5,364 research outputs found
Measure Recognition Problem
This is an article in mathematics, specifically in set theory. On the example
of the Measure Recognition Problem (MRP) the article highlights the phenomenon
of the utility of a multidisciplinary mathematical approach to a single
mathematical problem, in particular the value of a set-theoretic analysis. MRP
asks if for a given Boolean algebra \algB and a property of measures
one can recognize by purely combinatorial means if \algB supports a strictly
positive measure with property . The most famous instance of this problem
is MRP(countable additivity), and in the first part of the article we survey
the known results on this and some other problems. We show how these results
naturally lead to asking about two other specific instances of the problem MRP,
namely MRP(nonatomic) and MRP(separable). Then we show how our recent work D\v
zamonja and Plebanek (2006) gives an easy solution to the former of these
problems, and gives some partial information about the latter. The long term
goal of this line of research is to obtain a structure theory of Boolean
algebras that support a finitely additive strictly positive measure, along the
lines of Maharam theorem which gives such a structure theorem for measure
algebras
Diversification Preferences in the Theory of Choice
Diversification represents the idea of choosing variety over uniformity.
Within the theory of choice, desirability of diversification is axiomatized as
preference for a convex combination of choices that are equivalently ranked.
This corresponds to the notion of risk aversion when one assumes the
von-Neumann-Morgenstern expected utility model, but the equivalence fails to
hold in other models. This paper studies axiomatizations of the concept of
diversification and their relationship to the related notions of risk aversion
and convex preferences within different choice theoretic models. Implications
of these notions on portfolio choice are discussed. We cover model-independent
diversification preferences, preferences within models of choice under risk,
including expected utility theory and the more general rank-dependent expected
utility theory, as well as models of choice under uncertainty axiomatized via
Choquet expected utility theory. Remarks on interpretations of diversification
preferences within models of behavioral choice are given in the conclusion
Combinatorial Information Theory: I. Philosophical Basis of Cross-Entropy and Entropy
This study critically analyses the information-theoretic, axiomatic and
combinatorial philosophical bases of the entropy and cross-entropy concepts.
The combinatorial basis is shown to be the most fundamental (most primitive) of
these three bases, since it gives (i) a derivation for the Kullback-Leibler
cross-entropy and Shannon entropy functions, as simplified forms of the
multinomial distribution subject to the Stirling approximation; (ii) an
explanation for the need to maximize entropy (or minimize cross-entropy) to
find the most probable realization; and (iii) new, generalized definitions of
entropy and cross-entropy - supersets of the Boltzmann principle - applicable
to non-multinomial systems. The combinatorial basis is therefore of much
broader scope, with far greater power of application, than the
information-theoretic and axiomatic bases. The generalized definitions underpin
a new discipline of ``{\it combinatorial information theory}'', for the
analysis of probabilistic systems of any type.
Jaynes' generic formulation of statistical mechanics for multinomial systems
is re-examined in light of the combinatorial approach. (abbreviated abstract)Comment: 45 pp; 1 figure; REVTex; updated version 5 (incremental changes
From Wald to Savage: homo economicus becomes a Bayesian statistician
Bayesian rationality is the paradigm of rational behavior in neoclassical economics. A rational agent in an economic model is one who maximizes her subjective expected utility and consistently revises her beliefs according to Bayesâs rule. The paper raises the question of how, when and why this characterization of rationality came to be endorsed by mainstream economists. Though no definitive answer is provided, it is argued that the question is far from trivial and of great historiographic importance. The story begins with Abraham Waldâs behaviorist approach to statistics and culminates with Leonard J. Savageâs elaboration of subjective expected utility theory in his 1954 classic The Foundations of Statistics. It is the latterâs acknowledged fiasco to achieve its planned goal, the reinterpretation of traditional inferential techniques along subjectivist and behaviorist lines, which raises the puzzle of how a failed project in statistics could turn into such a tremendous hit in economics. A couple of tentative answers are also offered, involving the role of the consistency requirement in neoclassical analysis and the impact of the postwar transformation of US business schools.Savage, Wald, rational behavior, Bayesian decision theory, subjective probability, minimax rule, statistical decision functions, neoclassical economics
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