Related papers: Solving the 100 Swiss Francs Problem

In the context of auctions for digital goods, an interesting random sampling auction has been proposed by Goldberg, Hartline, and Wright [2001]. This auction has been analyzed by Feige, Flaxman, Hartline, and Kleinberg [2005], who have…

In 2008, Schmidt and Tuller stated a conjecture concerning optimal packing and covering of integers by translates of a given three-point set. In this note, we confirm their conjecture and relate it to several other problems in…

We give a new bound on the probability that the random sum $\xi_1 v_1 +...+ \xi_n v_n$ belongs to a ball of fixed radius, where the $\xi_i$ are iid Bernoulli random variables and the $v_i$ are vectors in $\R^d$. As an application, we prove…

The maximum likelihood degree (ML degree) measures the algebraic complexity of a fundamental optimization problem in statistics: maximum likelihood estimation. In this problem, one maximizes the likelihood function over a statistical model.…

The last success problem is an optimal stopping problem that aims to maximize the probability of stopping on the last success in a sequence of independent $n$ Bernoulli trials. In the classical setting where complete information about the…

We address the problem of extractive question answering using document-level distant super-vision, pairing questions and relevant documents with answer strings. We compare previously used probability space and distant super-vision…

We present a state-of-the-art model for fine-grained probability estimation of propositions conditioned on context. Recent advances in large language models (LLMs) have significantly enhanced their reasoning capabilities, particularly on…

In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of…

The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter…

The Frankl conjecture, also known as the union-closed sets conjecture, states that in any finite non-empty union-closed family, there exists an element in at least half of the sets. From an optimization point of view, one could instead…

In 1993, just about a century after the epoch of Classical Invariant Theory and almost 30 years after Mumford's seminal book on Geometric Invariant Theory, Bernd Sturmfels approached the subject from a new, algorithmic perspective in his…

A very popular class of models for networks posits that each node is represented by a point in a continuous latent space, and that the probability of an edge between nodes is a decreasing function of the distance between them in this latent…

100 prisoners and a light bulb is a long standing mathematical puzzle. The problem was studied mostly in 2002 [5], 2003 [1], and 2004 [3]. Solutions in published articles had average number of visits above 3850, but best solutions on forums…

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 propose an alternative refined de Sitter conjecture. It is given by a natural condition on a combination of the first and second derivatives of the scalar potential. We derive our conjecture in the same weak coupling, semi-classical…

In this paper, Sturm-Liouville problem for difference equations is considered with potential function q(n). The representations of solutions are obtained by variation of parameters method. These solutions are proved, using summation by…

In this paper, we study a game with positive or plus infinite expectation and determine the optimal proportion of investment for maximizing the limit expectation of growth rate per attempt. With this objective, we introduce a new pricing…

We establish connections between: the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by…

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional…

I describe a trick for training flow models using a prescribed rule as a surrogate for maximum likelihood. The utility of this trick is limited for non-conditional models, but an extension of the approach, applied to maximum likelihood of…