# Use likelihood based on one observation y bern π log

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Use likelihood based on one observation Y Bern( π ) log L ( θ ) = log( p ( y | π )) = Y log( π ) + (1 - Y ) log(1 - π ) I d d π log( L ( π )) = Y - (1 - Y ) / (1 - π ) d d π log( L ( π )) = - Y 2 + (1 - Y ) / (1 - π ) 2 E Y | π [ - d d π log( L ( π )] = E[ Y ] 2 - (1 - E [ Y ]) / (1 - π ) 2 = 1 π - 1 1 - π I I ( π ) = [ π (1 - π )] - 1 I p J ( π ) p I ( π ) = π - 1 / 2 (1 - π ) - 1 / 2 I Jeffreys’ prior for π is Beta(1 / 2 , 1 / 2)

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With the Natural Parameter θ I Find I ( θ ) and p J ( θ ) p I ( θ ) log( p ( y | θ )) = y θ - log(1 + exp( θ )) I Apply the change of variables h ( π ) = θ p ( π ) p I ( h ( π )) dh ( π ) d π π is Beta(1 / 2 , 1 / 2) Same result! Find Jeffreys’ prior in any parameterization and use change of variables gives the same result as if we start with that parametrization and find Jeffreys’ prior directly!
Plot > x = seq(0.001, 0.999, length = 1000) > plot(x, dbeta(x, 0.5, 0.5), type = "l", + main = "Beta(1/2,1/2 Density", xlab = expression(pi), + ylab = "density") 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 Beta(1/2,1/2 Density π density

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Which Prior? I Unless n and/or y is small little difference in posterior distributions for inference about π I prior sample size n 0 for Jeffreys’ prior is 1, versus 2 for uniform I both have prior expectation 1 / 2 I more dispersion with Jefferys’ prior I Different story in higher dimensions or with hypothesis testing
Conjugate Priors I All exponential families have a conjugate prior I Results under conjugate priors in exponential famililes can be sensitive to“outliers”(prior outliers and data outliers) due to linear updating of n 0 and t 0 with data I Mixtures of conjugate priors resolve most of these problems I Still conjugate or“closed under sampling”? (HW)

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