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A sequential measurement of two non-commuting quantum observables results in a joint probability distribution for all output combinations that can be explained in terms of an initial joint quasi-probability of the non-commuting observables,…
In this paper, we study the supports of measures in the free additive convolution semigroup $\{\mu^{\boxplus t}:t>1\}$, where $\mu$ is a Borel probability measure on $\mathbb{R}$. We give a formula for the density of the absolutely…
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…
We establish a general variational formula for the logarithmic potential of the free additive convolution of two compactly supported probability measure on $\R$. The formula is given in terms of the $R$-transform of the first measure, and…
This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and…
The fundamental principles of complementarity and uncertainty are shown to be related to the possibility of joint unsharp measurements of pairs of noncommuting quantum observables. A new joint measurement scheme for complementary…
We recall the notion of abstract bornology, and connect it with topological spaces and size functions. As a generalization of measures of non-compactness, we show how every size function can be mapped to a maxitive measure.
Let M denote the space of Borel probability measures on the real line. For every nonnegative t we consider the transformation $\mathbb B_t : M \to M$ defined for any given element in M by taking succesively the the (1+t) power with respect…
Suppose that a convolutionally encoded sequence is transmitted symbol by symbol over an AWGN channel using BPSK modulation. In this case, pairs of the signal (i.e., code symbol) and observation are not jointly Gaussian and therefore, a…
We introduce and discuss the notion of monotonicity for the complexity measures of general probability distributions, patterned after the resource theory of quantum entanglement. Then, we explore whether this property is satisfied by the…
The parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step. We prove that the deterministic parity decision tree complexity of any Boolean function is polynomially related to…
The Born rule assigns a probability to any possible outcome of a quantum measurement, but leaves open the question how these probabilities are to be interpreted and, in particular, how they relate to the outcome observed in an actual…
A map is given showing that convolutions of independent random variables over a finite group and matrix multiplications of doubly stochastic matrices are homomorphic. As an application, a short proof is given to the theorem that the…
In this paper, we introduce and develop the concept of conditional quantization for Borel probability measures on $\mathbb{R}^k,$ considering both constrained and unconstrained frameworks. For each setting, we define the associated…
We extend algorithmic conservation inequalities to probability measures. The amount of self information of a probability measure cannot increase when submitted to randomized processing. This includes (potentially non-computable) measures…
We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) S when the measures are indexed by another poset A. We give counterexamples…
We explore the relationship between possibility measures (supremum preserving normed measures) and p-boxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and…
In statistical modeling of computer experiments sometimes prior information is available about the underlying function. For example, the physical system simulated by the computer code may be known to be monotone with respect to some or all…
Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in…
The probability `measure' for measurements at two consecutive moments of time is non-additive. These probabilities, on the other hand, may be determined by the limit of relative frequency of measured events, which are by nature additive. We…