How to interpret this bayesian inference formula

Question

I know this simple Bayes formula:

$$ P(A|B)=P(B|A)*P(A)/P(B) $$

While I was seeing a video in MCMC I came across another variant

P ( x | t ) = ∑ c P ( x | c , t ) P ( c | t ) ?

How to interpret the above formula

Answer 1

$\mathsf P(x\mid t)$ is ussually a shorthand for $\mathsf P(X{=}x\mid T{=}t)$, where $X,T$ are discrete random variables and $x,t$ values within their support.

Let $C$ be some other random variable measured over the same sample space.   Then the Law of Total Probability, and the definition for conditional probability, gives.

$$\mathsf P(X{=}x\mid T{=}t){~=~\sum_{c} \mathsf P(X{=}x,C{=}c\mid T{=}t) \\ ~=~ \sum_c \mathsf P(X{=}x\mid C{=}c, T{=}t)\,\mathsf P(C{=}c\mid T{=}t)}$$

Or, as per the shorthand, $\mathsf P(x\mid t){~=~\sum_c \mathsf P(x,c\mid t)\\~=~\sum_c \mathsf P(x\mid c, t)\,\mathsf P(c\mid t)}$

---

In a similar manner, if $A,B,C$ are events in the same trial, we have

$$\mathsf P(A\mid B) {~=~ \mathsf P(A,C\mid B)+\mathsf P(A,C^\complement\mid B) \\ ~=~\mathsf P(A\mid B,C)\,\mathsf P(C\mid B)+\mathsf P(A\mid B,C^\complement)\,\mathsf P(C^\complement\mid B) }$$