Continuous m-dimensional distorted probabilities

Fuzzy measures, also known as non-additive measures, monotonic games, and capacities, have been used in many contexts. For example, in economics, risk analysis, in computer science, computer vision and machine learning and, in general, in mathematics. However, when looking at applications, one of the problems that still needs to be solved is how the measure should be defined in an easy and intuitive way. When the reference set is finite, a few families of measures have been established, e.g. distorted probabilities, k-additive and decomposable measures. But, when the reference set is infinite, the only family is distorted probabilities. In this paper we give a definition for m-dimensional distorted probabilities in the case that the reference set is not finite, and we study some properties of this family. We also give a definition for hierarchically decomposable m-dimensional distorted probabilities that relates to another family of measures defined for the finite case

Diffusion-limited reactions and mortal random walkers in confined geometries

Motivated by the diffusion-reaction kinetics on interstellar dust grains, we study a first-passage problem of mortal random walkers in a confined two-dimensional geometry. We provide an exact expression for the encounter probability of two walkers, which is evaluated in limiting cases and checked against extensive kinetic Monte Carlo simulations. We analyze the continuum limit which is approached very slowly, with corrections that vanish logarithmically with the lattice size. We then examine the influence of the shape of the lattice on the first-passage probability, where we focus on the aspect ratio dependence: Distorting the lattice always reduces the encounter probability of two walkers and can exhibit a crossover to the behavior of a genuinely one-dimensional random walk. The nature of this transition is also explained qualitatively.Comment: 18 pages, 16 figure

Beliefs, Doubts and Learning: Valuing Economic Risk

This paper explores two perspectives on the rational expectations hypothesis. One perspective is that of economic agents in such a model, who form inferences about the future using probabilities implied by the model. The other is that of an econometrician who makes inferences about the probability model that economic agents are presumed to use. Typically it is assumed that economic agents know more than the econometrician, and econometric ambiguity is often withheld from the economic agents. To understand better both of these perspectives and the relation between them, I appeal to statistical decision theory to characterize when learning or discriminating among competing probability models is challenging. I also use choice theory under uncertainty to explore the ramifications of model uncertainty and learning in environments in which historical data may be insufficient to yield precise probability statements. I use both tools to reassess the macroeconomic underpinnings of asset pricing models. I illustrate how statistical ambiguity can alter the risk-return tradeoff familiar from asset pricing; and I show that when real time learning is included risk premia are larger when macroeconomic growth is lower than average.

A classical interpretation of the Scrooge distribution

The Scrooge distribution is a probability distribution over the set of pure states of a quantum system. Specifically, it is the distribution that, upon measurement, gives up the least information about the identity of the pure state, compared with all other distributions having the same density matrix. The Scrooge distribution has normally been regarded as a purely quantum mechanical concept, with no natural classical interpretation. In this paper we offer a classical interpretation of the Scrooge distribution viewed as a probability distribution over the probability simplex. We begin by considering a real-amplitude version of the Scrooge distribution, for which we find that there is a non-trivial but natural classical interpretation. The transition to the complex-amplitude case requires a step that is not particularly natural but that may shed light on the relation between quantum mechanics and classical probability theory.Comment: 17 pages; for a special issue of Entropy: Quantum Communication--Celebrating the Silver Jubilee of Teleportatio