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Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition. [ABSTRACT FROM AUTHOR]

The Sleeping Beauty problem has spawned a debate between "thirders" and "halfers" who draw conflicting conclusions about Sleeping Beauty's credence that a coin lands heads. Our analysis is based on a probability model for what Sleeping Beauty knows at each time during the experiment. We show that conflicting conclusions result from different modeling assumptions that each group makes. Our analysis uses a standard "Bayesian" account of rational belief with conditioning. No special handling is used for self-locating beliefs or centered propositions. We also explore what fair prices Sleeping Beauty computes for gambles that she might be offered during the experiment. [ABSTRACT FROM AUTHOR]

The concept of coherent probabilities and conditional probabilities through a gambling argument and through a parallel argument based on a quadratic scoring rule was introduced by de Finetti (de Finetti, B. 1974. The Theory of Probability. John Wiley & Sons, New York). He showed that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules that might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring rule can replace the quadratic scoring rule while preserving the equivalence of de Finetti's two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti's fundamental theorem of probability to deal with conditional probabilities. [ABSTRACT FROM AUTHOR]

A class of inferential procedures (called user-oriented procedures) related to the equal probability test of Arrow (1960) is introduced for use in many univariate problems. These procedures require that experimenters be able to interpret the parameters about which they wish to make inference. The class of procedures include confidence intervals associated with the equal probability test as well as an analogue to the P value. The confidence intervals are easier to interpret than the usual confidence intervals, especially in the case of multiple comparisons. Limitations of user-oriented procedures are also considered. [ABSTRACT FROM AUTHOR]

Expounds on P values or significance probabilities. Alternative to hypothesis tests; Interval hypothesis; Logical flaws in the use of P values as evidence for hypotheses.

Explanation and examples are given to show how Milton's (1972) multivariate normal algorithm can give errors exceeding its supposed error bound. Theory is worked out to give a true bound on the error in calculating probabilities of rectangular regions. The resulting algorithms are tested for time and accuracy, and they are made available to potential users. [ABSTRACT FROM AUTHOR]

Presents a statistical algorithm for the computation of the non-central studentized maximum and related multiple-t probabilities. Cases wherein the algorithm can be used; Description and numerical method; Restrictions, time and accuracy.

The article presents the author's comments on an article on a one-sided goodness-of-fit test for an interesting problem involving multinomial proportions. Researcher I. Greenberg presents a likelihood ratio test for an interesting problem involving multinomial proportions. The solution was difficult to obtain and very complicated to describe, despite the extreme simplicity of the statement of the problem. But it is very often the case in statistics and mathematics that a problem that is simple to state requires a very complicated solution. In the problem considered by Greenberg, however, there is a solution of simplicity equal to that of the problem's statement if one adopts the Bayesian approach. Advantages of using the Bayesian approach in the problem are clear. First, the solution is as straightforward and simple as the problem. Second, the solution is easy to interpret as a degree of confidence in the hypothesis after seeing the data. There is no need to draw convoluted conclusions of the form "no sufficient evidence exists to reject the null hypothesis, although some indications of its falsity exist" or "the selection rule is consistent with each protected group's representation in the population but overrepresentation of one or more of the protected groups will be tolerated."

In this article, the author presents his views on a paper by David Oakes on statistical analysis. Oakes has shown that for each forecasting system, there exists a sequence of outcomes for which the system will not be calibrated. This implies that no forecasting system can be self-calibrating in the sense of A. P. Dawid. The situation with regard to the existence of calibrated forecasters is much more serious than either Dawid or Oakes makes it out to be. There are enough noncalibrable sequences so that noncalibrability is as much the normal state of affairs as calibrability. According to Oakes' theorem, the cardinality of the set of noncalibrable sequences is that of the continnum. There are at least two ways to interpret the preceding theorem and Oakes's result. The first is to claim that it is impossible to guarantee the existence of "empirically valid" (i.e., calibrated) sequential forecasts. A more reasonable interpretation is simply that these results are more evidence that (long run) calibration is not a useful measure of the goodness of forecasts.