Planning with Information-Processing Constraints and Model Uncertainty in Markov Decision Processes

Information-theoretic principles for learning and acting have been proposed to solve particular classes of Markov Decision Problems. Mathematically, such approaches are governed by a variational free energy principle and allow solving MDP planning problems with information-processing constraints expressed in terms of a Kullback-Leibler divergence with respect to a reference distribution. Here we consider a generalization of such MDP planners by taking model uncertainty into account. As model uncertainty can also be formalized as an information-processing constraint, we can derive a unified solution from a single generalized variational principle. We provide a generalized value iteration scheme together with a convergence proof. As limit cases, this generalized scheme includes standard value iteration with a known model, Bayesian MDP planning, and robust planning. We demonstrate the benefits of this approach in a grid world simulation.Comment: 16 pages, 3 figure

Marginal Likelihood Estimation with the Cross-Entropy Method

We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating independent draws instead of correlated MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving women's labor market participation and U.S. macroeconomic time series. In both applications the proposed CE method compares favorably to existing estimators

Informational Substitutes

We propose definitions of substitutes and complements for pieces of information ("signals") in the context of a decision or optimization problem, with game-theoretic and algorithmic applications. In a game-theoretic context, substitutes capture diminishing marginal value of information to a rational decision maker. We use the definitions to address the question of how and when information is aggregated in prediction markets. Substitutes characterize "best-possible" equilibria with immediate information aggregation, while complements characterize "worst-possible", delayed aggregation. Game-theoretic applications also include settings such as crowdsourcing contests and Q\&A forums. In an algorithmic context, where substitutes capture diminishing marginal improvement of information to an optimization problem, substitutes imply efficient approximation algorithms for a very general class of (adaptive) information acquisition problems. In tandem with these broad applications, we examine the structure and design of informational substitutes and complements. They have equivalent, intuitive definitions from disparate perspectives: submodularity, geometry, and information theory. We also consider the design of scoring rules or optimization problems so as to encourage substitutability or complementarity, with positive and negative results. Taken as a whole, the results give some evidence that, in parallel with substitutable items, informational substitutes play a natural conceptual and formal role in game theory and algorithms.Comment: Full version of FOCS 2016 paper. Single-column, 61 pages (48 main text, 13 references and appendix