Mathematical frameworks for data analysis and uncertainty, including statistical theory, probability theory, stochastic processes, statistical inference, and applications to data science
9 papersThis document serves as a concise digest of exponential family distributions, detailing their definitions, fundamental properties, and duality with Bregman divergences. It also reviews the Riemannian and information geometries associated with these statistical manifolds. The paper provides "flash cards" summarizing the properties and parameterizations of common exponential family distributions like Gaussian, Poisson, and Binomial.
This paper introduces a generalization of the Jensen-Shannon Divergence (JSD), a method used to measure the difference between probability distributions. It explores using different types of "means" (like arithmetic, geometric, and harmonic means) to create new JSD variations and provides closed-form formulas for these variations in specific cases like mixtures of Gaussian or Cauchy distributions.
This lecture note explores the "cutoff phenomenon" in Markov chains, where some processes abruptly transition to equilibrium instead of gradually converging. It introduces key concepts like mixing times, curvature, and varentropy, applying them to various models like card shuffling and random walks. The author presents a novel criterion based on varentropy for predicting cutoff behavior.
This study proposes the 'binomial-logit-t' model to improve analysis of bounded count data (data with a maximum value), particularly in scenarios with outliers like medication adherence. It handles outliers and accounts for overdispersion more effectively compared to existing methods, providing more accurate parameter estimates. The model is demonstrated on a medication adherence dataset and supported by simulations.
This paper presents a new mathematical proof for the sub-Gaussian norm concentration inequality, offering potentially tighter bounds than previous approaches using a novel "averaged moment generating function" (AMGF). The method applies to both vectors and matrices and directly analyzes the norm's concentration, unlike methods relying on the union bound.
The study proposes a framework for informal statistical inference with three key principles: generalizations beyond the data, data as evidence, and probabilistic language. Classroom episodes illustrate how teachers can foster inferential reasoning by emphasizing purposeful investigations, connecting conclusions to evidence, and promoting probabilistic language to express uncertainty.
The paper proposes the "exponentiated generalized" (EG) class of distributions, a new method of adding two shape parameters to existing continuous distributions using a double Lehmann alternative construction. This extends the flexibility of distributions, particularly in the tails, enabling improved modeling in various fields. The study explores mathematical properties, including moments, generating functions, and order statistics, and demonstrates applications to real datasets from diverse areas like agriculture and material science, finding superior fits compared to existing models.
This paper details the implementation of elastic net regularization for all generalized linear models (GLMs), Cox models with extended data types, and a simplified relaxed lasso within the glmnet R package. It also covers utility functions for evaluating fitted model performance.
This study argues that R-squared is a more informative and robust metric for evaluating regression models compared to SMAPE, MAE, MAPE, MSE, and RMSE. Through several synthetic use cases and analysis of two real medical datasets, the authors demonstrate that R-squared provides a more accurate assessment of model performance, particularly when dealing with skewed data or outliers. They propose using R-squared as the standard metric for regression analysis evaluation.