Decision Theory Lecture 2: a short report

During today’s class Dr Arif Ahmed went through some practical and philosophical issues related to decision theory, as presented by Savage’s model, starting from the concept of utility function. From a descriptive point of view, convenient assumptions over the form (concavity) of the utility function allow us to illustrate agents’ attitude towards risk. E.g., in the case of insurance, risk aversion can explain why it could be rational to willingly accept a certain monetary loss. Similar considerations can be applied to time-dependent preferences, i.e. the fact that agents prefer to obtain the same gain (like an amount of money) now than in the future, and that they are sensitive to future losses but indifferent to past losses. In oder to reckon with this psychological feature, the utility function is assumed to be a declining function of future time and a constant function of past time. But what about the normative side of the matter? As for risk aversion, it depends on the attitude of the agents involved. This led to more significant philosophical considerations.

Decision Theory Lecture 1: a short report

So yesterday we had the first lecture of the minicourse Foundations and Applications of Decision Theory provided by Dr Arif Ahmed. The audience was large and heterogeneous: philosophers, economists, social scientists, mathematicians. There were historians and linguists too!
We were introduced to the objects and the aim of decision theory and to the standard approach (due to Savage). The key-concepts of the theory are acts, states and outcomes and they are in the following relation: under certain circumstances (a given state), an act produces a certain outcome. An agent has a set of preferences on (all!) the possible acts, based on her beliefs and desires. Adopting an empiricist spirit, we might say that we can work out the preferences of the agent from her behaviour: what the agent do, that is her choices, tell us something (actually, everything!) about her preferences. In turn preferences, if satisfying certain axioms, allow us to extract agent’s degrees of belief or personal probabilities on the states, that is we can measure the agent’s confidence in each single state on the basis of her preferences (choices). Finally, a very interesting point is that preferences might be represented in terms of a utility function. This kind of function allows us to measure the expected value of an act (its desirability), in other terms it gives a numerical value to each outcome. The central result can be stated as follows: any agent whose preferences satisfy certain axioms may be represented as someone who maximizes expected utility.

Questo mese in ragionamento incerto

DA “THE REASONER“, Volume 7, Number 4 – April 2013

M. G. Kendall (1956: “Studies in the History of Probability and Statistics: II. The Beginnings of a Probability Calculus”, Biometrika, Vol. 43, No. 1/2 pp. 1-14) notes that

If any justification for the study of the history of probability and
statistics were required, it would be found simply and abundantly in
this, that a knowledge of the development of the subject would have
rendered superfluous much of what has been written about it in the
last thirty years.

Kendall’s contention is that the “doctrine of chance” should have been kept separate from the “art of conjecture”, as it used to be before Jacob Bernoulli recommended the application of the former to the analysis of the latter. This separation, in Kendall’s view, would have spared the foundations and applications of probability and statistics the “confusion [that ] has existed ever since and at the present time seems, if anything, to be getting worse”.

Questo mese in ragionamento incerto

DA “THE REASONER“, Volume 7, Number 3 – March 2013

Whilst the field of uncertain reasoning is readily associated with the quantification of uncertainty, the importance of some qualitative approaches to the problem should not be underestimated. Artificial intelligence is perhaps the field where the qualitative vs quantitative contrast is most familiar, as witnessed by the hugely successful ECSQARU conference series. Qualitative counterparts of rational degrees of belief are well-investigated also in the foundations of probability and statistical decision making. In his seminal Dutch-book theorem paper, de Finetti (1931: Sul significato soggettivo della probabilità. Fundamenta Mathematicae, 17, 289–329.) explicitly mentions the intuitive appeal of the qualitative notion “no less probable then” , on which –he claims– the whole foundations of probability can rest. In later work de Finetti went so far as to conjecture that the standard axiomatisation of   would be sufficient to define a (quantitative) probability measure representing  . Such a conjecture was proved false by C. Kraft, J. Pratt, and A. Seidenerg (1959: Intuitive Probability On Finite Sets. The Annals of Mathematical Statistics, 30(2), 408–419). However, this didn’t prevent the qualitative approach to rational degrees of belief from playing a major role in the subjectivist reaction to the sort of choice problem made famous by D. Ellsberg. See, for instance P. Fishburn (1983). Ellsberg revisited: A new look at comparative probability. The Annals of Statistics, 11(4), 1047–1059) which also provides an excellent access point to the fascinating literature on qualitative probability. With many pressing problems (from finance to climate) apparently resisting a robust probabilistic quantification of uncertainty, the concept of a qualitative measure of uncertainty is no less appealing today than it was at the beginning of the 1930s, when de Finetti regarded it as nothing but the intuitive notion of logical consistency formalised.

Against this background, I find it extremely interesting that A.C. Paseau articulates a quantitative measure of paradoxes in his (Paseau 2013: An exact measure of paradox. Analysis, 73 (1)).

Paseau assumes that a paradox amounts to a set of individually plausible premisses which jointly lead to an implausible, yet not necessarily (logically) inconsistent, conclusion. Thus a paradox is taken, in strict adherence with the origin of the word, to be the object of people’s wonder to the extent that it goes beyond their beliefs. Since belief is best characterised as coming in degrees, Paseau offers a quantitative characterisation of what it means for a set of propositions (or assertions, or principles, etc.) to be paradoxical. The formal definition is based on the idea that a paradox arises when the “collective degree of belief” in a set of propositions is in some sense lower than the aggregate of the individual degrees of belief. Further elements enter in the exact measure of paradox, for which I refer to the paper.

One thing which I find very intriguing is how Paseau’s measure relates with the subjective approach to the quantification of uncertainty. His framework does not require degrees of belief to be probabilities. But an interesting consequence follows from the assumption that they are, namely that probabilistic consistency is more fundamental, from a logical point of view, than logical consistency. This, in turn, suggests some sort of reversal of de Finetti’s grounding of the consistency of subjective degrees of belief in the notion of logical consistency. To some, this may sounds counterintuitive – after all, it is very reassuring to think of uncertain reasoning as a proper generalisation of classical logic. Yet this reversal certainly poses an interesting challenge to the basic assumptions underlying the ongoing work on qualitative probability mentioned above.