What follows is a sketch of my 2013 viewpoint on how statistical inference should be viewed by applied statisticians. The label DS is an acronym for “Dempster–Shafer” after the originators of the technical foundation of the theory. Our foundation remains essentially unchanged since the 1960s and 1970s when I and then Glenn Shafer were its initial expositors. Present issues concern why and how the theory has the potential to develop into a major competitor of the “frequentist” and “Bayesian” outlooks. This for me is a work in progress. My understanding has evolved substantially over the past eight years of my emeritus status, during which DS has been my major focus. It was also a major focus of mine over the eight years beginning in 1961 when I first had the freedom that came with academic tenure in the Harvard Statistics Department. Between the two periods I was more an observer and teacher in relation to DS than a primary developer. I do not attempt here to address the long history of how DS got to where I now understand it to be, including connections with R.A. Fisher’s controversial “fiducial” argument. DS draws on technical developments in fields such as stochastic modeling and Bayesian posterior computation, but my DS-guided perception of the nature of statistical inference is in different ways both narrower and broader than that of its established competitors. It is narrower because it maintains that what “frequentist” statisticians call “inference” is not inference in the natural language meaning of the word. The latter means to me direct situation-specific assessments of probabilistic uncertainties that I call “personal probabilities.” For example, I might predict on September 30, 2013 that with personal probability .31 the Dow Jones Industrials stock index will exceed 16,000 at the end of business on December 31, 2013.

We introduce an extension of nonparametric DS inference for arbitrary univariate CDFs to the case in which some failure times are (right)-censored, and then apply this to the problem of assessing evidence regarding assertions about relative risks across two populations. The approach enables exploration of the sensitivity of survival analyses to assumed independence of the missing data process and the failure proces. We present an application to the partially efficacious RV144 (HIV-1) vaccine trial, and show that the strength of conclusions of vaccine efficacy depend on assumptions about the maximum failure rates of the subjects lost-to-followup.

We present a Dempster-Shafer (DS) approach to estimating limits from Poisson counting data with nuisance parameters. Dempster-Shafer is a statistical framework that generalizes Bayesian statistics. DS calculus augments traditional probability by allowing mass to be distributed over power sets of the event space. This eliminates the Bayesian dependence on prior distributions while allowing the incorporation of prior information when it is available. We use the Poisson Dempster-Shafer model (DSM) to derive a posterior DSM for the "Banff upper limits challenge" three-Poisson model. The results compare favorably with other approaches, demonstrating the utility of the approach. We argue that the reduced dependence on priors afforded by the Dempster-Shafer framework is both practically and theoretically desirable.

A perspective on statistical inference is proposed that is broad enough to encompass modern Bayesian and traditional Fisherian thinking, and interprets frequentist theory in a way that gives appropriate weights to both science and mathematics, and to both objective and subjective elements. The aim is to inject new thinking into a field held back by a longstanding lack of consensus.

The Dempster–Shafer (DS) theory of probabilistic reasoning is presented in terms of a semantics whereby every meaningful formal assertion is associated with a triple (p,q,r) where p is the probability “for” the assertion, q is the probability “against” the assertion, and r is the probability of “don’t know”. Arguments are presented for the necessity of “don’t know”. Elements of the calculus are sketched, including the extension of a DS model from a margin to a full state space, and DS combination of independent DS uncertainty assessments on the full space. The methodology is applied to inference and prediction from Poisson counts, including an introduction to the use of join-tree model structure to simplify and shorten computation. The relation of DS theory to statistical significance testing is elaborated, introducing along the way the new concept of “dull” null hypothesis.

We present a Dempster-Shafer (DS) approach to finding confidence bounds on the mass of the Higgs boson. Dempster-Shafer is a statistical framework that generalizes Bayesian statistics. DS calculus augments traditional probability by allowing mass to be distributed over power sets of the event space. This eliminates the Bayesian dependence on prior distributions while allowing the incorporation of prior information when it is available. We use the Poisson DempsterShafer model (DSM) to derive a posterior DSM for the Banff threePoisson model, from which we make inferences about the unknown mass of the Higgs particle. The results compare favorably with other approaches, demonstrating the utility of the approach. We argue that the reduced dependence on priors afforded by the Dempster-Shafer framework is both practically and theoretically desirable.

We consider situations in which each individual member of a defined object set is characterized uniquely by a set of variables, and we propose models and associated methods that recognize or classify a newly observed individual. Inputs consist of uncertain observations on the new individual and on a memory bank of previously identified individuals. Outputs consist of uncertain inferences concerning degrees of agreement between the new object and previously identified objects or object classes, with inferences represented by Dempster–Shafer belief functions. We illustrate the approach using models constructed from independent simple support belief functions defined on binary variables. In the case of object recognition, our models lead to marginal belief functions concerning how well the new object matches objects in memory. In the classification model, we compute beliefs and plausibilities that the new object lies in defined subsets of an object set. When regarded as similarity measures, our belief and plausibility functions can be interpreted as candidate membership functions in the terminology of fuzzy logic. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 283–297, 2006.

Abstract Born 1909, Rutherglen, Scotland; died 1980 Orleans, MA. Leading contributor to the British–American school of applied statistics during a period of rapid development of the field across the middle decades of the twentieth century.

The origins and basic elements of the Dempster-Shafer theory of belief functions are explained, including the operations of combination of independent representations of uncertain evidence and propagation among margins of multivariate systems. The theory is described as a tool that scientists can use to formalize subjective uncertainties about objectively formalized unknowns. The future of the theory depends on the development of models that capture common situations and are amenable to modem computational methodologies.

Although not a traditional philosopher, John Tukey contributed much to our understanding of statistical science and empirical science more broadly. The former is represented by the light he shed on the relation of drawing conclusions to making decisions, and of how simple concepts like significance and confidence serve to back up or "confirm" empirical findings. Less successfully, he attempted inconclusively to sort out the ambiguities of R. A. Fisher's fiducial argument. His main effort, however, went to creating "exploratory data analysis" or EDA as a subfield of statistics with much to offer to ongoing developments in data mining and data visualization.

Circadian modulation of episodic bursts is recognized as the normal physiological pattern of diurnal variation in plasma cortisol levels. The primary physiological factors underlying these diurnal patterns are the ultradian timing of secretory events, circadian modulation of the amplitude of secretory events, infusion of the hormone from the adrenal gland into the plasma, and clearance of the hormone from the plasma by the liver. Each measured plasma cortisol level has an error arising from the cortisol immunoassay. We demonstrate that all of these three physiological principles can be succinctly summarized in a single stochastic differential equation plus measurement error model and show that physiologically consistent ranges of the model parameters can be determined from published reports. We summarize the model parameters in terms of the multivariate Gaussian probability density and establish the plausibility of the model with a series of simulation studies. Our framework makes possible a sensitivity analysis in which all model parameters are allowed to vary simultaneously. The model offers an approach for simultaneously representing cortisol's ultradian, circadian, and kinetic properties. Our modeling paradigm provides a framework for simulation studies and data analysis that should be readily adaptable to the analysis of other endocrine hormone systems.

Time series models of sampling error, true unobserved rates, and covariates can be used to pool data across time and space to reduce variance in a subnational estimator. We present such models along with associated hierarchical Bayesian analyses. Specifically, we present a joint time series model for a 51 U.S. state labor force series in a Bayesian framework. Data are input in the form of optimal composite estimates from a sampling error model. The basic time series model is constructed from fractional Gaussian noise processes. Covariation of the true series across states is modeled by having a common national component modified by individual state components, Markov chain Monte Carlo methods are applied to develop samplers for a high-dimensional system of 105 parameters. The results indicate substantial gains in the efficient use of CPS data for U.S. state employment and unemployment rates series.

This study presented a new classification method for single person’s motion, which is represented by Haar wavelet transform and classified by Hidden Markov Model. We tackle the challenge of detecting the feature points by Haar wavelet transform to improve classification accuracy. We extract binary silhouette and segment them by cycle after creating the background model. Then the low-level features are detected by Haar wavelet transform and principal vectors are determined by Principal Component Analysis. We utilize Hidden Markov Models to train and classify cycle sequences and demonstrate their usability. Compared with others, our approach is simple and effective in feature point detection, strength in scale-invariant and generalized in different motions. Therefore, the video surveillance based on our method is practicable in (but not limited to) many scenarios where the background is known.

Arguments are presented to support increased emphasis on logical aspects of formal methods of analysis, depending on probability in the sense of R. A. Fisher. Formulating probabilistic models that convey uncertain knowledge of objective phenomena and using such models for inductive reasoning are central activities of individuals that introduce limited but necessary subjectivity into science. Statistical models are classified into overlapping types called here empirical, stochastic and predictive, all drawing on a common mathematical theory of probability, and all facilitating statements with logical and epistemic content. Contexts in which these ideas are intended to apply are discussed via three major examples.

An approach to significance testing by the direct interpretation of likelihood is defined, developed and distinguished from the traditional forms of tail-area testing and Bayesian testing. The emphasis is on conceptual issues. Some theoretical aspects of the new approach are sketched in the two cases of simple vs. simple hypotheses and simple vs. composite hypotheses.