertices in each of the other classes.

The emergence of a giant component in random subgraphs of pseudo-random graphs

Let

Decentralized Indirect Methods for Learning Automata Games

G

Asymptotic properties of predictive recursion: Robustness and rate of convergence

be a