Related papers: Set-valued mapping and Rough Probability
The rough-set theory proposed by Pawlak, has been widely used in dealing with data classification problems. The original rough-set model is, however, quite sensitive to noisy data. Tzung thus proposed deals with the problem of producing a…
Stochastic localization is a pathwise analysis technique originating from convex geometry. This paper explores certain algorithmic aspects of stochastic localization as a computational tool. First, we unify various existing stochastic…
Here I discuss ideas that makes a synthesis of topology and probability theory. The idea is the following: given a set $X$, assign a number $p(A)\in [0,1]$ for any subset $A$ of $X$. We can interpret $p(A)$ as the probability of openness of…
This expository note aims at illustrating weak convergence of probability measures from a broader view than a previously published paper. Though the results are standard for functional analysts, this approach is rarely known by…
Given a raw knowledge in the form of a data table/a decision system, one is facing two possible venues. One, to treat the system as closed, i.e., its universe does not admit new objects, or, to the contrary, its universe is open on…
We investigate the stochastic modified equation which plays an important role in the stochastic backward error analysis for explaining the mathematical mechanism of a numerical method. The contribution of this paper is threefold. First, we…
A number of generalizations of stochastic and information-theoretic randomness are known in the literature. However, they are not compatible with handling meaning in vague and dynamic contexts of rough reasoning (and therefore explainable…
This text presents an unified approach of probability and statistics in the pursuit of understanding and computation of randomness in engineering or physical or social system with prediction with generalizability. Starting from elementary…
In this paper, we describe a representation for spatial information, called the stochastic map, and associated procedures for building it, reading information from it, and revising it incrementally as new information is obtained. The map…
Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to (i) very restricted model classes where exact or approximate…
We introduce set risk measures (SRMs), real-valued maps defined on the family of non-empty closed bounded sets of essentially bounded random variables. SRMs extend traditional scalar risk measures by assigning a single capital requirement…
Many hard problems in the computational sciences are equivalent to counting the leaves of a decision tree, or, more generally, summing a cost function over the nodes. These problems include calculating the permanent of a matrix, finding the…
Rough set theory models uncertainty by approximating target concepts through lower and upper sets induced by indiscernibility, or more generally, by granulation relations in data tables. This perspective captures vagueness caused by limited…
The study of mereology (parts and wholes) in the context of formal approaches to vagueness can be approached in a number of ways. In the context of rough sets, mereological concepts with a set-theoretic or valuation based ontology acquire…
This survey presents some historical background and recent developments in the area of selections for set-valued mappings along with several open questions. It was written with the hope that the presented material may pique an interest in…
In this paper, we analyze the behavior of stochastic approximation schemes with set-valued maps in the absence of a stability guarantee. We prove that after a large number of iterations if the stochastic approximation process enters the…
Stochastic approximation algorithm is a useful technique which has been exploited successfully in probability theory and statistics for a long time. The step sizes used in stochastic approximation are generally taken to be deterministic and…
This paper has two purposes. One is to demonstrate contextuality analysis of systems of epistemic random variables. The other is to evaluate the performance of a new, hierarchical version of the measure of (non)contextuality introduced in…
The paper analyzes theoretically and empirically the performance of likelihood weighting (LW) on a subset of nodes in Bayesian networks. The proposed scheme requires fewer samples to converge due to reduction in sampling variance. The…
We propose a generalization of the random matrix theory following the basic prescription of the recently suggested concept of superstatistics. Spectral characteristics of systems with mixed regular-chaotic dynamics are expressed as weighted…