priors - More on Prior Distributions September 24, 2010...

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Unformatted text preview: More on Prior Distributions September 24, 2010 Reading: Hoff Chapter 3 September 24, 2010 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 Conjugate Prior for θ p ( θ ) ∝ c ( θ ) n exp( θ n t ) ∝ 1 1 + exp( θ ) n exp( θ n t ) where θ = log( π/ (1- π )) and t is prior expected value for the probability that Y is 1. I Normalizing constant? Z ∞-∞ 1 1 + exp( θ ) n exp( θ n t ) d θ I Implied Prior distribution for π ? Normalizing Constant c = Z ∞-∞ 1 1 + exp( θ ) n exp( θ n t ) 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 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 exp( θ n t ) d θ = Z 1 π n t (1- π ) n- n t π...
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priors - More on Prior Distributions September 24, 2010...

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