Related papers: Probability Trees
We develop the complex-analytic viewpoint on the tree convolutions studied by the second author and Weihua Liu in "An operad of non-commutative independences defined by trees" (Dissertationes Mathematicae, 2020, doi:10.4064/dm797-6-2020),…
The occurrence and the distribution of patterns of trees associated to natural numbers are investigated. Bounds from above and below are proven for certain natural quantities.
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
Binary trees are fundamental objects in models of evolutionary biology and population genetics. Here, we discuss some of their combinatorial and structural properties as they depend on the tree class considered. Furthermore, the process by…
In this paper, we shall discuss the extendability of probability and non-probability measures on Cayley trees to a $\sigma$-additive measure on Borel fields which has a fundamental role in the theory of Gibbs measures.
We study here the topology of information on the space of probability measures over Polish spaces that was defined in [1]. We show that under this topology, a convergent sequence of probability measures satisfying a conditional independence…
This tutorial gives an overview of some of the basic techniques of measure theory. It includes a study of Borel sets and their generators for Polish and for analytic spaces, the weak topology on the space of all finite positive measures…
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…
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…
Given two measurable spaces $H$ and $D$ with countably generated $\sigma$-algebras, a perfect prior probability measure $P_H$ on $H$ and a sampling distribution $S: H \rightarrow D$, there is a corresponding inference map $I: D \rightarrow…
In the first part of this paper the notion of natural metric on the set of natural numbers is defined. It is such metric that the completion of N is a compact metric space that a probability borel measure exists in order that the sequence…
We study the problem of computing the tightest upper and lower bounds on the probability that the sum of $n$ dependent Bernoulli random variables exceeds an integer $k$. Under knowledge of all pairs of bivariate distributions denoted by a…
We describe the interface between measure theoretic probability and causal inference by constructing causal models on probability spaces within the potential outcomes framework. We find that measure theory provides a precise and instructive…
This contribution derives from a rather extensive study on the foundations of probability. We start by discussing critically the two main models of the random event in Probability Theroy and cast light over a number of incongruities. We…
There are several common ways to encode a tree as a matrix, such as the adjacency matrix, the Laplacian matrix (that is, the infinitesimal generator of the natural random walk), and the matrix of pairwise distances between leaves. Such…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
Decision trees built with data remain in widespread use for nonparametric prediction. Predicting probability distributions is preferred over point predictions when uncertainty plays a prominent role in analysis and decision-making. We study…
We prove an analogue of the portmanteau theorem on weak convergence of probability measures allowing measures which are unbounded on an underlying metric space but finite on the complement of any Borel neighbourhood of a fixed element.
The well known bunkbed conjecture about percolation on finite graphs is now resolved; Gladkov, Pak and Zimin, building upon work of Hollom, have constructed a counterexample. We revisit this conjecture and study it in the broader context of…
We introduce weights on the unrooted unlabelled plane trees as follows: let $\mu$ be a probability measure on the set of nonnegative integers whose mean is no larger than $1$; then the $\mu$-weight of a plane tree $t$ is defined as $\Pi \,…