Personal probabilities of probabilities

DeGroot, Morris H.; Marschak, Jacob; Borch, Karl; Chernoff, Herman; Groot, Morris De; Dorfman, Robert; Edwards, Ward; Ferguson, Thomas S.; Miyasawa, Koichi; Randolph, Paul H.; Savage, Leonard J.; Schlaifer, Robert; Winkler, Robert L.

doi:10.1007/bf00169102

Cited by 59 publications

(24 citation statements)

References 8 publications

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“…What can be expressed consist of fuzzy probabilities (as surprisingly acknowledged even by Luce and Raiffa [89] [7], and De Cooman [20]). Some authors even propose higher-order probabilities (for instance, Marschak [90]), which sounds like recursively solving a problem by creating the same problem one step higher.…”

Section: 3mentioning

confidence: 99%

Formal Representations of Uncertainty

Dubois¹,

Prade²

2009

Decision‐making Process

104

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Section: 3mentioning

confidence: 99%

Formal Representations of Uncertainty

Dubois¹,

Prade²

2009

Decision‐making Process

104

View full text Add to dashboard Cite

“…In fact, the notion of uncertainty about uncertainty has been discussed under a variety of rubrics; e.g., ambiguous probabilities, second-order uncertainty, and probabilities-for-probabilities (e.g., Marschak, 1975). Moreover, current work on fuzzy sets (Zadeh, 1978), Shafer's (1976) theory of evidence, Cohen's (1977) attempt to formalize uncertainty in legal settings, and the elicitation of probability ranges (Wallsten, Forsyth, & Budescu, 1983), all contain ideas regarding the vagueness that can underlie probabilities.…”

Section: The Nxuor Of Uncertamy and Ambigufrymentioning

confidence: 99%

Decision Making Under Ambiguity

Einhorn¹,

Hogarth²

1986

J BUS

489

231

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“…16, in which he asked how to pin down subjective beliefs on models from observables. In so doing, our analysis also shows that to study general data M, possibly linearly dependent, it is necessary to go beyond betting behavior on observables.…”

Section: Representationmentioning

confidence: 99%

“…By the Lyapunov theorem, there is a unanimous event E ∈ Σ, say with mðEÞ = 2 −1 for all m ∈ M. By order consistency, for each F ∈ Σ mðFÞ = mðEÞ ∀m ∈ M ⇒ PðFÞ = PðEÞ [16] and mðFÞ ≥ mðEÞ ∀m ∈ M ⇒ PðFÞ ≥ PðEÞ: [17] By ref. 21, Theorem 20, P belongs to the convex cone generated by M, because PðSÞ = mðSÞ = 1 for all m ∈ M, and then P ∈ coM and representation [9] holds.…”

Section: Appendix: Proofs and Related Analysismentioning

confidence: 99%

Classical subjective expected utility

Cerreia‐Vioglio

Maccheroni

Marinacci

et al. 2013

Proc. Natl. Acad. Sci. U.S.A.

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We consider decision makers who know that payoff-relevant observations are generated by a process that belongs to a given class M, as postulated in Wald [Wald A (1950) Statistical Decision Functions (Wiley, New York)]. We incorporate this Waldean piece of objective information within an otherwise subjective setting à la Savage [Savage LJ (1954) The Foundations of Statistics (Wiley, New York)] and show that this leads to a two-stage subjective expected utility model that accounts for both state and model uncertainty.C onsider a decision maker who is evaluating acts whose outcomes depend on some verifiable states, that is, on observations (workers' outputs, urns' drawings, rates of inflation, and the like). If the decision maker (DM) believes that observations are generated by some probability model, two sources of uncertainty affect his evaluation: model uncertainty and state uncertainty. The former is about the probability model that generates observations, and the latter is about the state that obtains (and that determines acts' outcomes).State uncertainty is payoff relevant and, as such, it is directly relevant for the DM's decisions. Model uncertainty, in contrast, is not payoff relevant and its role is instrumental relative to state uncertainty. Moreover, models cannot always be observed: Whereas in some cases they have a simple physical description (e.g., urns' compositions), often they do not have it (e.g., fair coins). For these reasons, the purely subjective choice frameworks à la Savage (1) focus on the verifiable and payoff-relevant state uncertainty. They posit an observation space S over which subjective probabilities are derived via betting behavior.In contrast, classical statistical decision theory à la Wald (2) assumes that the DM knows that observations are generated by a probability model that belongs to a given subset M, whose elements are regarded as alternative random devices that nature may select to generate observations. [As Wald (ref. 2, p. 1) writes, "A characteristic feature of any statistical decision problem is the assumption that the unknown distribution FðxÞ is merely known to be an element of a given class Ω of distributions functions. The class Ω is to be regarded as a datum of the decision problem."] In other words, Wald's approach posits a model space M in addition to the observation space S. In so doing, Wald adopted a key tenet of classical statistics, that is, to posit a set of possible data-generating processes (e.g., normal distributions with some possible means and variances), whose relative performance is assessed via available evidence [often collected with independent identically distributed (i.i.d.) trials] through maximum-likelihood methods, hypothesis testing, and the like. Although models cannot be observed, in Wald's approach their study is key to better understanding state uncertainty.Is it possible to incorporate this Waldean key piece of objective information within Savage's framework? Our work addresses this question and tries to embed this classical datum wi...

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Personal probabilities of probabilities

Cited by 59 publications

References 8 publications

Formal Representations of Uncertainty

Formal Representations of Uncertainty

Decision Making Under Ambiguity

Classical subjective expected utility

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