Question 1:
From some slides I saw that:
$p(x_{0:t+1}|y_{1:t+1}) = p(x_{0:t}, x_{t+1}|y_{1:t}, y_{t+1})$, and according to the product rule of probability theory $P(a,b) = P(a|b) P(b)$ we will have have:
$p(x_{0:t+1}|y_{1:t+1}) = p(x_{0:t}, x_{t+1}|y_{1:t}, y_{t+1}) = \frac{p(x_{0:t}, x_{t+1}, y_{t+1}|y_{1:t})}{p(y_{t+1}|y_{1:t})}$
I don't understand how the product rule of probability theory is applied for this example, what is $a$ and $b$ in this example ?
Note: $x_{0:t}$ is $\{x_0,x_1,x_2, \ldots, x_t\}$
Question 2:
Also, is it always true that: $P(X|Y)= \int P(X|Z) P(Z|Y)\, dz$ ?