Computing posterior density

Question

If I have two observations $X=n,X=m$, how do I then compute the posterior density? I can think of 2 ways but I don't know which one is the right one:

1) First compute posterior given $X=n$ $$p(\theta|X=n) = \frac{p(X=n|\theta)p(\theta)}{p(X=n)}$$

Then use this posterior as prior and compute the new posterior given $X=m$ $$p(\theta|X=m) = \frac{p(X=m|\theta)p(\theta|X=n)}{p(X=m)}$$

2) A joint probability approach like this: $$p(\theta|X=n,X=m) = \frac{p(X=n,X=m|\theta)p(\theta)}{p(X=n,X=m)}$$

Answer 1

no one of them.

posterior: $\pi(\theta | x_1,x_2)$

prior: $\pi (\theta)$

likelihood: $f(X|\theta)$

$\pi(\theta | x_1,x_2) =\frac{f(x_1,x_2|\theta) \pi(\theta)}{f(x_1,x_2)}$

$x_1,x_2$ are your observation.