Search

We consider how an unconditional, finite-valued, finitely additive probability P on a countable set may localize its non-conglomerability (non-disintegrability). Non-conglomerability, a characteristic of merely finitely additive probability, occurs when the unconditional probability of an event P(E) lies outside the closed interval of conditional probability values, [infhe pi P(E|h), suphe pp(E|h)], taken from a countable partition p = hj:j=1,...}. The problem we address is how to identify events and partitions where a finite-valued, finitely additive probability fails to satisfy conglomerability. We focus on the extreme case of 2-valued finitely additive probabilities that are not countably additive. These are, equivalently, non-principal ultrafilters. Evidently, the challenge we face is that given a countable partition, at most one of its elements has positive probability under P. Thus, we must find ways of regulating the coherent conditional probabilities, given null events, that cohere with the unconditional probability P. Our analysis of P proceeds by the use of combinatorial properties of the associated non-principal ultrafilter UP. We show that when ultrafilter UP is not minimal in the Rudin-Keisler partial order of b(w)\w, we may locate a partition in which P fails to satisfy the conglomerability principle by examining (at most) countably many partitions. This result is then applied to finitely additive probabilities that assume only finitely many values. By contrast, if ultrafilter UP is Rudin-Keisler minimal, then P is simultaneously conglomerable in each finite collection of partitions, though not simultaneously conglomerable in all partitions.

Bayesian hierarchical models are built to fit multiple health endpoints from a dose-response study of a toxic chemical, perchlorate. Perchlorate exposure results in iodine uptake inhibition in the thyroid, with health effects manifested by changes in blood hormone concentrations and histopathological effects on the thyroid. We propose linked empirical models to fit blood hormone concentration and thyroid histopathology data for rats exposed to perchlorate in the 90-day study of Springborn Laboratory Inc. (1998), based upon the assumed toxicological relationships between dose and the various endpoints. All of the models are fit in a Bayesian framework, and predictions about each endpoint in response to dose are simulated based on the posterior predictive distribution. A hierarchical model tries to exploit possible similarities between different combinations of sex and exposure duration, and it allows us to produce more stable estimates of dose-response curves. We also illustrate how the hierarchical model allows us to address additional questions that arise after the analysis.

Posterior consistency can be thought of as a theoretical justification of the Bayesian method. One of the most popular approaches to nonparametric Bayesian regression is to put a nonparametric prior distribution on the unknown regression function using Gaussian processes. In this paper, we study posterior consistency in nonparametric regression problems using Gaussian process priors. We use an extension of the theorem of Schwartz (1965) for nonidentically distributed observations, verifying its conditions when using Gaussian process priors for the regression function with normal or double exponential (Laplace) error distributions. We define a metric topology on the space of regression functions and then establish almost sure consistency of the posterior distribution. Our metric topology is weaker than the popular L1 topology. With additional assumptions, we prove almost sure consistency when the regression functions have L1 topologies. When the covariate (predictor) is assumed to be a random variable, we prove almost sure consistency for the joint density function of the response and predictor using the Hellinger metrics.

We contrast three decisions rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: -Maximin, Maximality, and -admissibility. The rules extend Expected Utility theory as they require that an option is inadmissible if there is another that carries greater expected utility for each probability in a (closed) convex set. If the convex set is a singleton, then each rule agrees with maximizing expected utility. We show that, even when the option set is convex, this pairwise comparison between acts may fail to identify those acts which are Bayes for some probability in a convex set that is not closed. This limitation affects two of the decision rules but not -admissibility, which is not a pairwise decision rule. -admissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes.

Maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on Gaussian processes. However, calculating the likelihood can be computationally infeasible for large datasets, requiring O(n^3) observations. This article proposes the method of covariance tapering to approximate the likelihood in this setting. In this approach, covariance matrices are ``tapered,'' or multiplied element-wise by a sparse correlation matrix. The resulting matrices can then be manipulated using efficient sparse matrix algorithms. We propose two approximations to the Gaussian likelihood using tapering. One simply replaces the model covariance with a tapered version; the other is motivated by the theory of unbiased estimating equations. Focusing on the particular case of the Matérn class of covariance functions, we give conditions under which estimators maximizing the tapering approximations are, like the maximum likelihood estimator, strongly consistent. Moreover, we show in a simulation study that the tapering estimators can have sampling densities quite similar to that of the maximum likelihood estimate, even when the degree of tapering is severe. We illustrate the accuracy and computational gains of the tapering methods in an analysis of yearly total precipitation anomalies at weather stations in the United States.

We extend de Finetti’s (1974) theory of coherence to apply also to unbounded random variables. We show that for random variables with mandated infinite prevision, such as for the St. Petersburg gamble, coherence precludes indifference between equivalent random quantities. That is, we demonstrate when the prevision of the difference between two such equivalent random variables must be positive. This result conflicts with the usual approach to theories of Subjective Expected Utility, where preference is defined over lotteries. In addition, we explore similar results for unbounded variables when their previsions, though finite, exceed their expected values, as is permitted within de Finetti’s theory. In such cases, the decision maker’s coherent preferences over random quantities is not even a function of probability and utility. One upshot of these findings is to explain further the differences between Savage’s theory (1954), which requires bounded utility for non-simple acts, and de Finetti’s theory, which does not. And it raises a question whether there is a theory that fits between these two.

It has long been known that the practice of testing all hypotheses at the same level (such as 0.05), regardless of the distribution of the data, is not consistent with Bayesian expected utility maximization. According to de Finetti’s “Dutch Book” argument, procedures that are not consistent with expected utility maximization are incoherent and they lead to gambles that are sure to lose no matter what happens. In this paper, we use a method to measure the rate at which incoherent procedures are sure to lose, so that we can distinguish slightly incoherent procedures from grossly incoherent ones. We present an analysis of testing a simple hypothesis against a simple alternative as a case‐study of how the method can work

The degree of incoherence, when previsions are not made in accordance with a probability measure, is measured by the rate at which an incoherent bookie can be made a sure loser. Each bet is rescaled by one of several normalizations to account for the overall sizes of bets. For each normalization, the sure loss for incoherent previsions is divided by the normalization to determine the rate of incoherence. We study several properties of normalizations and degrees of incoherence and present some examples. Potential applications include the degree of incoherence of classical statistical procedures.