arXiv subjects

Andrew K. Yang

Publications and source records attributed to Andrew K. Yang.

4 recordsLinked to original sources

arXiv2026

Applications of Improvements to the Pythagorean Won-Loss Expectation in Optimizing Rosters

Bill James' Pythagorean formula has for decades done an excellent job estimating a baseball team's winning percentage from very little data: if the average runs scored and allowed are denoted respectively by ${\rm RS}$ and ${\rm RA}$, there is some $γ\approx 2$ such that the winning percentage is approximately ${\rm RS}^γ/ ({\rm RS}^γ+ {\rm RA}^γ)$. One use case is to determine the value of potential signings to the team, as it allows us to estimate how many more wins one obtains over a season given an estimated change in run production and concession. We summarize earlier work on the subject, and extend the earlier theoretical model of Miller (who assumed the home and away teams' runs arise from independent Weibull distributions with the same shape parameter $γ$; this has been observed to describe the observed run data well and yields a win probability equivalent to that of James' formula). We extend this work to model runs scored and allowed as being drawn from independent Weibull distributions with different shape parameters, and then consider the first and second moments to solve a system of four equations in the four unknowns. Doing so fits the training data better, yielding a higher winning percentage over the last 30 MLB seasons (1994 to 2023). This comes at a small cost as we no longer have a closed form expression for the win probability, but must evaluate a two-dimensional integral of Weibull distributions and numerically estimate the solutions to the system of equations. These are trivial to do with simple computational programs.

stat.AP↗

arXiv2026

The German Tank Problem with Multiple Factories

During the Second World War, estimates of the number of tanks deployed by Germany were critically needed. The Allies adopted a successful statistical approach to estimate this information: assume that the tanks are sequentially numbered starting from, say, 1, and ending at an unknown positive integer $N$. If we observe the numbers of $k$ tanks, then the best linear unbiased estimator for $N$ is $M(1+1/k)-1$ where $M$ is the maximum observed serial number. While this approach was successful, there are many more adversarial situations where the approach for the original German Tank Problem falls short. Typically the number of ``factories'' is a possibly unknown $l>1$, and tanks produced by different factories may have serial numbers in disjoint ranges that are often separated by unknown amounts. Clark, Gonye and Miller (CGM) presented an unbiased estimator for $N$ when the minimum serial number is unknown. So if one can identify which samples correspond to which factory, one can then estimate each factory's range using CGM's method, and sum them for an estimate of the rival's total productivity. We present a procedure to estimate the total productivity and prove that it is effective when $\log l/\log k$ is sufficiently small. In the final section, we show that if we have a small number of samples, we can make an estimator that performs orders of magnitude better when given additional information about the size of the gaps.

math.ST↗

arXiv2024

Strong Data Processing Inequalities and their Applications to Reliable Computation

In 1952, von Neumann gave a series of groundbreaking lectures that proved it was possible for circuits consisting of 3-input majority gates that have a sufficiently small independent probability $δ> 0$ of malfunctioning to reliably compute Boolean functions. In 1999, Evans and Schulman used a strong data-processing inequality (SDPI) to establish the tightest known necessary condition $δ< \frac{1}{2} - \frac{1}{2\sqrt{k}}$ for reliable computation when the circuit consists of components that have at most $k$ inputs. In 2017, Polyanskiy and Wu distilled Evans and Schulman's SDPI argument to establish a general result on the contraction of mutual information in Bayesian networks. In this essay, we will first introduce the problem of reliable computation from unreliable components and establish the existence of noise thresholds. We will then provide an exposition of von Neumann's result with 3-input majority gates and extend it to minority gates. We will then provide an introduction to SDPIs, which have many applications, including in statistical mechanics, portfolio theory, and lower bounds on statistical estimation under privacy constraints. We will then use the introduced material to provide an exposition of Polyanskiy and Wu's 2017 result on Bayesian networks, from which the 1999 result of Evans-Schulman follows.

cs.IT↗

arXiv2023

The Reversed Zeckendorf Game

Zeckendorf proved that every natural number $n$ can be expressed uniquely as a sum of non-consecutive Fibonacci numbers, called its Zeckendorf decomposition. Baird-Smith, Epstein, Flint, and Miller created the Zeckendorf game, a two-player game played on partitions of $n$ into Fibonacci numbers which always terminates at a Zeckendorf decomposition, and proved that Player 2 has a winning strategy for $n\geq 3$. Since their proof was non-constructive, other authors have studied the game to find a constructive winning strategy, and lacking success there turned to related problems. For example, Cheigh, Moura, Jeong, Duke, Milgrim, Miller, and Ngamlamai studied minimum and maximum game lengths and randomly played games. We explore a new direction and introduce the reversed Zeckendorf game, which starts at the ending state of the Zeckendorf game and flips all the moves, so the reversed game ends with all pieces in the first bin. We show that Player 1 has a winning strategy for $n = F_{i+1} + F_{i-2}$ and solve various modified games.

math.CO↗