If we have a random variable $Y$ with pdf $P(Y|a,b)$, where $a$ and $b$ are parameters (with range $0$ to $\infty$).
As well as marginal posterior distributions for $a$ and $b$, these are $P(a\vert x)$ and $P(b\vert x)$, where $x$ is observed data.
Then would the predictive distribution of Y be
$$P(Y\vert x)=\int_0^\infty \left[\int_0^\infty P(Y|a,b)\cdot P(a|x) \,da\right] \cdot P(b\vert x) \,db$$
My confusion is because that we have two marginal distributions rather than a joint pdf.
Thanks in advance for any help it would be greatly appreciated.