Table of Links Abstract and 1. Introduction Abstract and 1. Introduction The Compound Decision Paradigm
Parametric Priors
Nonparametric Prior Estimation
Empirical Bayes Methods for Discrete Data
Empirical Bayes Methods for Panel Data
Conclusion The Compound Decision Paradigm The Compound Decision Paradigm Parametric Priors Parametric Priors Nonparametric Prior Estimation Nonparametric Prior Estimation Empirical Bayes Methods for Discrete Data Empirical Bayes Methods for Discrete Data Empirical Bayes Methods for Panel Data Empirical Bayes Methods for Panel Data Conclusion Conclusion Appendix A. Tweedie’s Formula Appendix A. Tweedie’s Formula Appendix B. Predictive Distribution Comparison Appendix B. Predictive Distribution Comparison References References 3. Parametric Priors The hyperparameters, v0 and s0 can be estimated by maximum likelihood. The null hypotheses H0 : µi = 0 have p-values, Another prominent option for the penalty, P, is treat the coordinates of β as if they were drawn iidly from the Cauchy distribution, as considered by Johnstone and Silverman (2004) and Castillo and van der Vaart (2012). Although such priors are generally structured to shrink coefficients toward zero, this is typically rationalized by some form of prior standardization of the design matrix. The empirical aspect of these procedures is generally restricted to choice of the tuning parameter λ representing the scale of the prior density. However, more flexibility can be achieved by permitting larger parametric families, for example Azevedo et al (2020) a class of Student priors for large-scale A/B testing settings with location, scale and degrees of freedom of the prior estimated by maximum likelihood. Returning to the simple Gaussian sequence model with scalar parameeters θi , in Figure 2 we contrast several forms of shrinkage: linear shrinkage with the classical Stein rule, the lasso procedure that shrinks moderately when y is near zero, and the Cauchy penalty that shrinks very aggressively near zero while large departures from zero are shrunken very little. It should be stressed that tuning the location and scale of these penalties offers some flexibility, but the selection of a functional form for such parametric priors involves a leap of Bayesian faith that may trouble some researchers. Thus far we have focused entirely on settings in which our base model, φ(y|θ) is Gaussian. Parametric mixture priors play an important role in many other corners of statistics. Poisson models are often paired with gamma mixing, and the modern literature on survival analysis, is permeated by parametric models of “frailty.” As anticipated by the pioneering critique of Heckman and Singer (1984), choosing a specific parametric model for frailty can be difficult so it is natural to turn to nonparametric methods for guidance. Authors:
(1) Roger Koenker;
(2) Jiaying Gu. Authors: Authors: (1) Roger Koenker; (2) Jiaying Gu. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

The Efron-Morris Rule: A Practical Application of the Empirical Bayes Paradigm

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

Untitled Story

Beyond Gaussian Mixtures: Applying Empirical Bayes to Discrete Data Problems

Bayesian Product Management: Getting Out of Your Own Head

Solving a Problem with Bayes’ Theorem and Decision Tree

What Matters to Trump Supporters/Opposers? A Bayesian Analysis of Polling Data

Empirical Bayes Methods for Compound Decision Problems: Principles and Applications

Beyond Gaussian Mixtures: Applying Empirical Bayes to Discrete Data Problems

Bayesian Product Management: Getting Out of Your Own Head

Solving a Problem with Bayes’ Theorem and Decision Tree

What Matters to Trump Supporters/Opposers? A Bayesian Analysis of Polling Data

Empirical Bayes Methods for Compound Decision Problems: Principles and Applications

Light-Mode

Classic

Newspaper

Dark-Mode

Neon Noir

Minty

HN StartUps