Conditional probability in Expectation Maximization (EM)
I think I found the solution.
In general we have:
$$p(\theta,x,j=1) = p(j = 1 | x, \theta) \cdot p(x,\theta) \\
\Leftrightarrow \frac{p(\theta, x, j=1)}{p(x, \theta)} = p(j=1| x,\theta)$$
By using this, we get:
$$p(j = 1 | x, \theta) = \frac{p(\theta, x, j=1)}{p(x,\theta)} = \frac{p(j=1,x|\theta) \cdot p(\theta)}{p(x|\theta) \cdot p(\theta)} = \frac{p(j=1,x | \theta)}{p(x|\theta)}$$