Teddy Seidenfeld

The Independence postulate links current preferences between called-off acts with current preferences between constant acts. Under the assumption that the chance-events used in compound von Neumann-Morgenstern lotteries are value-neutral, current preferences between these constant acts are linked to current preferences between hypothetical acts, conditioned by those chance events. Under an assumption of stability of preferences over time, current preferences between these hypothetical acts are linked to future preferences between what are then and there constant acts. Here, I show that a failure of Independence with respect to current preferences leads to an inconsistency in sequential decisions. Two called-off acts are constructed such that each is admissible in the same sequential decision and yet one is strictly preferred to the other. This responds to a question regarding admissibility posed by Rabinowicz ([2000] Preference stability and substitution of indifferents: A rejoinder to Seidenfeld, Theory and Decision 48: 311–318 [this issue])

De Finetti introduced the concept of coherent previsions and conditional previsions through a gambling argument and through a parallel argument based on a quadratic scoring rule. He shows that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules which 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 prevision to deal with conditional previsions.

Tiebreak rules are necessary for revealing indifference in non- sequential decisions. I focus on a preference relation that satisfies Ordering and fails Independence in the following way. Lotteries a and b are indifferent but the compound lottery f, 0.5b> is strictly preferred to the compound lottery f, 0.5a>. Using tiebreak rules the following is shown here: In sequential decisions when backward induction is applied, a preference like the one just described must alter the preference relation between a and b at certain choice nodes, i.e., indifference between a and b is not stable. Using this result, I answer a question posed by Rabinowicz (1997) concerning admissibility in sequential decisions when indifferent options are substituted at choice nodes

This paper (based on joint work with M.J.Schervish and J.B.Kadane) discusses some differences between the received theory of regular conditional distributions, which is the countably additive theory of conditional probability, and a rival theory of conditional probability using the theory of finitely additive probability. The focus of the paper is maximally "improper" conditional probability distributions, where the received theory requires, in effect, that P{a: P(a|a) = 0} = 1. This work builds upon the results of Blackwell and Dubins (1975).

DeFinetti took the concept of random variables as gambles very seriously, and used the concept to motivate the familiar concepts of probability and expectation. For each gamble X, he assumed that “You” would assign a value P (X), called the prevision of X so that you would be willing to accept the gamble β[X − P (X)] as fair for all positive and negative values β. The only constraint that deFinetti envisioned for you and your previsions is that you insisted that there be no positive amount that you had to lose for sure. For example, you would not be allowed to call a gamble fair if its supremum were negative. On the other hand, the criterion is weak enough to allow you call a gamble fair if its supremum is 0, even if all of its possible values are negative.

We extend a result of Dubins [3] from bounded to unbounded random variables. Dubins [3] showed that a finitely additive expectation over the collection of bounded random variables can be written as an integral of conditional expectations (disintegrability) if and only if the marginal expectation is always within the smallest closed interval containing the conditional expectations (conglomerability). We give a sufficient condition to extend this result to the collection Z of all random variables that have finite expected value and whose conditional expectations are finite and have finite expected value. The sufficient condition also allows the result to extend some, but not all, subcollections of Z. We give an example where the equivalence of disintegrability and conglomerability fails for a subcollection of Z that still contains all bounded random variables.

In 1936 R.A.Fisher asked the pointed question, "Has Mendel's Work Been Rediscovered?" The query was intended to open for discussion whether someone altered the data in Gregor Mendel's classic 1866 research report on the garden pea, "Experiments in Plant-Hybridization." Fisher concluded, reluctantly, that the statistical counts in Mendel's paper were doctored in order to create a better intuitive fit between Mendelian expected values and observed frequencies. That verdict remains the received view among statisticians, so I believe. Fisher's analysis is a tour de force of so-called "Goodness of Fit" statistical tests using c2 to calculate significance levels, i.e., P-values. In this presentation I attempt a defense of Mendel's report, based on several themes. (1) Mendel's experiments include some important sequential design features that Fisher ignores. (2) Fisher uses particular statistical techniques of Meta-analysis for pooling outcomes from different experiments. These methods are subject to critical debate. and (3) I speculate on a small modification to Mendelian theory that offers some relief from Fisher's harsh conclusion that Mendel's data are too good to be true.

When real-valued utilities for outcomes are bounded, or when all variables are simple, it is consistent with expected utility to have preferences defined over probability distributions or lotteries. That is, under such circumstances two variables with a common probability distribution over outcomes – equivalent variables – occupy the same place in a preference ordering. However, if strict preference respects uniform, strict dominance in outcomes between variables, and if indifference between two variables entails indifference between their difference and the status quo, then preferences over rich sets of unbounded variables, such as variables used in the St. Petersburg paradox, cannot preserve indifference between all pairs of equivalent variables. In such circumstances, preference is not a function only of probability and utility for outcomes. Then the preference ordering is not defined in terms of lotteries.

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti and Dubins, 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 our result, where we established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition

This paper offers a comparison between two decision rules for use when uncertainty is depicted by a non-trivial, convex2 set of probability functions Γ. This setting for uncertainty is different from the canonical Bayesian decision theory of expected utility, which uses a singleton set, just one probability function to represent a decision maker’s uncertainty. Justifications for using a non-trivial set of probabilities to depict uncertainty date back at least a half century (Good, 1952) and a foreshadowing of that idea can be found even in Keynes’ (1921), where he allows that not all hypotheses may be comparable by qualitative probability – in accord with, e.g., the situation where the respective intervals of probabilities for two events merely overlap with no further (joint) constraints, so that neither of the two events is more, or less, or equally probable compared with the other

On pp. 55–58 of Philosophical Problems of Statistical Inference, I argue that in light of unsatisfactory after-trial properties of “best” Neyman-Pearson confidence intervals, we can strengthen a traditional criticism of the orthodox N-P theory. The criticism is that, once particular data become available, we see that the pre-trial concern for tests of maximum power may then misrepresent the conclusion of such a test. Specifically, I offer a statistical example where there exists a Uniformly Most Powerful test, a test of highest N-P credentials, which generates a system of “best” confidence intervals with exact confidence coefficients. But the [CIλ] intervals have the unsatisfactory feature that, for a recognizable set of outcomes, the interval estimates cover all parameter values consistent with the data, at strictly less than 100% confidence.

This paper examines definitions of independence for events and variables in the context of full conditional measures; that is, when conditional probability is a primitive notion and conditioning is allowed on null events. Several independence concepts are evaluated with respect to graphoid properties; we show that properties of weak union, contraction and intersection may fail when null events are present. We propose a concept of “full” independence, characterize the form of a full conditional measure under full independence, and suggest how to build a theory of Bayesian networks that accommodates null events.

We review de Finetti’s two coherence criteria for determinate probabilities: coherence1defined in terms of previsions for a set of events that are undominated by the status quo – previsions immune to a sure-loss – and coherence2 defined in terms of forecasts for events undominated in Brier score by a rival forecast. We propose a criterion of IP-coherence2 based on a generalization of Brier score for IP-forecasts that uses 1-sided, lower and upper, probability forecasts. However, whereas Brier score is a strictly proper scoring rule for eliciting determinate probabilities, we show that there is no real-valuedstrictly proper IP-score. Nonetheless, with respect to either of two decision rules – Γ-maximin or E-admissibility-+-Γ-maximin – we give a lexicographic strictly proper IP-scoring rule that is based on Brier score.

We contrast three decision rules that extend Expected Utility to contexts where a convex set of probabilities is used to depict uncertainty: Γ-Maximin, Maximality, and E-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 E-admissibility, which is not a pairwise decision rule. E-admissibility can be used to distinguish between two convex sets of probabilities that intersect all the same supporting hyperplanes.

We discuss several features of coherent choice functions—where the admissible options in a decision problem are exactly those that maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty—where only the probability component of S is indeterminate and utility for two privileged outcomes is determinate. Coherent choice distinguishes between each pair of sets of probabilities regardless the “shape” or “connectedness” of the sets of probabilities. We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility.