Bayesian Decision Theory and the Representation of Beliefs

2002

Bayes decision problems require subjective elicitation of inputs: beliefs and preferences. Elicitation methods are far from simple, and elicited quantities cannot perfectly represent the judgements of the Decision Maker. Several foundations suggest overlaying this problem using robust approaches. In these models, beliefs are modelled by a class of probability distributions, and preferences by a class of loss functions. Then we have a Pareto order. So, the solution concept is an efficient set. In this paper, we deal with the problem of existence of a solution. Moreover, we study the relations between Bayes actions and nondominated ones.

Artificial Intelligence, 2004

In the current discussion about the capacity of Bayesianism in reasoning under uncertainty, there is a conceptual and notational confusion between the explicit condition and the implicit condition of a probability evaluation. Consequently, the limitation of Bayesianism is often seriously underestimated. To represent the uncertainty of a belief system where revision is needed, it is not enough to assign a probability value to each belief.

2006

In this paper, Knight's distinction between risk and uncertainty, and its significance for economic analysis are examined. The paper consists of a survey of some recent developments on the theory of choice under uncertainty and some applications of these theories to problems for which Bayesian Decision Theory has not proved entirely satisfactory. Two problems are examined in detail. The first is that of finance and insurance and the second is that of risktaking behaviour with special emphasis on lotteries.

2003

Studies in Cognitive Systems, 1990

Ahstrurl-Questions of consensus among Bayesian investigators from tno perspectives are discussed: I) Inference: What are the agreement\ in posterior probilities that result from increasing shared data? 2) Decisions: What are the shared (strict) preferences of Bayesian decisionmakers? When do their shared preferences support coherent compromises? Concerning topic 1, results about asymptotic consensus and certaint! with increasing evidence are reported. The findings deal with extending a conclusion of Blackwell and Dubins from pairs of agents to larger communities. The discussion is in a setting rather more general than that of existing results, as i.i.d. data is not required. nor are conditional probability given only non-null events restricted. Increasing evidence creates (almost sure) certainty of the truth and, depending upon the size of the communih. it leads to varieties of concensus for conditional (posterior) distributions. Concerning topic 2). we report results on the shared agreements of two Bayesian decision makers who have some differences in their probabilities for events and some differences in their utilities for outcomes. Our results are couched in a setting where acts are "horse lotteries," with "coherence" defined by aiomatic restrictions on preferences over such acts. Subject to a weak Pareto condition, it is established that the only coherent preference schemes are the two agents' preferences themselves. Moreover, stronger Pareto conditions exclude even these extreme solutions. This negative finding is immune to familiar maneuvers with Arrow's "impossibilit>" theorem. In particular, it is insensitive to restrictions on the domain of (pairs of) agents considered. And it is unaffected by allowing "interpersonal utilities" precluded by Arrow's multipartite "Independence of Irrelevant Alternatives" clause. Thus, the answers to the two sets of questions are v e p different. I. INTRODUCTION UESTIONS OF CONSENSUS among Bayesian in-Q vestigators are discussed from two perspectives: 1) Regarding inference: What are the agreements in (posterior) probabilities that result from increasing shared data? When do these agreements converge on practical certainties? 2) Regarding decisions: What are the shared, strict preferences of Bayesian decision makers? When do these shared preferences support "coherent" compromises? Manuscript received February 1. 19x9: revised October 17. 19x9. This work was supported in part by the Buhl Foundation. by National Science Foundation grant numbered DMS-XX05676. and in part by Office of Naval Research contract number N00014-XX-KO013. This work was prcsented at the Illinois Interdisciplinary Workshop on Decision Making. University of Illinois at Champaign-Urbana, June 19XX. I t was previously presented at the Society for Exact Philosophy. University of Rochester. May 1YXX.

AIP Conference Proceedings, 2001

We discuss precise assumptions entailing Bayesianism in the line of investigations started by Cox, and relate them to a recent critique by Halpern. We show that every finite model which cannot be rescaled to probability violates a natural and simple refinability principle. A new condition, separability, was found sufficient and necessary for rescalability of infinite models. We finally characterize the acceptable ways to handle uncertainty in infinite models based on Cox's assumptions. Certain closure properties must be assumed before all the axioms of ordered fields are satisfied. Once this is done, a proper plausibility model can be embedded in an ordered field containing the reals, namely either standard probability (field of reals) for a real valued plausibility model, or extended probability (field of reals and infinitesimals) for an ordered plausibility model. The end result is that if our assumptions are accepted, all reasonable uncertainty management schemes must be based on sets of extended probability distributions and Bayes conditioning.

2018

This work pries into the analytic and systematic approach to the study of decision making in the light of Edwin Jaynes’ Probability Logic. At Jaynes’ time of writing, the speculative sciences were beginning to assume a mathematical bend vis-a-vis the representation of the process of reasoning quantitatively. Within this framework; the use of Logic and Set Theories, was key to understanding the foundations of mathematics. But all mathematical formulation attempts to put forward a consistent and complete representation of the real world, often fails to satisfy Kurt Gödel’s incompleteness theorem. Our mathematical probability model however different, is a powerful analytical tool that guarantees detection of inconsistencies or violations of our other desiderata, derivable from its axioms. Since most Decision problems hinges on uncertainty crisis in enumerating the range of possibilities. How then can one deal with this epistemological inadequacy with its nuances? Jaynes’ Objective-Bayesianism expressed in the MaxEnt principle serves as the foundation of Probability theory for modern science. This is somewhat instrumental in resolving the imaginary distinction between Probability Theory and Statistical Theory. Thus achieving a logical unity, simplicity and flexibility in application of MaxEnt decision model. This resultant apparatus leads to the idea of assigning distributions to represent complete ignorance, and to accommodate new information; showing the most honest description of what we actually know without assuming any information. The decision rule L(D, I) = −log[p(I |D)] curtails information loss, conditions our choice as it is suggestive of decision Di when the specific information I is present. Finally, the essence of any decision rule is that the decision must be made on the basis of observed data, (and all relevant prior knowledge) which contains all the information sufficient to arriving at the decision. However, in spite of the formal simplicity of these rules, the major critique leveled against it is that it lacks practical relevance (for a non-ideal agent in a non-ideal situation), and so worthless, if people actually fail to attain its ideals. I see this as an epistemological recidivism, - a travesty of an ideal on the bases of inadequacies. Its normative outlook contributes a great deal to practical reasoning. So as Jaynes suggests; “a good deal of leading by the hand is necessary here, in order to develop a mastery of method.”

2008

This paper deals with the intellectual environment in which George L. S. Shackle's theory of decision making was formulated and first discussed. Shackle's approach had a great impact on decision theory in late 1940s and early 1950s being the single formalised attempt to discard the probability framework in the description of behaviour under uncertainty-a goal shared by Knight and Keynes. Against Shackle, Arrow defended the use of probability theory in decision making, by denying that the Knightian distinction between risk and uncertainty had any behavioural significance, and paving the way to Savage's Foundations of Statistics as the new mainstream reference. Through an assessment of the reception of Shackle's theory the paper presents the way a number of theoretical economists, psychologists, and mathematicians were interested in the viability of a formally structured alternative to theories of behaviour using probability statements to describe uncertainty. The paper aims to show that the lively but concentrated discussion on alternative decisional criteria Shackle was part of is crucial to understand the multifarious developments observed in modern decision theory in the last twenty years or so. Indeed, as discussed in a twin paper by Basili and Zappia, Shackle's theory was a much more viable alternative to subjective expected utility than both its contemporary critics and modern decision theorists have recognised.

1989

The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co∼event. The theory of experience and chance is the theory of co∼events. To study the co∼events, it is not enough to study the experience and to study the chance. For this, it is necessary to study the experience and chance as a single entire, a co∼event. In other words, it is necessary to study their interaction within a co∼event. The new co∼event axiomatics and the theory of co∼events following from it were created precisely for these purposes. In this work, I am going to demonstrate the effectiveness of the new theory of co∼events in a studying the logic of uncertainty. I will do this by the example of a co∼event splitting of the logic of the Bayesian scheme, which has a long history of fierce debates between Bayesionists and frequentists. I hope the logic of the theory of experience and chance will make its modest contribution to the application of these old dual debaters. Keywords: Eventology, event, probability, probability theory, Kolmogorov’s axiomatics, experience, chance, cause, consequence, co∼event, set of co∼events, bra-event, set of bra-events, ket-event, set of ket-events, believability, certainty, believability theory, certainty theory, theory of co∼events, theory of experience and chance, co∼event dualism, co∼event axiomatics, logic of uncertainty, logic of experience and chance, logic of cause and consequence, logic of the past and the future, Bayesian scheme.