Bayes rules with more than two variables

Question

In my lectures I saw this equation.

$$ p(j=1, x|\theta) = p(x|j=1, \theta)p(j=1) $$

I tried to apply Bayes rule on the left hand side, but I couldn't get the right hand side of the equation.

Answer 1

The right equation is $$ p(j=1, x|\theta) = p(x|j=1, \theta){\color{red}{p(j=1|\theta)}} $$ since $\theta$ is considered a global information in this equation and must hold anywhere. A simple proof follows by using the definition of conditional probability with the events $x|\theta$ and $j=1|\theta$.