homework1-solution - CSE 555 Spring 2010 Homework 1:...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CSE 555 Spring 2010 Homework 1: Bayesian Decision Theory Jason J. Corso Computer Science and Engineering SUNY at Buffalo SUNY jcorso@buffalo.edu Date Assigned 13 Jan 2010 Date Due 1 Feb 2010 Homework must be submitted in class. No late work will be accepted. Problem 1: Bayesian Decision Rule (30%) Suppose the task is to classify the input signal x into one of K classes { 1 , 2 , . . . , K } such that the action ( x ) = i means classifying x into class i . The Bayesian decision rule is to maximize the posterior probability Bayes ( x ) = * = arg max p ( | x ) . Suppose we replace it by a randomized decision rule , which classifies x to class i following the posterior probability p ( = i | x ) , i.e., rand ( x ) = p ( | x ) . Solution: Maximizing the posterior probability is equivalent to minimizing the overall risk. Using the zero-one loss function, the overall risk for the Bayes Decision Rule is: R Bayes = I R ( Bayes ( x ) | x ) p ( x ) dx = I n 1- max P ( j | x ) | j = 1 , ..., k o p ( x ) dx For simplicity, the class with max posterior probability is abbreviated as max , and we get: R Bayes = I (1- P ( max | x )) p ( x ) dx. 1. What is the overall risk R rand for this decision rule? Derive it in terms of the posterior probability using the zero-one loss function. Solution: 1 For any given x , the probability of each class j = 1 , ..., k being the correct class is P ( j | k ) . With the randomized algorithm, it will select the correct class with probability P ( j | k ) , which means that it will select the wrong class with probability 1- P ( j | k ) . Thus, the zero-one conditional risk will become j P ( j | x ) 1- P ( j | x ) on average. Thus, R rand = I n X j P ( j | x ) 1- P ( j | x ) o p ( x ) dx = I n X j P ( j | x )- P ( j | x ) 2 o p ( x ) dx = I h 1- X j P ( j | x ) 2 i p ( x ) dx 2. Show that this risk R rand is always no smaller than the Bayes risk R Bayes . Thus, we cannot benefit from the randomized decision. Solution: Proving R rand R Bayes is equivalent to proving j P ( j | x ) 2 P ( max | x ) : X j P ( j | x ) 2 X j P ( j | x ) P ( max | x ) = P ( max | x ) , thus proved. R rand is always no smaller than R Bayes ....
View Full Document

This note was uploaded on 11/24/2010 for the course STAT 201a taught by Professor Wu during the Spring '10 term at Pasadena City College.

Page1 / 7

homework1-solution - CSE 555 Spring 2010 Homework 1:...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online