Marginalization of multiple random variables

borracho · #1 $Old$ July 9th, 2009, 09:47 AM

Can someone enlighten me how to marginalize so that I get p(x5) from p(x1, x2, x3, x4, x5)? When summing over x2, x3, x4, and x5 to marginalize, I'm not sure how to write out the conditional probabilities from the joint probability to get down to p(x5).

Thanks!

Moo · #2 $Old$ July 9th, 2009, 09:58 AM

Hello,

For example :

You want $\rho_{(x_1,x_2,x_3,x_4)}(a_1,a_2,a_3,a_4)=\mathbb{P}(x_1=a_1,x_2=a_2,x_3=a_3,x_4=a_4)$

Intersect the set $\{x_1=a_1,x_2=a_2,x_3=a_3,x_4=a_4\}$ by the partition of the universe : $\bigcup_{i\in\mathbb{N}} \{x_5=b_i\}$

So, after some set operations, we get
$\rho_{(x_1,x_2,x_3,x_4)}(a_1,a_2,a_3,a_4)$ $=\sum_{i\in\mathbb{N}} \mathbb{P}(x_1=a_1,x_2=a_2,x_3=a_3,x_4=a_4 \mid x_5=b_i)\mathbb{P}(x_5=b_i)$

Does this help ?