MIT15_097S12_lec15

When we specify for example a prior of 7 and 3 it is

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Unformatted text preview: (α − 1) log θ + (β − 1) log(1 − θ) − log B (α, β )) . ˆ Differentiating and setting to zero at θMAP , mH m − mH α−1 β−1 − + − =0 ˆ ˆ ˆ ˆ θMAP 1 − θMAP 1 − θMAP θMAP mH + α − 1 ˆ . θMAP = m+β−1+α−1 (9) This is a very nice result illustrating some interesting properties of the MAP estimate. In particular, comparing the MAP estimate in (9) to the ML esti­ mate in (5) which was mH ˆ , θML = m we see that the MAP estimate is equivalent to the ML estimate of a data set with α − 1 additional Heads and β − 1 additional Tails. When we specify, for example, a prior of α = 7 and β = 3, it is literally as if we had begun the 6 coin tossing experiment with 6 Heads and 2 Tails on the record. If we truly believed before we started flipping coins that the probability of Heads was around 6/8, then this is a good idea. This can be very useful in reducing the variance of the estimate for small samples. For example, suppose the data contain only one coin flip, a Heads. The ML ˆ estimate wil...
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This note was uploaded on 03/24/2014 for the course MIT 15.097 taught by Professor Cynthiarudin during the Spring '12 term at MIT.

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