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

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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
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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...
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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.

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