Quiz7 - m successes are recorded. Find the MML rule. You...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
ISyE8843 Brani Vidakovic Friday 29/10/04 Name: Quiz 7 Wallace-Freeman MML Estimator. Recall that the Minimum Message Length (MML) estimate, based on X 1 ,...,X n f ( x | θ ) is defined as argmin θ [ - log π ( θ ) - log n Y i =1 f ( x i | θ ) + 1 2 log |I ( θ ) | ] , where π ( θ ) is the prior and I ( θ ) is the Fisher information matrix. This is equivalent to maximizing weighted posterior π ( θ ) Q n i =1 f ( x i | θ ) |I ( θ ) | 1 / 2 . Problem. Suppose a single observation X | θ is coming from the Negative Binomial NB ( m,θ ) , with p.m.f. f ( x | θ ) = ± m + x - 1 x θ m (1 - θ ) x , and that the prior on θ is Beta B e ( α,β ) . For example, the observation X could be interpreted as the number of failures incurred until
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: m successes are recorded. Find the MML rule. You will need the following facts: • The expectation of X ∼ NB ( m,θ ) , is EX = m (1-θ ) θ . • The Fisher information h-E ∂ 2 ∂θ log f ( x | θ ) i for θ from the Negative Binomial NB ( m,θ ) distribution is I ( θ ) = m θ 2 (1-θ ) . Prove this! • The mode of Beta B e ( a,b ) distribution is a-1 a + b-2 . Also, X | θ ∼ NB ( m,θ ) , and B e ( α,β ) are conjugate; the posterior is straightforward. 1...
View Full Document

This note was uploaded on 10/23/2011 for the course ISYE 8843 taught by Professor Vidakovic during the Spring '11 term at Georgia Institute of Technology.

Ask a homework question - tutors are online