# Pattern recognition and machine learning problem 2.7

1. Aug 6, 2012

### karse

I'm working my way through pattern recognition and machine learning using this site as a guide.

1. The problem statement, all variables and given/known data
We have to prove that a binomial random variable x, with a prior distribution for $\mu$ given by a beta distribution, has a posterior mean value that is x that lies between the pror mean and the maximum likelihood estimate for $\mu$.

$\underbrace{\frac{a}{a+b}}_{prior-mean}<\underbrace{\frac{m+a}{m+a+l+b}}_{posterior-mean}< \underbrace{\frac{m}{m+l}}_{ml-estimate-of-\mu} (eq. 1)$

where a hint in the book state that it is equal to solving:

$\frac{m+a}{m+a+l+b}= \lambda\cdot \frac{a}{a+b}+(1-\lambda)\cdot\frac{m}{m+l}, 0<=\lambda<=1 \text{ (eq. 2)}$

m and l is the numer of observed values where x=1 and x=0 respectively. a and b specifies our prior belief via the beta distribution.

My question is about the hint. how do i get from eq. 1 to eq. 2.? Is it always "legal" to solve eq. 2 instead of eq. 1??

(i'm not looking for a solution to the original problem :) )

2. Aug 6, 2012

### Staff: Mentor

Look at $\lambda$ = 0 and $\lambda$ = 1.

3. Aug 6, 2012

Thanks ;)