Suppose there are 3 random variables: $A_t\in\{0,1\},B_t,\text{and}\; B_{t-1}$. I would like to write a joint probability distribution to express the idea:
At time step $t$, $B_{t-1}=b$ has been observed. We sample $A_t$ from $P(A_t)$ first, if $A_t=1$ then we sample $B_t$ from $P(B_t|A_t=1)$; If $A_t=0$, then let $B_t=B_{t-1}=b$.
$$P(A_t,B_t|B_{t-1}=b)= P(A_t)P(B_t|A_t)\mathbb{1}[B_t=B_{t-1}]^{A_t}$$
Is this a valid distribution? What is a formal way to express the above idea? I checked this post Conditional probability and indicator function but did not come up with an formulation.