Related papers: Urns & Tubes
During last two decades it has been discovered that the statistical properties of a number of microscopically rather different random systems at the macroscopic level are described by {\it the same} universal probability distribution…
A cyclic urn is an urn model for balls of types $0,\ldots,m-1$. The urn starts at time zero with an initial configuration. Then, in each time step, first a ball is drawn from the urn uniformly and independently from the past. If its type is…
If you color a table using k colors, and throw a needle randomly on it, for some proper definition, you get a certain probability that the endpoints will fall on different colors. How can one make this probability maximal? This problem is…
Predicting the winner of an election is of importance to multiple stakeholders. To formulate the problem, we consider an independent sequence of categorical data with a finite number of possible outcomes in each. The data is assumed to be…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
Finite frames can be viewed as mass points distributed in $N$-dimensional Euclidean space. As such they form a subclass of a larger and rich class of probability measures that we call probabilistic frames. We derive the basic properties of…
We propose an approach to analyze the asymptotic behavior of P\'olya urns based on the contraction method. For this, a new combinatorial discrete time embedding of the evolution of the urn into random rooted trees is developed. A…
Coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They are more general than the very important class of context-specific Bayesian networks. In this paper, we…
Geometry constitutes a core set of intuitions present in all humans, regardless of their language or schooling [1]. Could brain's built in machinery for processing geometric information take part in uncertainty representation? For decades…
This study analyzes pass networks in football (soccer) using a stochastic model known as the P\'olya urn. By focusing on preferential selection, it theoretically demonstrates that the time evolution of networks can be characterized by a…
Consider an election where N seats are distributed among parties with proportions p_1,...,p_m of the votes. We study, for the common divisor and quota methods, the asymptotic distribution, and in particular the mean, of the seat excess of a…
Sufficient conditions are developed for a class of generalized Polya urn schemes ensuring exchangeability. The extended class includes the Blackwell-MacQueen Polya urn and the urn schemes for the two-parameter Poisson-Dirichlet process and…
Using the known possibility to associate the completely positive maps with density matrices and recent results on expressing the density matrices with sets of classical probability distributions of dichotomic random variables we construct…
Transformers are often the go-to architecture to build foundation models that ingest a large amount of training data. But these models do not estimate the probability density distribution when trained on regression problems, yet obtaining…
Elections involving a very large voter population often lead to outcomes that surprise many. This is particularly important for the elections in which results affect the economy of a sizable population. A better prediction of the true…
We consider the general version of P\'olya urns recently studied by Bandyopadhyay and Thacker (2016+) and Mailler and Marckert (2017), with the space of colours being any Borel space $S$ and the state of the urn being a finite measure on…
In this expository note, we discuss a ``balls-and-urns'' probability puzzle posed by Daniel Litt.
In the present paper, we consider the two-color nonlinear unbalanced urn model, under a drawing rule reinforced by an $\mathbb{R}^+$-valued concave function and an unbalanced replacement matrix. The large deviation inequalities for the…
We study a networked system of innovation processes, where each process is modeled as an urn with infinitely many colors-a classical framework for capturing the emergence of novelties. Extending this paradigm, we analyze a model of…
Many real life situations require a set of items to be repeatedly placed in a random sequence. In such circumstances, it is often desirable to test whether such randomization indeed obtains, yet this problem has received very limited…