The Fermi's Bayes Theorem

It is curious to learn that Enrico Fermi knew how to base probabilistic inference on Bayes theorem, and that some influential notes on statistics for physicists stem from what the author calls elsewhere, but never in these notes, {\it the Bayes Theorem of Fermi}. The fact is curious because the large majority of living physicists, educated in the second half of last century -- a kind of middle age in the statistical reasoning -- never heard of Bayes theorem during their studies, though they have been constantly using an intuitive reasoning quite Bayesian in spirit. This paper is based on recollections and notes by Jay Orear and on Gauss' ``Theoria motus corporum coelestium'', being the {\it Princeps mathematicorum} remembered by Orear as source of Fermi's Bayesian reasoning.Comment: 4 pages, to appear in the Bulletin of the International Society of Bayesian Analysis (ISBA). Related links and documents are available in http://www.roma1.infn.it/~dagos/history

Bayes' theorem and quantum retrodiction

We derive on the basis of Bayes' theorem a simple but general expression for the retrodicted premeasurement state associated with the result of any measurement. The retrodictive density operator is the normalized probability operator measure element associated with the result. We examine applications to quantum optical cryptography and to the optical beam splitter

Statistical Science and Philosophy of Science: Whrer Do / Should They Meet in 2011 (and Beyond)?

philosophy of science, philosophy of statistics, decision theory, likelihood, subjective probability, Bayesianism, Bayes theorem, Fisher, Neyman and Pearson, Jeffreys, induction, frequentism, reliability, informativeness

Universal efficiency at optimal work with Bayesian statistics

If the work per cycle of a quantum heat engine is averaged over an appropriate prior distribution for an external parameter $a$, the work becomes optimal at Curzon-Ahlborn efficiency. More general priors of the form $\Pi(a) \propto 1/a^{\gamma}$ yield optimal work at an efficiency which stays close to CA value, in particular near equilibrium the efficiency scales as one-half of the Carnot value. This feature is analogous to the one recently observed in literature for certain models of finite-time thermodynamics. Further, the use of Bayes' theorem implies that the work estimated with posterior probabilities also bears close analogy with the classical formula. These findings suggest that the notion of prior information can be used to reveal thermodynamic features in quantum systems, thus pointing to a new connection between thermodynamic behavior and the concept of information.Comment: revtex4, 5 pages, abstract changed and presentation improved; results unchanged. New result with Bayes Theorem adde

Bayes and empirical-Bayes multiplicity adjustment in the variable-selection problem

This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham's-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable differences between the two approaches are found. In particular, we prove a theorem that characterizes a surprising aymptotic discrepancy between fully Bayes and empirical Bayes. This discrepancy arises from a different source than the failure to account for hyperparameter uncertainty in the empirical-Bayes estimate. Indeed, even at the extreme, when the empirical-Bayes estimate converges asymptotically to the true variable-inclusion probability, the potential for a serious difference remains.Comment: Published in at http://dx.doi.org/10.1214/10-AOS792 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org