Can any one give me any intuation like I'm 10 years old... why below is legal?
The definition of conditional probability states that the joint probability of two events $A$ and $B$ can be expressed as the product of the conditional probability of $A$ given $B$ and the probability of $B$: $$P(A, B) = P(A \mid B)P(B)$$
- Thus $A=\theta$ and $B=y \mid x$, why $P(\theta \mid y \mid x) = P(\theta \mid x,y)$????
In step 2, we apply this definition to the joint probability $P(\theta, y \mid x)$ in two different ways. First, we can treat $\theta$ as event $A$ and $y$ as event $B$, giving us: $$P(\theta, y \mid x) = P(\theta \mid x, y)P(y \mid x)$$
Alternatively, we can treat $y$ as event $A$ and $\theta$ as event $B$, giving us: $$P(\theta, y \mid x) = P(y \mid x, \theta)P(\theta \mid x)$$
- And how this one is come out?
Both of these expressions are valid ways to apply the definition of conditional probability to the joint probability $P(\theta, y \mid x)$.
Arrange $A\cap B\cap C$ in two ways to see that:
$$(A\cap B)\cap C = A \cap (B\cap C)$$
Using the conditional probability definition you cite and treating $A\cap B$ as the first event and $C$ as the second event we see that
$$\Pr((A\cap B)\cap C) = \Pr(A\cap B\mid C)\Pr(C)$$
Now, do this again but this time with $A$ as the first event and $B\cap C$ as the second event but keep going and expand the resulting $\Pr(B\cap C)$ as well.
$$\Pr(A\cap (B\cap C)) = \Pr(A\mid B\cap C)\Pr(B\cap C) = \Pr(A\mid B\cap C)\Pr(B\mid C)\Pr(C)$$
Divide both expressions by $\Pr(C)$ and recall that they are equal. We have as a result:
$$\Pr(A\cap B\mid C) = \Pr(A\mid B\cap C)\Pr(B\mid C)$$
or, using the variable names you chose, replace $A$ with $\theta$, replace $B$ with $y$ and $C$ with $x$.