Teddy Seidenfeld

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.

In this paper, we show how to represent a non-Archimedean preference over a set of random quantities by a nonstandard utility function. Non-Archimedean preferences arise when some random quantities have no fair price. Two common situations give rise to non-Archimedean preferences: random quantities whose values must be greater than every real number, and strict preferences between random quantities that are deemed closer in value than every positive real number. We also show how to extend a non-Archimedean preference to a larger set of random quantities. The random quantities that we consider include real-valued random variables, horse lotteries, and acts in the theory of Savage. In addition, we weaken the state-independent utility assumptions made by the existing theories and give conditions under which the utility that represents preference is the expected value of a state-dependent utility with respect to a probability over states.

A familiar defense of Personalist or Subjective Bayesian theory is that, under a variety of sufficient conditions, asymptotically—with increasing shared evidence—almost surely, each non-extreme, countably additive Bayesian opinion, when updated by conditionalization, converges to certainty that is veridical about the truth/falsity of hypotheses of interest. Then, with probability 1 over possible evidential histories, personal probabilities track the truth. In this note we examine varieties of failures of these asymptotics. In an extreme case, conditional probabilities are deceptive when they converge to certainty for a false hypothesis. We establish that proposals for so-called “modest” credences, offered by Elga (2016) and by Nielsen and Stewart (2019) in response to a concern about Bayesian orgulity raised by Belot (2013), instead support deceptive credences. We argue that deceptive credences are not modest, but for a reason different than Belot adduces.

We consider Geanakoplos and Polemarchakis’s generalization of Aumman’s famous result on “agreeing to disagree", in the context of imprecise probability. The main purpose is to reveal a connection between the possibility of agreeing to disagree and the interesting and anomalous phenomenon known as dilation. We show that for two agents who share the same set of priors and update by conditioning on every prior, it is impossible to agree to disagree on the lower or upper probability of a hypothesis unless a certain dilation occurs. With some common topological assumptions, the result entails that it is impossible to agree not to have the same set of posterior probabilities unless dilation is present. This result may be used to generate sufficient conditions for guaranteed full agreement in the generalized Aumman-setting for some important models of imprecise priors, and we illustrate the potential with an agreement result involving the density ratio classes. We also provide a formulation of our results in terms of “dilation-averse” agents who ignore information about the value of a dilating partition but otherwise update by full Bayesian conditioning.

We discuss two general issues concerning diverging sets of Bayesian (conditional) probabilities—divergence of “posteriors”—that can result with increasing evidence. Consider a setof probabilities typically, but not always, based on a set of Bayesian “priors.” Incorporating sets of probabilities, rather than relying on a single probability, is a useful way to provide a rigorous mathematical framework for studying sensitivity and robustness in Classical and Bayesian inference. See: Berger (1984, 1985, 1990); Lavine (1991); Huber and Strassen (1973); Walley (1991); and Wasserman and Kadane (1990). Also, sets of probabilities arise in group decision problems. See: Levi (1982); and Seidenfeld, Kadane, and Schervish (1989). Third, sets of probabilities are one consequence of weakening traditional axioms for uncertainty. See: Good (1952); Smith (1961); Kyburg (1961); Levi (1974); Fishburn (1986); Seidenfeld, Schervish, and Kadane (1990); and Walley (1991).

In the longstanding foundational debate whether to require that probability is countably additive, in addition to being finitely additive, those who resist the added condition raise two concerns that we take up in this paper. (1) Existence: Settings where no countably additive probability exists though finitely additive probabilities do. (2) Complete Additivity: Where reasons for countable additivity don’t stop there. Those reasons entail complete additivity—the (measurable) union of probability 0 sets has probability 0, regardless the cardinality of that union. Then probability distributions are discrete, not continuous. We use Easwaran’s (Easwaran, Thought 2:53–61, 2013) advocacy of the Comparative principle to illustrate these two concerns. Easwaran supports countable additivity, both for numerical probabilities and for finer, qualitative probabilities, by defending a condition he calls the Comparative principle [C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}]. For numerical probabilities, principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} contrasts pairs, P1 and P2, defined over a common partition ∏\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod$$\end{document} = {ai: i ∈ I} of measurable events. C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} requires that no P1 may be pointwise dominated, i.e., no (finitely additive) probability P2 exists such that for each i ∈ I, P2(ai) > P1(ai). By design, the cardinality of ∏\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\prod$$\end{document} is not limited in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}, which Easwaran asserts is important when arguing that the principle does not require more, or less, than that probability is countably additive. We agree that a numerical probability P satisfies principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} in all partitions just in case P is countably additive. However, we show that for numerical probabilities, by considering the size of the algebra of events to which probability is applied, principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} is subject to each of the above concerns, (1) and (2). Also, Easwaran considers principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} with non-numerical, qualitative probabilities, where a qualitative probability may be finer than an almost agreeing numerical probability P. A qualitative probability is regular if possible events are strictly more likely than impossible events. Easwaran motivates and illustrates regular qualitative probabilities using a continuous, almost agreeing quantitative probability that is uniform on the unit interval. We make explicit the conditions for applying principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} with qualitative probabilities and show that C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document} restricts regular qualitative probabilities to those whose almost agreeing quantitative probabilities are completely additive. For instance, Easwaran’s motivating example of a regular qualitative probability is precluded by principle C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}}$$\end{document}.

The “Dutch Book” argument, tracing back to Ramsey (1926) and deFinetti (1974), offers prudential grounds for action in conformity with personal probability. Under several structural assumptions about combinations of stakes (that is, assumptions about the combination of wagers), your betting policy is consistent (coherent) only if your fair-odds are probabilities. The central question posed here is the following one: Besides providing an operational test of coherent betting, does the “Book” argument also provide for adequate measurement (elicitation) of the agent’s degrees of beliefs? That is, are an agent’s fairodds also his/her personal probabilities for those events?We argue the answer is “No!” The problem is created by state-dependent utilities. The methods of elicitation proposed by Ramsey, by deFinetti and by Savage (1954), are inadequate to the challenge of state-dependent values.1.

This paper has two main parts. In the first part, we motivate a kind of indeterminate, suppositional credences by discussing the prospect for a subjective interpretation of a causal Bayesian network, an important tool for causal reasoning in artificial intelligence. A CBN consists of a causal graph and a collection of interventional probabilities. The subjective interpretation in question would take the causal graph in a CBN to represent the causal structure that is believed by an agent, and interventional probabilities in a CBN to represent suppositional credences. We review a difficulty noted in the literature with such an interpretation, and suggest that a natural way to address the challenge is to go for a generalization of CBN that allows indeterminate credences. In the second part, we develop a decision-theoretic foundation for such indeterminate suppositional credences, by generalizing a theory of coherent choice functions to accommodate some form of act-state dependence. The upshot is a decision-theoretic framework that is not only rich enough to, so to speak, ground the probabilities in a subjectively interpreted causal network, but also interesting in its own right, in that it accommodates both act-state dependence and imprecise probabilities.

In his recent monograph [7], Professor Shafer has offered us an alternative to Bayesian inference with his novel theory of belief functions and, in his current paper [8], has characterized his position by pointing to two basic differences it shares with Bayesianism. First, belief functions are non-additive so that the degree of belief assigned to the disjunction ‘A1 or A2’ may be larger than the sum of the degrees of belief assigned to the separate disjuncts. Second, the theory of belief functions has its own rule for determining the commitments to changes in degrees of belief when evidence is compounded. So, instead of the Bayesian postulate of conditionalization, that is, in place of using Bayes’ theorem to identify the commitments to changes in probability when evidence accumulates, the theory advocated by Professor Shafer relies on a proposal he traces to A.P.Dempster, which he calls Dempster’s rule for combination of belief functions.

We study the representation of sets of desirable gambles by sets of probability mass functions. Sets of desirable gambles are a very general uncertainty model, that may be non-Archimedean, and therefore not representable by a set of probability mass functions. Recently, Cozman (2018) has shown that imposing the additional requirement of even convexity on sets of desirable gambles guarantees that they are representable by a set of probability mass functions. Already more that 20 years earlier, Seidenfeld et al. (1995) gave an axiomatisation of binary preferences—on horse lotteries, rather than on gambles—that also leads to a unique representation in terms of sets of probability mass functions. To reach this goal, they use two devices, which we will call ‘SSK–Archimedeanity’ and ‘SSK–extension’. In this paper, we will make the arguments of Seidenfeld et al. (1995) explicit in the language of gambles, and show how their ideas imply even convexity and allow for conservative reasoning with evenly convex sets of desirable gambles, by deriving an equivalence between the SSK–Archimedean natural extension, the SSK–extension, and the evenly convex natural extension.

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.

This important collection of essays is a synthesis of foundational studies in Bayesian decision theory and statistics. An overarching topic of the collection is understanding how the norms for Bayesian decision making should apply in settings with more than one rational decision maker and then tracing out some of the consequences of this turn for Bayesian statistics. There are four principal themes to the collection: cooperative, non-sequential decisions; the representation and measurement of 'partially ordered' preferences; non-cooperative, sequential decisions; and pooling rules and Bayesian dynamics for sets of probabilities. The volume will be particularly valuable to philosophers concerned with decision theory, probability, and statistics, statisticians, mathematicians, and economists.

Gordon Belot argues that Bayesian theory is epistemologically immodest. In response, we show that the topological conditions that underpin his criticisms of asymptotic Bayesian conditioning are self-defeating. They require extreme a priori credences regarding, for example, the limiting behavior of observed relative frequencies. We offer a different explication of Bayesian modesty using a goal of consensus: rival scientific opinions should be responsive to new facts as a way to resolve their disputes. Also we address Adam Elga’s rebuttal to Belot’s analysis, which focuses attention on the role that the assumption of countable additivity plays in Belot’s criticisms.

SummaryThe objective Bayesian program has as its fundamental tenet (in addition to the three Bayesian postulates) the requirement that, from a given knowledge base a particular probability function is uniquely appropriate. This amounts to fixing initial probabilities, based on relatively little information, because Bayes' theorem (conditionalization) then determines the posterior probabilities when the belief state is altered by enlarging the knowledge base. Moreover, in order to reconstruct orthodox statistical procedures within a Bayesian framework, only privileged ‘ignorance’ probability functions will work.To serve all these ends objective Bayesianism seeks additional principles for specifying ‘ignorance’ and ‘partial information’ probabilities. H. Jeffreys' method of invariance (or Jaynes' modification thereof) is used to solve the former problem, and E. Jaynes' rule of maximizing entropy (subject to invariance for continuous distributions) has recently been thought to solve the latter. I have argued that neither policy is acceptable to a Bayesian since each is inconsistent with conditionalization. Invariance fails to give a consistent representation to the state of ignorance professed. The difficulties here parallel familiar weaknesses in the old Laplacean principle of insufficient reason. Maximizing entropy is unsatisfactory because the ‘partial information’ it works with fails to capture the effect of uncertainty about related nuisance factors. The result is a probability function that represents a state richer in empirical content than the belief state targeted for representation. Alternatively, by conditionalizing on information about a nuisance parameter one may move from a distribution of lower to higher entropy, despite the obvious increase in information available.Each of these two complaints appear to me to be symptoms of the program's inability to formulate rules for picking privileged probability distributions that serve to represent ignorance or near ignorance. Certainly the methods advocated by Jeffreys, Jaynes and Rosenkrantz are mathematically convenient idealizations wherein specified distributions are elevated to the roles of ‘ignorance’ and ‘partial information’ distributions. But the cost that goes with the idealization is a violation of conditionalization, and if that is the ante that we must put up to back objective Bayesianism then I propose we look for a different candidate to earn our support.31.

It is a familiar argument that advocates accommodating the so-called paradoxes of decision theory by abandoning the “independence” postulate. After all, if we grant that choice reveals preference, the anomalous choice patterns of the Allais and Ellsberg problems violate postulate P2 of Savage's system. The strategy of making room for new preference patterns by relaxing independence is adopted in each of the following works: Samuelson, Kahneman and Tversky's “Prospect Theory”, Allais and Hagen, Fishburn, Chew and MacCrimmon, McClennen, and in closely argued essays by Machina

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.

The "Dutch Book" argument, tracing back to Ramsey and to deFinetti, offers prudential grounds for action in conformity with personal probability. Under several structural assumptions about combinations of stakes, your betting policy is coherent only if your fair odds are probabilities. The central question posed here is the following one: Besides providing an operational test of coherent betting, does the "Book" argument also provide for adequate measurement of the agents degrees of beliefs? That is, are an agent's fair odds also his/her personal probabilities for those events? We argue the answer is "No!" The problem is caused by the possibility of state dependent utilities