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priors - More on Prior Distributions Reading Ho Chapter 3...

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More on Prior Distributions September 24, 2010 Reading: Hoff Chapter 3 September 24, 2010
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Binomial as an Exponential Family Rearrange Bernoulli (a Binomial with n = 1) to get in to exponential family form: p ( y | π ) = π y (1 - π ) 1 - y = π 1 - π y (1 - π ) = exp log π 1 - π y (1 - π ) = c ( θ ) exp( θ y ) where θ = log( π/ (1 - π )) c ( θ ) = (1 + exp( θ )) - 1 Natural parameter θ is the log-odds
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Conjugate Prior for θ p ( θ ) c ( θ ) n 0 exp( θ n 0 t 0 ) 1 1 + exp( θ ) n 0 exp( θ n 0 t 0 ) where θ = log( π/ (1 - π )) and t 0 is prior expected value for the probability that Y is 1. I Normalizing constant? Z -∞ 1 1 + exp( θ ) n 0 exp( θ n 0 t 0 ) d θ I Implied Prior distribution for π ?
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Normalizing Constant c = Z -∞ 1 1 + exp( θ ) n 0 exp( θ n 0 t 0 ) d θ Change back to π I substitute θ = log( π/ (1 - π )) I find new range of integration I need to account for change in units via the Jacobian of the transformation“ d θ = Δ( π ) d π If θ = h ( π ), then the Jacobian is dh d π
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continued d log( π 1 - π ) d π = 1 - π π 1 1 - π + π (1 - π ) 2 = [ π (1 - π )] - 1 d θ = π - 1 (1 - π ) - 1 d π 00 So c = Z -∞ 1 1 + exp( θ ) n 0 exp( θ n 0 t 0 ) d θ = Z 1 0 π n 0 t 0 (1 - π ) n 0 - n 0 t 0 π - 1 (1 - π ) - 1 d π = Z 1 0 π n 0 t 0 - 1 (1 - π ) n 0 - n 0 t 0 - 1 d π = B ( n 0 t 0 , n 0 - n 0 t 0 )
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Change of Variables 1 = Z -∞ 1 B ( n 0 t 0 , n 0 - n 0 t 0 ) 1 1 + exp( θ ) n 0 exp( θ n 0 t 0
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