Attempts to replicate probabilistic reasoning in expert systems have typically overlooked a critical ingredient of that process. Probabilistic analysis typically requires extensive judgments regarding interdependencies among hypotheses and data, and regarding the appropriateness of various alternative models. The application of such models is often an iterative process, in which the plausibilit...

The d-dimensional Slepian Gaussian random field {S(t), t ∈ R+} is a mean zero Gaussian process with covariance function ES(s)S(t) = ∏d i=1 max(0, ai − |si − ti|) for ai > 0 and t = (t1, · · · , td) ∈ R+. Small ball probabilities for S(t) are obtained under the L2-norm on [0, 1]d, and under the sup-norm on [0, 1]2 which implies Talagrand’s result for the Brownian sheet. The method of proof for t...

Recently, some variants of the l1 norm, particularly matrix norms such as the l1,2 and l1,∞ norms, have been widely used in multi-task learning, compressed sensing and other related areas to enforce sparsity via joint regularization. In this paper, we unify the l1,2 and l1,∞ norms by considering a family of l1,q norms for 1 < q ≤ ∞ and study the problem of determining the most appropriate spars...

We show the shortest vector problem in the l2 norm is NP-hard (for randomized reductions) to approximate within any constant factor less than p2. We also give a deterministic reduction under a reasonable number theoretic conjecture. Analogous results hold in any lp norm (p 1). In proving our NP-hardness result, we give an alternative construction satisfying Ajtai’s probabilistic variant of Saue...

There is given an axiomatization of the hybrid probabilistic-possibilistic mixture, based on a pair of t-conorm S and t-norm T satisfying (CD) condition, and the corresponding S-measure.

The purpose of the present notes is to examine following issues related Chernoff estimate: (1) For contractions on a Banach space we modify $\sqrt{n}$-estimate and apply it in proof product formula for $C_0$-semigroups \textit{strong} operator topology. (2) We use idea {probabilistic} approach, proving estimate strong topology, uplift \textit{operator-norm} \textit{quasi-sectorial} contraction ...

For a simple probabilistic language we present a semantics based on linear operators on infinite dimensional Hilbert spaces. We show the equivalence of this semantics with a standard operational one and we discuss its relationship with the well-known denotational semantics introduced by Kozen. For probabilistic programs, it is typical to use Banach spaces and their norm topology to model the pr...

There is given an overview on the results related to the generalization of the decision theory to non probabilistic uncertainty based on the characterization of the families of operations involved in generalized mixtures. It turns out that this is based on a previous result on the characterization of the pair of continuous t-norm T and t-conorm S such that the former is conditionally distributi...

We propose a norm of consistency for a mixed set X of defeasible and strict sen­ tences, based on a probabilistic interpre­ tation of these sentences. This norm es­ tablishes a clear distinction between knowl­ edge bases depicting exceptions and those containing outright contradictions. We then define a notion of entailment based also on probabilistic considerations and provide a characterizati...