10 the log likelihood of θ is then given by n

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(10) The log-likelihood of θ is then given by n summationdisplay i =1 log p ( x i | θ ) = n summationdisplay i =1 ( x i ) log θ - (11) Setting the derivate wrt θ to 0, we get the MLE ˆ θ as, ˆ θ = n i =1 x i n (12) 6.2 The log-likelihood of μ is summationdisplay i log P ( x i | f ( μ )) = summationdisplay i ( x i ) log( f ( μ )) - nf ( μ ) Setting the derivative wrt μ to zero, we have for the MLE ˆ μ , f μ ) = n i =1 x i n = ˆ θ 6.3 Let θ have a uniform prior and consider parametrization θ = f ( μ ) = μ 2 (13) Let y be the number of heads, z - number of tails, then MAP estimate of θ is ˆ θ = y y + z (14) Now let us find the MAP of μ . Because the prior on θ is uniform, probability density of μ is f ( μ ) = 2 μ, μ [0 , 1] ( = 2 = 2 μdμ ) (15) so we need to find arg max μ 2( μ 2 ) y (1 - μ 2 ) z μ = arg max μ μ 2 y +1 (1 - μ 2 ) z (16) and setting the derivative to zero we get (2 y +1) μ 2 y (1 - μ 2 ) z - z 2 μμ 2 y +1 (1 - μ 2 ) z = 0 μ 2 y (1 - μ 2 ) z - 1 ((2 y +1)(1 - μ 2 ) - 2 2 ) = 0 (17) μ 2 = 2 y + 1 2 y + 2 z + 1 negationslash = y y + z = ˆ θ (18) 4
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