Related papers: Solving the 100 Swiss Francs Problem
Statistical models with latent structure have a history going back to the 1950s and have seen widespread use in the social sciences and, more recently, in computational biology and in machine learning. Here we study the basic latent class…
We prove a conjecture of Sturmfels, Timme and Zwiernik on the ML-degrees of linear covariance models in algebraic statistics. As in our previous works on linear concentration models, the proof ultimately relies on the computation of certain…
This paper presents an initial exploration of high frequency records of extreme wind speed in two steps. The first consists in finding the suitable extreme distribution for $120$ measuring stations in Switzerland, by comparing three known…
Maximum likelihood estimation is a fundamental computational problem in statistics. In this note, we give a bound for the maximum likelihood degree of algebraic statistical models for discrete data. As usual, such models are identified with…
We have analyzed the risks of possible development of bubbles in the Swiss residential real estate market. The data employed in this work has been collected by comparis.ch, and carefully cleaned from duplicate records through a procedure…
The saddlepoint approximation to the likelihood, and its corresponding maximum likelihood estimate (MLE), offer an alternative estimation method when the true likelihood is intractable or computationally expensive. However, maximizing this…
We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many…
This paper formalizes a latent variable inference problem we call {\em supervised pattern discovery}, the goal of which is to find sets of observations that belong to a single ``pattern.'' We discuss two versions of the problem and prove…
Gilmer has recently shown that in any nonempty union-closed family $\mathcal F$ of subsets of a finite set, there exists an element contained in at least a proportion $.01$ of the sets of $\mathcal F$. We improve the proportion from $.01$…
It is shown that an equiprobability hypothesis leads to a scenario in which it is possible to predict the outcome of a single toss of a fair coin with a success probability greater than 50%. We discuss whether this hypothesis might be…
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…
This study provides the solution to the equity premium puzzle. The new model was developed by including the behavior of investors toward risk in financial markets in prior studies. The calculations of this newly tested model show that the…
How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We…
We revisit the problem of the existence of the maximum likelihood estimate for multi-class logistic regression. We show that one method of ensuring its existence is by assigning positive probability to every class in the sample dataset. The…
The maximum likelihood degree of a statistical model refers to the number of solutions, where the derivative of the log-likelihood function is zero, over the complex field. This paper examines the maximum likelihood degree of the parameter…
The skew-normal and the skew-$t$ distributions are parametric families which are currently under intense investigation since they provide a more flexible formulation compared to the classical normal and $t$ distributions by introducing a…
In this paper we study the frequentist convergence rate for the Latent Dirichlet Allocation (Blei et al., 2003) topic models. We show that the maximum likelihood estimator converges to one of the finitely many equivalent parameters in…
We collect here various conjectures on congruences made by the author in a series of papers, some of which involve binary quadratic forms and other advanced theories. Part A consists of 100 unsolved conjectures of the author while…
The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice $L$ with more than one element contains a…
This expository note describes how to apply the method of maximum likelihood to estimate the parameters of the ``$q$-exponential'' distributions introduced by Tsallis and collaborators. It also describes the relationship of these…