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The notion of probability plays an important role in almost all areas of science and technology. In modern mathematics, however, probability theory means nothing other than measure theory, and the operational characterization of the notion…
Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal…
Following an argument advanced by Feynman, we consider a method for obtaining the effective action which generates the sum of tree diagrams with external physical particles. This technique is applied, in the unbroken \lambda \phi^4 theory,…
Tree-based ensemble methods such as random forests, gradient-boosted trees, and Bayesianadditive regression trees have been successfully used for regression problems in many applicationsand research studies. In this paper, we study ensemble…
A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq \alpha\leq\beta\leq1$, there is a group $\Gamma$ with measure $\mu$ equidistributed on a finite…
Bayesian networks are popular probabilistic models that capture the conditional dependencies among a set of variables. Inference in Bayesian networks is a fundamental task for answering probabilistic queries over a subset of variables in…
Entropic uncertainty relations in a finite dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for the sum of R\'enyi entropies describing probability distributions associated…
The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by…
In this paper, we propose an optimization-based mechanism to explain power law distributions, where the function that the optimization process is seeking to optimize is derived mathematically, then the behavior and interpretation of this…
Probabilistic-driven classification techniques extend the role of traditional approaches that output labels (usually integer numbers) only. Such techniques are more fruitful when dealing with problems where one is not interested in…
Taking a Feynman categorical perspective, several key aspects of the geometry of surfaces are deduced from combinatorial constructions with graphs. This provides a direct route from combinatorics of graphs to string topology operations via…
In Chapter 1 we fully characterise pairs of finite graphs which form a gap in the full homomorphism order. This leads to a simple proof of the existence of generalised duality pairs. We also discuss how such results can be carried to…
A Multinomial Processing Tree (MPT) is a directed tree with a probability associated with each arc. Here we consider an additional parameter associated with each arc, a measure such as the time required to select the arc. MPTs are often…
A general theory of stochastic decision forests is developed to bridge two concepts of information flow: decision trees and refined partitions on the one side, filtrations from probability theory on the other. Instead of the traditional…
We consider a recursively defined random set of points and its barycenter, where the random set is constructed by the following inductive rule: Given a realization of $n-1$ points, one of them is picked at random and serves as a source the…
The time process of transport on randomly evolving trees is investigated. By introducing the notions of living and dead nodes a model of random tree evolution is constructed which describes the spreading in time of objects corresponding to…
Trees are partial orders in which every element has a linearly ordered set of predecessors. Here we initiate the exploration of the structural theory of trees with the study of different notions of \emph{branching in trees} and of…
We study the N-step binary stationary ergodic Markov chain and analyze its differential entropy. Supposing that the correlations are weak we express the conditional probability function of the chain through the pair correlation function and…
The most fundamental problem in statistical causality is determining causal relationships from limited data. Probability trees, which combine prior causal structures with Bayesian updates, have been suggested as a possible solution. In this…
We apply the principle of maximum entropy to select a unique joint probability distribution from the set of all joint probability distributions specified by a credal network. In detail, we start by showing that the unique joint distribution…