Introduction to the operational subjective theory of probability of Bruno de Finetti

• published: 29 Jun 2012
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Presentation slides: http://www.math.canterbury.ac.nz/~r.sainudiin/primernotes/20120504_WhatIsSubjectiveProbability_FrankLad_PrimerNotes.pdf Details: PRIMER SERIES Friday, 4 May 2012 4.00pm KB04 (Kirkwood Village) Frank Lad (University of Canterbury) What's it all about, Subjective Probability? In this brief Primer I'll talk through the structure of the constructive mathematics of operational subjective statistical methods, in the mode of Bruno de Finetti. We'll address his constructive formulation of probability (more generally, prevision which includes both probability and expectation) via quantities, their realms, coherent prevision, the fundamental theorem of prevision, coherent conditional prevision, conditional quantities, the notion of regarding quantities exchangeably and the representation of exchangeable distributions, sequential forecasting, and the theory of proper scoring rules. Whew, that will be a mouthful in a 30 minute chat!. It should allow you to get a grip on the distinction between the objectivist conception of Statistics as a study of empirical methods for estimating unobservable probabilistic generating structures and the subjectivist conception of Statistics as a procedure for learning about quantities about which you are uncertain from the observation of quantities about which you were uncertain before observing them. Participating students will be offered a choice of the topic of the seminar for the following week: 1. How big is your Mathematics? ... presenting the solution to a famous urn problem you will be asked to solve before the seminar, and its relevance to the fundamental theorem of probability and a billion dollar problem on identifying asbestosis in a lung via x-ray methods. Presumes stage two probability. 2. Why do whales strand when they do? ... presenting a subjectivist statistical analysis of the timing of whale strandings relative to the phases of the moon ... exhibiting an application of exchangeable assessments to mixture trigonometric forecasting distributions based on the New Zealand whale stranding data base. Presumes stage two probability. 3. Completing the logarithmic scoring rule for assessing probability distributions: presenting a leading edge reassessment of the logarithmic scoring rule, with an analysis that motivates the discovery of extropy as a complementary dual of entropy. Presumes stage two probability and basic familiarity with entropy. MOTIVATION: Why is Statistics the most universally detested and distressing course of university students and graduate students throughout the world? Whereas the classical subjectivists, Huyghens, Bernoulli, Montmort, Condorcet, Laplace, and Augustus DeMorgan, who ended the era, were content to assess probability in the words of Laplace as merely "common-sense reduced to mathematics," ... the objectivist revisions since the latter half of the 19th century have constructed a forbidding metaphysical contortion driven by undefinable relations of cause, titillated by in-principle unobservable probabilities, ... dooming statistical practice to such Quixotic ventures as the estimation of the in-principle-unobservable effects of in-principle-unobservable independent variables on an unobservable dependent variable which can only be measured with unobservable error! The unfortunate lecturing statistician is left holding the bag, touting our self-proclaimed powers to sort out these intricacies on the basis of the world of X and Y in which we live, finite and discrete ... under the questioning gaze of 18-year-olds for whom life has wildly departed from the conjured vision of repetitions of identical conditions, and the amazed looks of adults back to school for whom such metaphysical contortions are never going to fly! It is not the lack of effort or pedagogical skill ... IT'S THE CONTENT! Mainstream probability theory and statistical practice, including its epicyclic objective Bayesian appendages, is misleading and chimeric. Operational subjective statistical methods provide a satisfying alternative. What are Primers? Primers in Mathematics and Statistics are short expository introductions to a field of research by someone who is active in the area. Strengthen your research; learn from your colleagues; develop a common language. http://www.math.canterbury.ac.nz/primers