binbagsss
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Question
Find the posterior probability that the next two observations y4 and y5 will both be zero? Where the prior distribution is a gamma with parameters (a,b) and the sample is of size of 3 taking from a poisson disribution with parameter V.
So far I have shown that the posterior distribution is also gamma with parameters a*=a+t, where t=y1+y2+...yn, and b*=b+n, n the sample size.
Attempt at the solution
In my notes, a different question but all i have on my notes on this- is that the probability of exactly the next observation being 1 is given as:
##P(y=1 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV,## So wheree P(V|D) is the updated posterior gamma distribution,
where D is the data, the observations y1+...+yn.
This is for one sample and I'm unsure how to approach for 2 samples. The ideas I have are:
1) Compute ##P(y=0 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV## , and then since we want consecutvie 0 to just square this.
2) To compute ##P(y4=0 an y5=0)=\int^{\infty}_{0} P(y4=1 and y5=1|V) P(V|D) dV## , so i.e. is proportional to ##e^{-2V}## ,
Are any of these methods correct? Could someone please explain which is right or wrong, or if both or wrong and what I should be doing,
Thanks for any help, really appreciated.
Find the posterior probability that the next two observations y4 and y5 will both be zero? Where the prior distribution is a gamma with parameters (a,b) and the sample is of size of 3 taking from a poisson disribution with parameter V.
So far I have shown that the posterior distribution is also gamma with parameters a*=a+t, where t=y1+y2+...yn, and b*=b+n, n the sample size.
Attempt at the solution
In my notes, a different question but all i have on my notes on this- is that the probability of exactly the next observation being 1 is given as:
##P(y=1 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV,## So wheree P(V|D) is the updated posterior gamma distribution,
where D is the data, the observations y1+...+yn.
This is for one sample and I'm unsure how to approach for 2 samples. The ideas I have are:
1) Compute ##P(y=0 | D)= \int^{\infty}_{0} P(y=1|V) P(V|D) dV## , and then since we want consecutvie 0 to just square this.
2) To compute ##P(y4=0 an y5=0)=\int^{\infty}_{0} P(y4=1 and y5=1|V) P(V|D) dV## , so i.e. is proportional to ##e^{-2V}## ,
Are any of these methods correct? Could someone please explain which is right or wrong, or if both or wrong and what I should be doing,
Thanks for any help, really appreciated.