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: $$p(\theta|X=n,X=m) = \frac{p(X=n,X=m|\theta)p(\theta)}{p(X=n,X=m)}$$

  • $\begingroup$ what is your observation? $X_1=n,X_2=m$ is your observation? $\endgroup$ – masoud Mar 31 '19 at 15:16
  • $\begingroup$ @masoud Yes, I removed the index so that it is clear that $X_1$ and $X_2$ are not two different variables but rather 2 different time points (First observation and second observation.) $\endgroup$ – Matriz Mar 31 '19 at 15:25
  • $\begingroup$ in method 2, in right side of equation , replace $p(\theta|X=n,X=m)$ with prior. $\endgroup$ – masoud Mar 31 '19 at 15:39

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.

  • $\begingroup$ Thanks. That was my method 2, but I accidently wrote the prior wrong. I somewhere read though that one can use the posterior as a new prior, do you know when this is the case? $\endgroup$ – Matriz Mar 31 '19 at 15:40

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