Ergodicity, Decisions, and Partial Information

In the simplest sequential decision problem for an ergodic stochastic process X, at each time n a decision u_n is made as a function of past observations X_0,...,X_{n-1}, and a loss l(u_n,X_n) is incurred. In this setting, it is known that one may choose (under a mild integrability assumption) a decision strategy whose pathwise time-average loss is asymptotically smaller than that of any other strategy. The corresponding problem in the case of partial information proves to be much more delicate, however: if the process X is not observable, but decisions must be based on the observation of a different process Y, the existence of pathwise optimal strategies is not guaranteed. The aim of this paper is to exhibit connections between pathwise optimal strategies and notions from ergodic theory. The sequential decision problem is developed in the general setting of an ergodic dynamical system (\Omega,B,P,T) with partial information Y\subseteq B. The existence of pathwise optimal strategies grounded in two basic properties: the conditional ergodic theory of the dynamical system, and the complexity of the loss function. When the loss function is not too complex, a general sufficient condition for the existence of pathwise optimal strategies is that the dynamical system is a conditional K-automorphism relative to the past observations \bigvee_n T^n Y. If the conditional ergodicity assumption is strengthened, the complexity assumption can be weakened. Several examples demonstrate the interplay between complexity and ergodicity, which does not arise in the case of full information. Our results also yield a decision-theoretic characterization of weak mixing in ergodic theory, and establish pathwise optimality of ergodic nonlinear filters.Comment: 45 page

A Polynomial Optimization Approach to Constant Rebalanced Portfolio Selection

We address the multi-period portfolio optimization problem with the constant rebalancing strategy. This problem is formulated as a polynomial optimization problem (POP) by using a mean-variance criterion. In order to solve the POPs of high degree, we develop a cutting-plane algorithm based on semidefinite programming. Our algorithm can solve problems that can not be handled by any of known polynomial optimization solvers.

The Value of Information for Populations in Varying Environments

The notion of information pervades informal descriptions of biological systems, but formal treatments face the problem of defining a quantitative measure of information rooted in a concept of fitness, which is itself an elusive notion. Here, we present a model of population dynamics where this problem is amenable to a mathematical analysis. In the limit where any information about future environmental variations is common to the members of the population, our model is equivalent to known models of financial investment. In this case, the population can be interpreted as a portfolio of financial assets and previous analyses have shown that a key quantity of Shannon's communication theory, the mutual information, sets a fundamental limit on the value of information. We show that this bound can be violated when accounting for features that are irrelevant in finance but inherent to biological systems, such as the stochasticity present at the individual level. This leads us to generalize the measures of uncertainty and information usually encountered in information theory

Nonparametric Entropy Estimation for Stationary Processes and Random Fields, with Applications to English Text

We discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a Doeblin-type mixing condition. The estimators are Ces`aro averages of longest match-lengths, and their consistency follows from a generalized ergodic theorem due to Maker. We provide examples of their performance on English text, and we generalize our results to countable alphabet processes and to random fields. Index Terms -- Entropy rate, pattern matching, universal data compression, entropy of English. 1 I. Kontoyiannis and P. Algoet are with the Information Systems Laboratory (Durand 141A), Electrical Engineering Department, Stanford University, Stanford CA 94305. Email: [email protected], and [email protected]. Yu.M. Suhov is with the Statistical Laboratory, DPMMS, Cambridge University, and also with the Institute for Problems in Information Transmission, Moscow. Email: [email protected]. A.J. Wyner is with the Stat..

A maximal entropy stochastic process for a timed automaton

International audienceSeveral ways of assigning probabilities to runs of timed automata (TA) have been proposed recently. When only the TA is given, a relevant question is to design a probability distribution which represents in the best possible way the runs of the TA. This question does not seem to have been studied yet. We give an answer to it using a maximal entropy approach. We introduce our variant of stochastic model, the stochastic process over runs which permits to simulate random runs of any given length with a linear number of atomic operations. We adapt the notion of Shannon (continuous) entropy to such processes. Our main contribution is an explicit formula defining a process $Y^*$ which maximizes the entropy. This formula is an adaptation of the so-called Shannon-Parry measure to the timed automata setting. The process $Y^*$ has the nice property to be ergodic. As a consequence it has the asymptotic equipartition property and thus the random sampling wrt. $Y^*$ is quasi uniform

The evolution of portfolio rules and the capital asset pricing model

Evolution, Market selection, Wealth dynamics, Portfolio rules, CAPM, C61, D81, G11,