Compact-Friendly Multiplication Operators on Banach Function Spaces

AbstractAnswering a question posed by Abramovich et al. (Indag. Math.(N.S.)10 (1999), 161–171, we prove that a positive multiplication operator on an arbitrary Banach function space X is compact-friendly if and only if the multiplier is constant on a set of positive measure

Cyclic Directed Probabilistic Graphical Model: A Proposal Based on Structured Outcomes

In the process of building (structural learning) a probabilistic graphical model from a set of observed data, the directional, cyclic dependencies between the random variables of the model are often found. Existing graphical models such as Bayesian and Markov networks can reflect such dependencies. However, this requires complicating those models, such as adding additional variables or dividing the model graph into separate subgraphs. Herein, we describe a probabilistic graphical model - probabilistic relation network - that allows the direct capture of directional cyclic dependencies during structural learning. This model is based on the simple idea that each sample of the observed data can be represented by an arbitrary graph (structured outcome), which reflects the structure of the dependencies of the variables included in the sample. Each of the outcomes contains only a part of the graphical model structure; however, a complete graph of the probabilistic model is obtained by combining different outcomes. Such a graph, unlike Bayesian and Markov networks, can be directed and can have cycles. We explored the full joint distribution and conditional distribution and conditional independence properties of variables in the proposed model. We defined the algorithms for constructing of the model from the dataset and for calculating the conditional and full joint distributions. We also performed a numerical comparison with Bayesian and Markov networks. This model does not violate the probability axioms, and it supports learning from observed data. Notably, it supports probabilistic inference, making it a prospective tool in data analysis and in expert and design-making applications.Comment: 41 pages, 11 figures, arXiv:2206.06089v