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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online