186,807 research outputs found
Shackle versus Savage: non-probabilistic alternatives to subjective probability theory in the 1950s
G.L.S Shackle’s rejection of the probability tradition stemming from Knight's definition of uncertainty was a crucial episode in the development of modern decision theory. A set of methodological statements characterizing Shackle’s stance, abandoned for long, especially after Savage’s Foundations, have been re-discovered and are at the basis of current non-expected utility theories, in particular of the non-additive probability approach to decision making. This paper examines the discussion between Shackle and his critics in the 1950s. Drawing on Shackle’s papers housed at Cambridge University Library as well as on printed matter, we show that some critics correctly understood two aspects of Shackle’s theory which are of the utmost importance in our view: the non-additive character of the theory and the possibility of interpreting Shackle’s ascendancy functions as a specific distortion of the weighting function of the decision maker. It is argued that Shackle neither completely understood criticisms nor appropriately developed suggestions put forward by scholars like Kenneth Arrow, Ward Edwards, Nicholas Georgescu- Roegen. Had he succeeded in doing so, we contend, his theory might have been a more satisfactory alternative to Savage’s theory than it actually was.uncertainty, decision theory, non-additive measures
On vector measures with values in
We study some aspects of countably additive vector measures with values in
and the Banach lattices of real-valued functions that are
integrable with respect to such a vector measure. On the one hand, we prove
that if is a total set not containing sets
equivalent to the canonical basis of , then there is a
non-countably additive -valued map defined on a
-algebra such that the composition is countably
additive for every . On the other hand, we show that a Banach lattice
is separable whenever it admits a countable positively norming set and both
and are order continuous. As a consequence, if is a countably
additive vector measure defined on a -algebra and taking values in a
separable Banach space, then the space is separable whenever
is order continuous
Introduzione
EnGiven a set , a finitely additive probability measure on is considered. Let be "strongly" non-atomic: we prove that there exists a sequence of subsets of (mutually disjoint and with ) whose union has measure equal to an arbitrarily given (with ) and such that is countably additive on them. As a simple corollary, the following property (well-known for countably additive measures)is deduced: the range of is the whole interval [0,1]. In the last part of the paper, some aspects of a decomposition theorem by B. De Finetti (for an arbitrary ) are deepened
The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures
My studies and my Ph.D. thesis deal with topics that recently emerged in the field of decisions under risk and uncertainty. In particular, I deal with the "target-based approach" to utility theory. A rich literature has been devoted in the last decade to this approach to economic decisions: originally, interest had been focused on the "single-attribute" case and, more recently, extensions to "multi-attribute" case have been studied. This literature is still growing, with a main focus on applied aspects. I will, on the contrary, focus attention on some aspects of theoretical type, related with the multi-attribute case.
Various mathematical concepts, such as non-additive measures, aggregation functions, multivariate probability distributions, and notions of stochastic dependence emerge in the formulation and the analysis of target-based models. Notions in the field of non-additive measures and aggregation functions are quite common in the modern economic literature. They have been used to go beyond the classical principle of maximization of expected utility in decision theory. These notions, furthermore, are used in game theory and multi-criteria decision aid.
Along my work, on the contrary, I show how non-additive measures and aggregation functions emerge in a natural way in the frame of the target-based approach to classical utility theory, when considering the multi-attribute case. Furthermore they combine with the analysis of multivariate probability distributions and with concepts of stochastic dependence.
The concept of copula also constitutes a very important tool for this work, mainly for two purposes. The first one is linked to the analysis of target-based utilities, the other one is in the comparison between classical stochastic order and the concept of "stochastic precedence". This topic finds its application in statistics as well as in the study of Markov Models linked to waiting times
to occurrences of words in random sampling of letters from an alphabet. In this work I give a generalization of the concept of stochastic precedence and we discuss its properties on the basis of properties of the connecting copulas of the variables. Along this work I also trace connections to reliability theory, whose aim is studying the lifetime of a system through the analysis of
the lifetime of its components. The target-based model finds an application in representing the behavior of the whole system by means of the interaction of its components
The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures
My studies and my Ph.D. thesis deal with topics that recently emerged in the field of decisions under risk and uncertainty. In particular, I deal with the "target-based approach" to utility theory. A rich literature has been devoted in the last decade to this approach to economic decisions: originally, interest had been focused on the "single-attribute" case and, more recently, extensions to "multi-attribute" case have been studied. This literature is still growing, with a main focus on applied aspects. I will, on the contrary, focus attention on some aspects of theoretical type, related with the multi-attribute case.
Various mathematical concepts, such as non-additive measures, aggregation functions, multivariate probability distributions, and notions of stochastic dependence emerge in the formulation and the analysis of target-based models. Notions in the field of non-additive measures and aggregation functions are quite common in the modern economic literature. They have been used to go beyond the classical principle of maximization of expected utility in decision theory. These notions, furthermore, are used in game theory and multi-criteria decision aid.
Along my work, on the contrary, I show how non-additive measures and aggregation functions emerge in a natural way in the frame of the target-based approach to classical utility theory, when considering the multi-attribute case. Furthermore they combine with the analysis of multivariate probability distributions and with concepts of stochastic dependence.
The concept of copula also constitutes a very important tool for this work, mainly for two purposes. The first one is linked to the analysis of target-based utilities, the other one is in the comparison between classical stochastic order and the concept of "stochastic precedence". This topic finds its application in statistics as well as in the study of Markov Models linked to waiting times
to occurrences of words in random sampling of letters from an alphabet. In this work I give a generalization of the concept of stochastic precedence and we discuss its properties on the basis of properties of the connecting copulas of the variables. Along this work I also trace connections to reliability theory, whose aim is studying the lifetime of a system through the analysis of
the lifetime of its components. The target-based model finds an application in representing the behavior of the whole system by means of the interaction of its components
A step beyond Tsallis and Renyi entropies
Tsallis and R\'{e}nyi entropy measures are two possible different
generalizations of the Boltzmann-Gibbs entropy (or Shannon's information) but
are not generalizations of each others. It is however the Sharma-Mittal
measure, which was already defined in 1975 (B.D. Sharma, D.P. Mittal,
J.Math.Sci \textbf{10}, 28) and which received attention only recently as an
application in statistical mechanics (T.D. Frank & A. Daffertshofer, Physica A
\textbf{285}, 351 & T.D. Frank, A.R. Plastino, Eur. Phys. J., B \textbf{30},
543-549) that provides one possible unification. We will show how this
generalization that unifies R\'{e}nyi and Tsallis entropy in a coherent picture
naturally comes into being if the q-formalism of generalized logarithm and
exponential functions is used, how together with Sharma-Mittal's measure
another possible extension emerges which however does not obey a
pseudo-additive law and lacks of other properties relevant for a generalized
thermostatistics, and how the relation between all these information measures
is best understood when described in terms of a particular logarithmic
Kolmogorov-Nagumo average
On the support of the Ashtekar-Lewandowski measure
We show that the Ashtekar-Isham extension of the classical configuration
space of Yang-Mills theories (i.e. the moduli space of connections) is
(topologically and measure-theoretically) the projective limit of a family of
finite dimensional spaces associated with arbitrary finite lattices. These
results are then used to prove that the classical configuration space is
contained in a zero measure subset of this extension with respect to the
diffeomorphism invariant Ashtekar-Lewandowski measure.
Much as in scalar field theory, this implies that states in the quantum
theory associated with this measure can be realized as functions on the
``extended" configuration space.Comment: 22 pages, Tex, Preprint CGPG-94/3-
Predictability of evolutionary trajectories in fitness landscapes
Experimental studies on enzyme evolution show that only a small fraction of
all possible mutation trajectories are accessible to evolution. However, these
experiments deal with individual enzymes and explore a tiny part of the fitness
landscape. We report an exhaustive analysis of fitness landscapes constructed
with an off-lattice model of protein folding where fitness is equated with
robustness to misfolding. This model mimics the essential features of the
interactions between amino acids, is consistent with the key paradigms of
protein folding and reproduces the universal distribution of evolutionary rates
among orthologous proteins. We introduce mean path divergence as a quantitative
measure of the degree to which the starting and ending points determine the
path of evolution in fitness landscapes. Global measures of landscape roughness
are good predictors of path divergence in all studied landscapes: the mean path
divergence is greater in smooth landscapes than in rough ones. The
model-derived and experimental landscapes are significantly smoother than
random landscapes and resemble additive landscapes perturbed with moderate
amounts of noise; thus, these landscapes are substantially robust to mutation.
The model landscapes show a deficit of suboptimal peaks even compared with
noisy additive landscapes with similar overall roughness. We suggest that
smoothness and the substantial deficit of peaks in the fitness landscapes of
protein evolution are fundamental consequences of the physics of protein
folding.Comment: 14 pages, 7 figure
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