Why $P(t,z|\theta)=P(t|z,\theta)P(z|\theta)$?

Question

In this post, I can understand the joint probability $P(A,B)=P(A|B)P(B)$.

But why

$P(t,z|\theta)=P(t|z,\theta)P(z|\theta)$

$P(t,z|\theta)$, in this case, if $(z|\theta)$ denotes $B$, the equation should be like

$P(t,z|\theta)=P(t|z,\theta)P(z,\theta)$

Why we write $p(B)$ as $P(z|\theta)$ instead of $P(z, \theta)$?

Answer 1

\begin{align*} P(t,z | \theta) &= \frac{P(t, z, \theta)}{P(\theta)}\\ &= \frac{P(\theta) P(z|\theta) P(t | z, \theta) }{P(\theta)}\\ &= P(z | \theta) P(t| z, \theta) \end{align*}

Multiplication rule: $$P\left(\bigcap_{i = 1}^n A_i\right) = P(A_1) P(A_2 | A_1) P(A_3 | A_1 \cap A_2) \cdots P(A_n | \cap_{i=1}^{n-1} A_i)$$

Write $P(t, z, \theta) = P(T = t \cap Z = z \cap \Theta = \theta)$

and apply the multiplication rule. This rules applies to both discrete and continuous random variables.