Conditionally-Independent-Joint-Distribution

05-20-2022 || 15:51
Tags: #probability

conditionally-independent-joint-distribution

For a conditionally independent joint distribution, there are some variables which are dependent on each other and on the other hand, there are some variables which are independent. You can say it a partial dependency/indepedency.

As an example, if there are 3 random variables $(x_1, x_2, x_3)$ and $x_1$, $x_2$ are independent and $x_3$ depends on $x_1$ then the joint probability equation will be,

$$p(x_1, x_2, x_3) = p(x_1) p(x_2) p(x_3 | x_1)$$
if they can take, 2, 3, 4 variables respectively, then,

For $p(x_1)$, there are 2 parameters,
for $p(x_2)$, there are 3 parameters,
and for $p(x_3|x_1)$ there are 2 * 4 parameters

In total there will be 2 + 3 + 2 * 4 = 13 parameters.