Some alternatives to probability in the literature include sets of probability measure [Huber 1981; Walley 1991], Dempster-Shafer belief functions [Shafer 1976] and the closely related nonadditive measures [Schmeidler 1989], and possibility measures [Dubois and Prade 1990]. In this paper, we consider the notion of expectation for all these representations of uncertainty. We do not take a stand here on what the "right" way is to represent uncertainty; we simply investigate characterizations of expectation and reasoning about expectation, both for probability and for other representations of uncertainty. It is well known that a probability measure determines a unique expectation function that is linear (i.e., E (aX + bY) = aE (X) + bE (Y)), monotone (i.e., X Y implies E ( X) E (Y)), and maps constant functions to their value. Conversely, given an expectation function E (that is, a function from random variables to the reals) that is linear, monotone, and maps constant functions to their value, there is a unique probability measure µ such that E = E µ. That is, there is a 1-1 mapping from probability measures to (probabilistic) expectation functions. One of the goals of this paper is to provide similar characterizations of expectation for other representations of uncertainty. Some work along these lines has already been done, particulary with regard to sets of probability measures [Huber 1981; Walley 1991; 1981]. 1 However, there seems to be surprisingly little work on characterizing expectation in the context of other measures of uncertainty, such as belief functions [Shafer 1976] and possibility measures [Dubois and Prade 1990].

Probability-like parameters appearing in some statistical models, and their prior distributions, are reinterpreted through the notion of `circumstance', a term which stands for any piece of knowledge that is useful in assigning a probability and that satisfies some additional logical properties. The idea, which can be traced to Laplace and Jaynes, is that the usual inferential reasonings about the probability-like parameters of a statistical model can be conceived as reasonings about equivalence classes of `circumstances' - viz., real or hypothetical pieces of knowledge, like e.g. physical hypotheses, that are useful in assigning a probability and satisfy some additional logical properties - that are uniquely indexed by the probability distributions they lead to.

An approach to induction is presented, based on the idea of analysing the context of a given problem into `circumstances'. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti's representation theorem and on the notion of infinite exchangeability. In particular, it gives an alternative interpretation of those formulae that apparently involve `unknown probabilities' or `propensities'. Various advantages and applications of the presented approach are discussed, especially in comparison to that based on exchangeability. Generalisations are also discussed.

The multi-armed bandit is a concise model for the problem of iterated decision-making under uncertainty. In each round, a gambler must pull one of $K$ arms of a slot machine, without any foreknowledge of their payouts, except that they are uniformly bounded. A standard objective is to minimize the gambler's regret, defined as the gambler's total payout minus the largest payout which would have been achieved by any fixed arm, in hindsight. Note that the gambler is only told the payout for the arm actually chosen, not for the unchosen arms. Almost all previous work on this problem assumed the payouts to be non-adaptive, in the sense that the distribution of the payout of arm $j$ in round $i$ is completely independent of the choices made by the gambler on rounds $1, \dots, i-1$. In the more general model of adaptive payouts, the payouts in round $i$ may depend arbitrarily on the history of past choices made by the algorithm. We present a new algorithm for this problem, and prove nearly optimal guarantees for the regret against both non-adaptive and adaptive adversaries. After $T$ rounds, our algorithm has regret $O(\sqrt{T})$ with high probability (the tail probability decays exponentially). This dependence on $T$ is best possible, and matches that of the full-information version of the problem, in which the gambler is told the payouts for all $K$ arms after each round. Previously, even for non-adaptive payouts, the best high-probability bounds known were $O(T^{2/3})$, due to Auer, Cesa-Bianchi, Freund and Schapire. The expected regret of their algorithm is $O(T^{1/2}) for non-adaptive payouts, but as we show, $Ω(T^{2/3})$ for adaptive payouts.

Relational representation of knowledge makes it possible to perform all the computations and decision making in a uniform relational way by means of special relational compositions called triangle and square products. In this paper some applications in manufacturing related to cost analysis are described. Testing fuzzy relational structures for various relational properties allows us to discover dependencies, hierarchies, similarities, and equivalences of the attributes characterizing technological processes and manufactured artifacts in their relationship to costs and performance. A brief overview of mathematical aspects of BK-relational products is given in Appendix 1 together with further references in the literature.

A fuzzy logic based classification engine has been developed for classifying mass spectra obtained with an imaging internal source Fourier transform mass spectrometer (I^2LD-FTMS). Traditionally, an operator uses the relative abundance of ions with specific mass-to-charge (m/z) ratios to categorize spectra. An operator does this by comparing the spectrum of m/z versus abundance of an unknown sample against a library of spectra from known samples. Automated positioning and acquisition allow I^2LD-FTMS to acquire data from very large grids, this would require classification of up to 3600 spectrum per hour to keep pace with the acquisition. The tedious job of classifying numerous spectra generated in an I^2LD-FTMS imaging application can be replaced by a fuzzy rule base if the cues an operator uses can be encapsulated. We present the translation of linguistic rules to a fuzzy classifier for mineral phases in basalt. This paper also describes a method for gathering statistics on ions, which are not currently used in the rule base, but which may be candidates for making the rule base more accurate and complete or to form new rule bases based on data obtained from known samples. A spatial method for classifying spectra with low membership values, based on neighboring sample classifications, is also presented.

For dimension reduction in $l_1$, the method of {\em Cauchy random projections} multiplies the original data matrix $\mathbf{A} \in\mathbb{R}^{n\times D}$ with a random matrix $\mathbf{R} \in \mathbb{R}^{D\times k}$ ($k\ll\min(n,D)$) whose entries are i.i.d. samples of the standard Cauchy C(0,1). Because of the impossibility results, one can not hope to recover the pairwise $l_1$ distances in $\mathbf{A}$ from $\mathbf{B} = \mathbf{AR} \in \mathbb{R}^{n\times k}$, using linear estimators without incurring large errors. However, nonlinear estimators are still useful for certain applications in data stream computation, information retrieval, learning, and data mining. We propose three types of nonlinear estimators: the bias-corrected sample median estimator, the bias-corrected geometric mean estimator, and the bias-corrected maximum likelihood estimator. The sample median estimator and the geometric mean estimator are asymptotically (as $k\to \infty$) equivalent but the latter is more accurate at small $k$. We derive explicit tail bounds for the geometric mean estimator and establish an analog of the Johnson-Lindenstrauss (JL) lemma for dimension reduction in $l_1$, which is weaker than the classical JL lemma for dimension reduction in $l_2$. Asymptotically, both the sample median estimator and the geometric mean estimators are about 80% efficient compared to the maximum likelihood estimator (MLE). We analyze the moments of the MLE and propose approximating the distribution of the MLE by an inverse Gaussian.

The quality of the ranking function is an important factor that determines the quality of the Information Retrieval system. Each document is assigned a score by the ranking function; the score indicates the likelihood of relevance of the document given a query. In the vector space model, the ranking function is defined by a mathematic expression. We propose a fuzzy logic (FL) approach to defining the ranking function. FL provides a convenient way of converting knowledge expressed in a natural language into fuzzy logic rules. The resulting ranking function could be easily viewed, extended, and verified: * if (tf is high) and (idf is high) > (relevance is high); * if (overlap is high) > (relevance is high). By using above FL rules, we are able to achieve performance approximately equal to the state of the art search engine Apache Lucene (deltaP10 +0.92%; deltaMAP -0.1%). The fuzzy logic approach allows combining the logic-based model with the vector model. The resulting model possesses simplicity and formalism of the logic based model, and the flexibility and performance of the vector model.

Despite its widespread acceptance, there are some problems in usin g probability to represent uncertainty. Perhaps the most serious is that p robability is not good at representing ignorance.

The management and combination of uncertain, imprecise, fuzzy and even paradoxical or high conflicting sources of information has always been, and still remains today, of primal importance for the development of reliable modern information systems involving artificial reasoning. In this introduction, we present a survey of our recent theory of plausible and paradoxical reasoning, known as Dezert-Smarandache Theory (DSmT) in the literature, developed for dealing with imprecise, uncertain and paradoxical sources of information. We focus our presentation here rather on the foundations of DSmT, and on the two important new rules of combination, than on browsing specific applications of DSmT available in literature. Several simple examples are given throughout the presentation to show the efficiency and the generality of this new approach.