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We study the statistics of height and balanced height in the binary search tree problem in computer science. The search tree problem is first mapped to a fragmentation problem which is then further mapped to a modified directed polymer…
Here we introduce probabilistic weighted and unweighted multilayer networks as derived from information theoretical correlation measures on large multidimensional datasets. We present the fundamentals of the formal application of…
As suggested by Currie, we apply the probabilistic method to problems regarding pattern avoidance. Using techniques from analytic combinatorics, we calculate asymptotic pattern occurrence statistics and use them in conjunction with the…
This report introduces general ideas and some basic methods of the Bayesian probability theory applied to physics measurements. Our aim is to make the reader familiar, through examples rather than rigorous formalism, with concepts such as:…
Bayesian networks can be used to extract explanations about the observed state of a subset of variables. In this paper, we explicate the desiderata of an explanation and confront them with the concept of explanation proposed by existing…
Probabilistic programs encode stochastic models as ordinary-looking programs with primitives for sampling numbers from predefined distributions and conditioning. Their applications include, among many others, machine learning and modeling…
A new nonparametric approach, based on a decision tree algorithm, is proposed to calculate the overlap between two probability distributions. The devised framework is described analytically and numerically. The convergence of the estimated…
Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A…
We investigate the application of Weak Poincar\'e Inequalities (WPI) to Markov chains to study their rates of convergence and to derive complexity bounds. At a theoretical level we investigate the necessity of the existence of WPIs to…
The general use of subjective probabilities to model belief has been justified using many axiomatic schemes. For example, ?consistent betting behavior' arguments are well-known. To those not already convinced of the unique fitness and…
With origins in game theory, probabilistic values like Shapley values, Banzhaf values, and semi-values have emerged as a central tool in explainable AI. They are used for feature attribution, data attribution, data valuation, and more.…
A random forest prediction can be computed by the scalar product of the labels of the training examples and a set of weights that are determined by the leafs of the forest into which the test object falls; each prediction can hence be…
Comprehensible explanations of probabilistic reasoning are a prerequisite for wider acceptance of Bayesian methods in expert systems and decision support systems. A study of human reasoning under uncertainty suggests two different…
How do we ascribe subjective probability? In decision theory, this question is often addressed by representation theorems, going back to Ramsey (1926), which tell us how to define or measure subjective probability by observable preferences.…
We explore a generating function trick which allows us to keep track of infinitely many statistics using finitely many variables, by recording their individual distributions rather than their joint distributions. Building on previous work…
A cluster tree provides a highly-interpretable summary of a density function by representing the hierarchy of its high-density clusters. It is estimated using the empirical tree, which is the cluster tree constructed from a density…
Given i.i.d. data from an unknown distribution, we consider the problem of predicting future items. An adaptive way to estimate the probability density is to recursively subdivide the domain to an appropriate data-dependent granularity. A…
Bayesian inference requires specification of a single, precise prior distribution, whereas frequentist inference only accommodates a vacuous prior. Since virtually every real-world application falls somewhere in between these two extremes,…
While for deterministic systems, a counterexample to a property can simply be an error trace, counterexamples in probabilistic systems are necessarily more complex. For instance, a set of erroneous traces with a sufficient cumulative…
Variational inference is a popular method for estimating model parameters and conditional distributions in hierarchical and mixed models, which arise frequently in many settings in the health, social, and biological sciences. Variational…