%%% Title of object: Marginal Probability. Its use in Bayesian %%% Statistics as %%% the Evidence of Models and Bayes Factors %%% Canonical Name: MarginalProbabilityItsUseInBayesianStatisticsAsTheEvidenceOfModelsAndBayesFactors3 %%% Type: Topic %%% Created on: 2010-08-24 22:47:03 %%% Modified on: 2010-08-24 22:47:03 %%% Creator: pericchi %%% Modifier: jkimmel %%% %%% Classification: msc:62F03, msc:62F15 %%% Preamble: \documentclass[10pt]{article} % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %%% Content: %\documentclass[11pt]{article} %\usepackage[activeacute,spanish]{babel} %\usepackage{graphicx} %\usepackage{amsmath,amsthm,amssymb} %\oddsidemargin .3 cm \evensidemargin .3 cm \textheight 21 cm %\topmargin -1 cm \textwidth 6.3 in %\newcommand{\R}{\mathbb{R}} \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\bx}{\bf{x}} \newcommand{\btheta}{\bf{\theta}} %\theoremstyle{definition} %\newtheorem{thm}{Theorem}[section] %\newtheorem{cor}[thm]{Corollary} %\newtheorem{lem}[thm]{Lemma} %\newtheorem{exmp}{Example} %\newtheorem{res}{Result} %\newtheorem*{defin}{Definition} \begin{document} \title{Marginal Probability. Its use in Bayesian Statistics as the Evidence of Models and Bayes Factors} \author{Luis Ra\'ul Pericchi, Department of Mathematics and Biostatistics and Bioinformatics Center,\\ University of Puerto Rico, Rio Piedras, San Juan, Puerto Rico.\thanks{email address: luarpr@uprrp.edu,This work sponsored in part by NIH Grant: P20-RR016470}} \date{} \maketitle \textbf{Keywords:}\emph{Bayes Factors, Evidence of Models, Intrinsic Bayes Factors, Intrinsic Priors, Posterior Model Probabiities} \section{Definition} Suppose that we have vectors of random variables $[\textbf{v,w}]=[v_1,v_2,\ldots,v_I,w_1,\ldots,w_J]$ in $\Re^{(I+J)}$. Denote as the \textbf{joint} density function: $f_{\textbf{v,w}}$, which obeys:$f_{\textbf{v,w}}(v,w) \ge 0$ and \\ $\int^{\infty}_{-\infty}\ldots\int^{\infty}_{-\infty} f_{\textbf{v,w}}(v,w) dv_1\ldots dv_I dw_1\ldots dw_I=1$. Then the probability of the set $[A_v,B_w]$ is given by \[ P(A_v,B_w)=\int \ldots \int_{A_v,B_w} f_{\textbf{v,w}}(v,w) \textbf{dv} \textbf{dw}. \] The the \textbf{marginal} density $f_{\textbf{v}}$ is obtained as \[ f_{\textbf{v}}(v)=\int^{\infty}_{-\infty}\ldots \int^{\infty}_{-\infty}f_{\textbf{v,w}}(v,w) dw_1\ldots dw_I. \] The the \textbf{marginal probability} of the set $A_v$ is then obtained as, \[ P(A_v)=\int \ldots \int_{A_v} f_{\textbf{v}}(v) dv. \] We have assumed that the random variables are continuous. When they are discrete, integrals are substituted by sums.\\ We proceed to present an important application of marginal densities to construct the \emph{Evidence of the Model} and marginal probabilities for measuring the \emph{Bayesian Probability of a Model}. \section{Measuring the Evidence in Favor of a Model} In Statistics, a parametric model, is denoted as $f(x_1,\ldots,x_n|\theta_1,\ldots,\theta_k)$, where $\textbf{x}=(x_1,\ldots, x_n)$ is the vector of $n$ observations and $\btheta=(\theta_1,\ldots,\theta_k)$ is the vector of $k$ parameters. For instance we may have $n$ observations normally distributed and the vector of parameters is $(\theta_1,\theta_2)$ the location and scale respectively, denoted by $f_{\textit{Normal}}(\textbf{x}|\btheta)=\prod_{i=1}^n \frac{1}{\sqrt{2 \pi} \theta_2} \exp(-\frac{1}{2 \theta^2_2} (x_i-\theta_1)^2)$.\\ Assume now that there is reason to suspect that the location is zero. As a second example, it may be suspected that the sampling model which usually has been assumed Normally distributed, is instead a Cauchy, $f_{\textit{Cauchy}}(\textbf{x}|\btheta)=\prod_{i=1}^n \frac{1}{\pi \theta_2}\frac{1}{(1+(\frac{x_i-\theta_1}{\theta_2})^{2})}$. The first problem is a \emph{hypothesis test} denoted by \[H_0: \theta_1=0 \mbox{ VS } H_1: \theta_1 \neq 0, \] and the second problem is a \emph{model selection} problem: \[ M_0: f_{\textit{Normal}} \mbox{ VS } M_1: f_{\textit{Cauchy}}. \] How to measure the evidence in favor of $H_0$ or $M_0$? Instead of maximizing likelihoods as it is done in traditional significance testing, in Bayesian statistics the central concept is \textit{the evidence} or \textit{marginal probability density} \[ m_j({\bx})=\int f_j({\bx}|\btheta_j) \pi(\btheta_j) d\btheta_j, \] where $j$ denotes either model or hypothesis $j$ and $\pi(\btheta_j)$ denotes the prior for the parameters under model or hypothesis $j$.\\ Marginal probabilities embodies the likelihood of a model or hypothesis in great generality and can be claimed it is the natural probabilistic quantity to compare models. \section{Marginal Probability of a Model} Once the marginal densities of the model j, for $j=1,\ldots,J$ models have been calculated and assuming the prior model probabilities $P(M_j), j=1,\ldots, J$ with $\sum_{j=1}^J P(M_j)=1$ then, using Bayes Theorem, \textit{the marginal probability of a model} $P(M_j|\bx)$ can be calculated as, \[ P(M_j|\bx)=\frac{m_j({\bx}) \cdot P(M_j)}{\sum_{i=1}^n m_i({\bx}) \cdot P(M_i)}. \] We have then the following formula for any two models or hypotheses: \[ \frac{P(M_j|\bx)}{P(M_i|\bx)}= \frac{P(M_j)}{P(M_i)} \times \frac{m_j({\bx})}{m_i({\bx})}, \] or in words: Posterior Odds equals Prior Odds times Bayes Factor, where the Bayes Factor of $M_j$ over $M_i$ is \[ B_{j,i}=\frac{m_j({\bx})}{m_i({\bx})}, \] Jeffreys (1961).\\ In contrast to \textit{p-values}, which have interpretations heavily dependent on the sample size $n$, and its definition is not the same as the scientific question, the posterior probabilities and Bayes Factors address the scientific question: "how probable is model or hypothesis j as compared with model or hypothesis i?", and the interpretation is the same for any sample size, Berger and Pericchi (1996a, 2001). Bayes Factors and Marginal Posterior Model Probabilities have several advantages, like for example large sample consistency, that is as the sample size grows the Posterior Model Probability of the sampling model tends to one. Furthermore, if the goal is to predict future observations $y_f$ it is \textbf{not} necessary to select one model as \textit{the} predicting model since we may predict by the so called Bayesian Model Averaging, which if quadratic loss is assumed, the optimal predictor takes the form, \[ E[Y_f|\bx]= \sum_{j=1}^J E[Y_f|\bx, M_j] \times P(M_j|\bx), \] where $E[Y_f|\bx,M_j]$ is the expected value of a future observation under the model or hypothesis $M_j$. \section{Intrinsic Priors for Model Selection and Hypothesis Testing} Having said some of the advantages of the marginal probabilities of models, the question arises: how to assign the conditional priors $\pi(\theta_j)$? In the two examples above which priors are sensible to use? The problem is \textbf{not} a simple one since it is not possible to use the usual Uniform priors since then the Bayes Factors are undetermined. To solve this problem with some generality, Berger and Pericchi (1996a,b) introduced the concepts of Intrinsic Bayes Factors and Intrinsic Priors. Start by splitting the sample in two sub-samples $\bx=[\bx(l),\bx(-l)]$ where the training sample $\bx(l)$ is as small as possible such that for $j=1,\ldots,J: 0