Exponential Families in Theory and Practice

During the past half-century, exponential families have attained a position at the center of parametric statistical inference. Theoretical advances have been matched, and more than matched, in the world of applications, where logistic regression by itself has become the go-to methodology in medical statistics, computer-based prediction algorithms, and the social sciences. This book is based on a one-semester graduate course for first year Ph.D. and advanced master's students. After presenting the basic structure of univariate and multivariate exponential families, their application to generalized linear models including logistic and Poisson regression is described in detail, emphasizing geometrical ideas, computational practice, and the analogy with ordinary linear regression. Connections are made with a variety of current statistical methodologies: missing data, survival analysis and proportional hazards, false discovery rates, bootstrapping, and empirical Bayes analysis. The book connects exponential family theory with its applications in a way that doesn't require advanced mathematical preparation.

在過去的半個世紀中，指數家族已經成為參數統計推論的核心。在應用領域中，理論進展已經得到了匹配，甚至超越了匹配，其中單獨的邏輯回歸已成為醫學統計、基於計算機的預測算法和社會科學中的首選方法。本書是基於一個學期的研究生課程，針對第一年博士和高級碩士學生。在介紹單變量和多變量指數家族的基本結構後，詳細描述了它們在廣義線性模型中的應用，包括邏輯回歸和泊松回歸，強調幾何思想、計算實踐和與普通線性回歸的類比。與眾多當前統計方法建立了聯繫：缺失數據、生存分析和比例風險、虛假發現率、自助法和經驗貝葉斯分析。本書以一種不需要高級數學準備的方式將指數家族理論與應用相結合。