I am having trouble understanding how to apply Bayes the problem of calculating the probability of a significant weather event (earthquake, hurricane, flood, fire, etc.) repeating $r$ times.
I understand the concept of "Return Period" and it's probabilistic interpretation using a Poisson distribution. For example, what's the probability of a 500 year hurricane occurring twice within a 50 year period?
Start with the counting rate, $\mu$:
$$\mu = \frac{m}{n+1}$$
$m$ = number of recorded occurrences
$n$ = number of years on record
$$\mu = \frac{1}{500+1} = 0.0020$$
Take the Poisson distribution
$$P_t(r) = {\frac{(\mu t)^r}{r!}}{e^{-\mu t}}$$
$r$ = number of occurrences we are calculating for = 2
$t$ = time period = 50
$\mu$ = counting rate = 0.0020
$$P_t(2) = {\frac{(0.0020 \cdot 50)^2}{2!}}{e^{-0.0020 \cdot 50}} = 0.0045$$
Enter Bayes:
$$P(H|A) = \frac{P(A|H)P(H)}{P(A)}$$
I am confused about how to apply this. Is the prior probability $P(H)$ what I calculated above $P_t(2) = 0.0045$ or the simpler ratio $\frac{1}{500} = 0.0020$? I think it's the latter.
I think the conditional probability $P(A|H)$ is what I calculated above $P_t(2) = 0.0045$.
Which leaves me with $P(A)$
I'm really confused at this point. I'd appreciate a shove in the right direction.