We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include generating tables with fixed row and column sums and higher dimensional analogs. The algorithms involve finding bases for associated polynomial ideals and so an excursion into computational algebraic geometry.

The study of scientific discovery—where do new ideas come from?—has long been denigrated by philosophers as irrelevant to analyzing the growth of scientific knowledge. In particular, little is known about how cognitive theories are discovered, and neither the classical accounts of discovery as either probabilistic induction (e.g., Reichenbach, 1938) or lucky guesses (e.g., Popper, 1959), nor the stock anecdotes about sudden “eureka ” moments deepen the insight into discovery. A heuristics approach is taken in this review, where heuristics are understood as strategies of discovery less general than a supposed unique logic of discovery but more general than lucky guesses. This article deals with how scientists’ tools shape theories of mind, in particular with how methods of statistical inference have turned into metaphors of mind. The tools-to-theories heuristic explains the emergence of a broad range of cognitive theories, from the cognitive revolution of the 1960s up to the present, and it can be used to detect both limitations and new lines of development in current cognitive theories that investigate the mind as an “intuitive statistician.” Scientific inquiry can be viewed as “an ocean, continuous everywhere and without a break or division ” (Leibniz, 1690/1951, p. 73). Hans Reichenbach (1938) nonetheless divided this ocean into two great seas, the context of discovery and the context of justification. Philosophers, logicians,

Despite the widespread use of key concepts of the Neyman–Pearson (N–P) statistical paradigm—type I and II errors, significance levels, power, confidence levels—they have been the subject of philosophical controversy and debate for over 60 years. Both current and long-standing problems of N–P tests stem from unclarity and confusion, even among N–P adherents, as to how a test’s (pre-data) error probabilities are to be used for (post-data) inductive inference as opposed to inductive behavior. We argue that the relevance of error probabilities is to ensure that only statistical hypotheses that have passed severe or probative tests are inferred from the data. The severity criterion supplies a meta-statistical principle for evaluating proposed statistical inferences, avoiding classic fallacies from tests that are overly sensitive, as well as those not sensitive enough to particular errors and discrepancies.

Conformal prediction uses past experience to determine precise levels of confidence in new predictions. Given an error probability ε, together with a method that makes a prediction ˆy of a label y, it produces a set of labels, typically containing ˆy, that also contains y with probability 1 − ε. Conformal prediction can be applied to any method for producing ˆy: a nearest-neighbor method, a support-vector machine, ridge regression, etc. Conformal prediction is designed for an on-line setting in which labels are predicted successively, each one being revealed before the next is predicted. The most novel and valuable feature of conformal prediction is that if the successive examples are sampled independently from the same distribution, then the successive predictions will be right 1 − ε of the time, even though they are based on an accumulating data set rather than on independent data sets. In addition to the model under which successive examples are sampled independently, other on-line compression models can also use conformal prediction. The widely used Gaussian linear model is one of these. This tutorial presents a self-contained account of the theory of conformal prediction and works through several numerical examples. A more comprehensive treatment of the topic is provided in

Methods of sample size planning are developed from the accuracy in parameter approach in the multiple regression context in order to obtain a sufficiently narrow confidence interval for the population squared multiple correlation coefficient when regressors are random. Approximate and exact methods are developed that provide necessary sample size so that the expected width of the confidence interval will be sufficiently narrow. Modifications of these methods are then developed so that necessary sample size will lead to sufficiently narrow confidence intervals with no less than some desired degree of assurance. Computer routines have been developed and are included within the MBESS R package so that the methods discussed in the article can be implemented. The methods and computer routines are demonstrated using an empirical example linking innovation in the health services industry with previous innovation, personality factors, and group climate characteristics.

Bayes (1763) introduced the observed likelihood function to statistical inference and provided a weight function to calibrate the parameter; he also introduced a confidence distribution on the parameter space but restricted attention to models now called location models; of course the names likelihood and confidence did not appear until much later: Fisher (1922) for likelihood and Neyman (1937) for confidence. Lindley (1958) showed that the Bayes and the confidence results were different when the model was not location. This paper examines the occurrence of true statements from the Bayes approach and from the confidence approach, and shows that the proportion of true statements in the Bayes case depends critically on the presence of linearity in the model; and with departure from this linearity the Bayes approach can be seriously misleading. Bayesian integration of weighted likelihood provides a first order linear approximation to confidence, but without linearity can give substantially incorrect results. 1

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While Bayes’ theorem has a 250-year history, and the method of inverse probability that flowed from it dominated statistical thinking into the twentieth century, the adjective “Bayesian” was not part of the statistical lexicon until relatively recently. This paper provides an overview of key Bayesian developments, beginning with Bayes’ posthumously published 1763 paper and continuing up through approximately 1970, including the period of time when “Bayesian” emerged as the label of choice for those who advocated Bayesian methods.

In time series regression with nonparametrically autocorrelated errors, it is now standard empirical practice to construct con…dence intervals for regression coe ¢ cients on the basis of nonparametrically studentized t-statistics. The standard error used in the studentization is typically estimated by a kernel method that involves some smoothing process over the sample autocovariances. The underlying parameter (M) that controls this tuning process is a bandwidth or truncation lag and it plays a key role in the …nite sample properties of tests and the actual coverage properties of the associated con…dence intervals. The present paper develops a bandwidth choice rule for M that optimizes the coverage accuracy of interval estimators in the context of linear GMM regression. The optimal bandwidth balances the asymptotic variance with the asymptotic bias of the robust standard error estimator. This approach contrasts with the conventional bandwidth choice rule for nonparametric estimation where the focus is the nonparametric quantity itself and the choice rule balances asymptotic variance with squared asymptotic bias. It turns out that the optimal bandwidth for interval estimation has a di¤erent expansion rate and is typically substantially larger