week3-notes - Poisson distribution BAYESIAN ANALYSIS Week 3...

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Poisson distribution BAYESIAN ANALYSIS: Week 3 and 4 - Poisson Distribution July 3rd and 17th 2008 This week we are interested in the Poisson distribution. We first look at the probabil- ity density function and discuss some of its properties. We then compute the likelihood associated with the Poisson distribution and work out why the gamma distribution is its conjugate prior. Finally, we discuss how the negative binomial distribution arises as a gamma mixture of Poisson distributions. R code for the example is included in the Ap- pendix. 1 Poisson distribution The probability density function for a single observation of the Poisson distribution is: P ( x | λ ) = λ x e - λ x ! (1) Assuming that a series of observations x 1 ,...,x n are sampled from P ( x | λ ) and that each observation is independent and is indentically distributed (i.i.d), the joint probability density function for x 1 n is the product of the individual pdfs: P ( x 1 ,...x n | λ ) = n Y i =1 P ( x i | λ ) = n Y i =1 λ x i e - λ x i ! = λ P n i =1 x i e - Q n i =1 x i ! (2) The likelihood function for λ will be proportional to the joint pdf and only needs the terms that involve λ (i.e. we can drop the denominator of equation 2): L ( λ | x 1 n ) P ( x 1 n | λ ) λ P n i =1 x i e - (3) 1 Notes: D. Ricard. Bayesian Group:W. Blanchard, D. Ricard, D.P. Tittensor, C. Minto, T. Davies, S. Anderson
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Poisson distribution The maximum likelihood estimate of λ (called ˆ λ hereafter) can be found by first taking the log of the likelihood: log ( L ( λ | x 1 ,...,x n )) = l ( λ | x 1 n ) n X i =1 x i log ( λ ) - λn (4) and then taking the derivative with respect to λ : dl n X i =1 x i 1 λ - n (5) setting the derivative equal to 0 and finally solving for λ : n X i =1 x i 1 λ - n = 0 ˆ λ = n i =1 x i n (6) In order to ascertain that this is indeed the maximum likelihood estimate, we would also take the second order derivative. It can be shown that the second derivative satifies the criteria for global optimality for λ .
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This note was uploaded on 10/18/2010 for the course IEOR 4702 taught by Professor Kou during the Spring '10 term at Columbia.

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week3-notes - Poisson distribution BAYESIAN ANALYSIS Week 3...

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