EM Algorithm question

Dear friends:

Please, see page 3 of this document. I don't undestand equation(1). I understand the index separation of sum {1,...,m} + {m+1,...,n}, but I don't know why the first term of second line (the sum over {1,...,m}).

Thanks a lot.


If you read example, it mentioned that yi; i = 1, ...,m are observed and yi; i = m + 1, ...,n are missing (at random) so u know exactly that what are y1,..., ym but you dn't know about y(m+1),...,yn. That's a reason in second line of equation No.(1) you have summation on y1,..., ym because you already observed them and you have them but for second summation that you dn't know about them because you didnot observed them you have (n-m); means number of missing data multiply their estimations.

I hope that will be usefull for u.


TS Contributor
Refer to the paragraph above equation 1
"Suppose \( y_i, i = 1, 2, ..., m \) are observed and
\( y_i, i = m+1, m+2, ... , n \) are missing (at random) ...."

So given the observations, if we take expectation on
\( \sum_{i = 1}^n y_i = \sum_{i = 1}^m y_i + \sum_{i = m+1}^n y_i \)

, we have
\( E[\sum_{i = 1}^m y_i + \sum_{i = m+1}^n y_i|\sum_{i = 1}^m y_i] \)
\( = \sum_{i = 1}^m y_i + (n - m)\mu^{(t)} \)