Total belief theorem and conditional belief functions

In this paper new theoretical results for reasoning with belief functions are obtained and discussed. After a judicious decomposition of the set of focal elements of a belief function, we establish the Total Belief Theorem (TBT) which is the direct generalization of the Total Probability Theorem when working in the framework of belief functions. The TBT is also generalized for dealing with different frames of discernments thanks to Cartesian product space. From TBT, we can derive and define formally the expressions of conditional belief functions which are consistent with the bounds of imprecise conditional probability. This work provides a direct establishment and solid justification of Fagin-Halpern belief conditioning formulas. The well-known Bayes' Theorem of Probability Theory is then generalized in the framework of belief functions and we illustrate it with an example at the end of this paper

Optimality of Universal Bayesian Sequence Prediction for General Loss and Alphabet

Various optimality properties of universal sequence predictors based on Bayes-mixtures in general, and Solomonoff's prediction scheme in particular, will be studied. The probability of observing $x_t$ at time $t$, given past observations $x_1...x_{t-1}$ can be computed with the chain rule if the true generating distribution $\mu$ of the sequences $x_1x_2x_3...$ is known. If $\mu$ is unknown, but known to belong to a countable or continuous class \M one can base ones prediction on the Bayes-mixture $\xi$ defined as a $w_\nu$-weighted sum or integral of distributions \nu\in\M. The cumulative expected loss of the Bayes-optimal universal prediction scheme based on $\xi$ is shown to be close to the loss of the Bayes-optimal, but infeasible prediction scheme based on $\mu$. We show that the bounds are tight and that no other predictor can lead to significantly smaller bounds. Furthermore, for various performance measures, we show Pareto-optimality of $\xi$ and give an Occam's razor argument that the choice $w_\nu\sim 2^{-K(\nu)}$ for the weights is optimal, where $K(\nu)$ is the length of the shortest program describing $\nu$. The results are applied to games of chance, defined as a sequence of bets, observations, and rewards. The prediction schemes (and bounds) are compared to the popular predictors based on expert advice. Extensions to infinite alphabets, partial, delayed and probabilistic prediction, classification, and more active systems are briefly discussed.Comment: 34 page

On the `Semantics' of Differential Privacy: A Bayesian Formulation

Differential privacy is a definition of "privacy'" for algorithms that analyze and publish information about statistical databases. It is often claimed that differential privacy provides guarantees against adversaries with arbitrary side information. In this paper, we provide a precise formulation of these guarantees in terms of the inferences drawn by a Bayesian adversary. We show that this formulation is satisfied by both "vanilla" differential privacy as well as a relaxation known as (epsilon,delta)-differential privacy. Our formulation follows the ideas originally due to Dwork and McSherry [Dwork 2006]. This paper is, to our knowledge, the first place such a formulation appears explicitly. The analysis of the relaxed definition is new to this paper, and provides some concrete guidance for setting parameters when using (epsilon,delta)-differential privacy.Comment: Older version of this paper was titled: "A Note on Differential Privacy: Defining Resistance to Arbitrary Side Information

Deterministic Bayesian Information Fusion and the Analysis of its Performance

This paper develops a mathematical and computational framework for analyzing the expected performance of Bayesian data fusion, or joint statistical inference, within a sensor network. We use variational techniques to obtain the posterior expectation as the optimal fusion rule under a deterministic constraint and a quadratic cost, and study the smoothness and other properties of its classification performance. For a certain class of fusion problems, we prove that this fusion rule is also optimal in a much wider sense and satisfies strong asymptotic convergence results. We show how these results apply to a variety of examples with Gaussian, exponential and other statistics, and discuss computational methods for determining the fusion system's performance in more general, large-scale problems. These results are motivated by studying the performance of fusing multi-modal radar and acoustic sensors for detecting explosive substances, but have broad applicability to other Bayesian decision problems