Related papers: The EL Theorem
A fundamental tool in network information theory is the covering lemma, which lower bounds the probability that there exists a pair of random variables, among a give number of independently generated candidates, falling within a given set.…
We find that the probability distribution for the largest intervals $p(l)$ exhibits universal properties for different systems including random walk and random cutting models. In particular, $p(l)$ has an infinite set of singularities at…
The problem addressed concerns the determination of the average number of successive attempts of guessing a word of a certain length consisting of letters with given probabilities of occurrence. Both first- and second-order approximations…
Among the set of hypersurfaces of degree $d$ and dimension $\ell$ defined by the vanishing of a homogeneous polynomial with coefficients $\pm 1$, we investigate the probability that a hypersurface contains a rational point as $d$ and $\ell$…
Four new probability models are derived which generalize the common univariate continuous distributions. Classical distributional measures are derived from Hoel, et al., Introduction to Probability Theory, 1971. Measures include probability…
Generalized linear mixed models are powerful tools for analyzing clustered data, where the unknown parameters are classically (and most commonly) estimated by the maximum likelihood and restricted maximum likelihood procedures. However,…
We establish global rates of convergence of the Maximum Likelihood Estimator (MLE) of a multivariate distribution function in the case of (one type of) "interval censored" data. The main finding is that the rate of convergence of the MLE in…
We construct a hierarchy of supersymmetric string theories by showing that the general N-extended superstrings may be viewed as a special class of the (N+1)-extended superstrings. As a side result, we find a twisted (N+2) superconformal…
In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded…
We consider a generic system composed of a fixed number of particles distributed over a finite number of energy levels. We make only general assumptions about system's properties and the entropy. System's constraints other than fixed number…
This article investigates the difference between the true detection probability and the subjective probability of a uniformly optimal search plan. Its main contributions are multi-fold. First, it provides a set of examples to show that, in…
Do LMs infer the semantics of text from co-occurrence patterns in their training data? Merrill et al. (2022) argue that, in theory, sentence co-occurrence probabilities predicted by an optimal LM should reflect the entailment relationship…
The class of $\ell$-maximum distance separable ($\ell$-MDS) codes {is a} generalization of maximum distance separable (MDS) codes {that} has attracted a lot of attention due to its applications in several areas such as secret sharing…
The $\lambda$-exponential family generalizes the standard exponential family via a generalized convex duality motivated by optimal transport. It is the constant-curvature analogue of the exponential family from the information-geometric…
We study a new generalization of palindromes and gapped palindromes called block palindromes. A block palindrome is a string that becomes a palindrome when identical substrings are replaced with a distinct character. We investigate several…
Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. In this paper, MLE for statistical models with discrete data is studied from an algebraic statistics viewpoint. A reformulation of the MLE problem in…
Word embeddings -- distributed representations of words -- in deep learning are beneficial for many tasks in natural language processing (NLP). However, different embedding sets vary greatly in quality and characteristics of the captured…
We generalize the optimal coupling theorem to multiple random variables: Given a collection of random variables, it is possible to couple all of them so that any two differ with probability comparable to the total-variation distance between…
Consider a collection of m competing machine learning algorithms. Given their performance on a benchmark of datasets, we would like to identify the best performing algorithm. Specifically, which algorithm is most likely to ``win'' (rank…
A novel notion of unpredictable strings is revealed and utilized to define deterministic unpredictable sequences on a finite number of symbols. We prove the first law of large strings for random processes in discrete time, which confirms…