[BMI776] A note on likelihoods

[Colin Dewey]

Hi all,

I've had a number of questions regarding the likelihood that is being  
maximized in EM and Gibbs, so I thought I would post this clarification.

- The log likelihood of the data and a set of motif positions given  
the parameters in the OOPS model is given on slide 3 of lecture 6.   
In this equation, the Z_i,j are binary values.

- During the EM algorithm, you are maximizing the *expected* log  
likelihood, which is also given by the formula on slide 3 of lecture  
6, but with the Z_i,j now representing expected values of the binary  
Z_i,j (so the Z_i,j terms will be between 0 and 1).  You should  
calculate the expected log likelihood after each EM iteration and  
stop when it doesn't change by much (e.g., if the difference isn't  
greater than 0.01).

- During Gibbs, you calculate just the log likelihood, because you  
have set motif positions at each iteration (i.e., the Z_i,j are binary).

- In the log likelihood formula, the term P[X_i | Z_i,j = 1, theta]  
is maybe more easily seen on slide 13 of lecture 5.

Hope that helps,
Colin