I am reading a text book on probabilistic modeling, and I came across the follow formala:
$P(x_1, x_2, j, k) = P(k | x_1, x_2, j) P(x_1, x_2, j)$
$P(x_1, x_2, j, k) = P(k_{i<j}| x_1, x_2, j)P(k_{i\ge j} | x_1, x_2, j) P(x_1) P(x_2) P(j)$
I don't understand how to go from the first formula to the second one, It seems to be me that the only way to do so would be to assume independence on between $x_1$, $x_2$ and $j$. But how can you split a conditional distribution like this? whats the justification?