{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ps4_sol - 1 Introduction to Probabilistic Graphical Models...

This preview shows pages 1–2. Sign up to view the full content.

1 Introduction to Probabilistic Graphical Models December 26, 2006 Problem set 4 - Solutions Lecturer: Eran Segal 1. (a) This can be done by summing out the variables that we wish to remove and incorpo- rating their effects into the leak noise parameter. The important thing to see here is that the summing out can be done efficiently. P ( F i = f 0 i , D 1 , ..., D ) = X D +1 ,...,D k P ( F i = f 0 i , D 1 . . . D k ) = X D +1 ,...,D k P ( F i = f 0 i | D 1 ...D k ) · P ( D 1 . . . D k ) = X D +1 ,...,D k P ( F i = f 0 i | D 1 ...D k ) · k Y i =1 P ( D i ) = X D +1 ,...,D k (1 - λ 0 ) k Y j =1 (1 - λ j ) d j · k Y i =1 P ( D i ) = (1 - λ 0 ) Y j =1 (1 - λ j ) d j P ( D j ) X D +1 ,...,D k k Y i = +1 (1 - λ i ) d i · P ( D i ) For each D i where i > ‘ the term in parentheses can be simplified as follows: (1 - λ i ) P ( D i = 1) + P ( D i = 0) = 1 - λ i P ( D i = 1) And so the entire term in parentheses can be written as: k Y i = +1 (1 - λ i P ( D i = 1)) We will denote this term as A . Notice that A can be computed efficiently in the price of O ( k ) instead of a trivial summing out in F i ’s CPD, in the price of O (2 k ).

This preview has intentionally blurred sections. Sign up to view the full version.

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

{[ snackBarMessage ]}