p.24 of CT6-2 (Bayesian statistics)

[MissAussie]

Hi,

I have 2 quick questions on the table shown on p. 24 of Chapter 2 (ie. the one that contains 15 posteriors for different combinations of priors/likelihoods)

1) When attempting to derive number 7 (ie. the one where the likelihood is Geo(p), I used the fact that the geometric distribution is a special case of the negative binomial distribution. However I didn't know whether to use the Type 1 or Type 2 negative binomial distribution. When I used the Type 1 negative binomial, I got the wrong expression for the posterior, while the Type 2 yielded the right expression. This leads to 2 sub-questions:
a) Can a geometric distribution be a Type 1 negative binomial one with k = 1?
b) If so, surely one should specify whether to use Type 1 or Type 2 in these derivations or statements?


2nd question is, after deriving number 8 (where prior and likelihood are both normal), I think there is an "n" missing in the formula. I think there should be an "n" attached to the (sigma x) term...?

Thanks in advance!

 

[MissAussie]

Hmm, ignore Q2, I think I might have made a mistake in my derivation. Question 1 still stands though... regarding Type I or Type II negative binomials...

 

[didster]

Miss Aussie

The negative binomial type 1 and type 2 distributions have the following differences:

1. Type 2 can take any (positive) value for k, whereas Type 1 takes integers for k
2. (according to my actuarial tables) the count for x (the number of trials need to get k successes) starts at k for type 1 and at 0 for type 2, ie the count for type 1 includes the trials which count as successes

My guess is that point 2 is why your method didn't work, ie you need to adjust the value of x so that the count is consistent.

After adjusting for the count convention, type 1 should just be a special case of type 2.

To be specific regarding your questions
a)
(X-1) = Y~negative binomial Type 1 (k=1,p) (as per actuarial tables) is a geometric distribution (Y=0,1,2,..)

b)
Yes I would certainly specify which one you use especially if you quote formulae from the tables.

Anyway, hopefully that gives you enough to solve the problem. If not, maybe someone else could help (or I could dust out the cobwebs from prior/posterior distributions).


[On a side note, I believe that there is also a Type 1 and Type 2 geometric distribution as well, with the difference being whether the X values include the success or not, ie X can take 0,1,2,... or X can take 1,2,3,4...]

 

[MissAussie]

Thanks! I just used the type I without thinking of the conditions on x, but you're right, i may need to be more careful about that. I'll go back and have a look. Thankee very much!!