Re: Anyone found an Elementary Bayesian Book yet?


David Jones wrote:
Reef Fish wrote:

That's why, in the form of a Beta density, the r and n of
the likelihood function must be parametrized as (r+1)
and (n-r+1). The posterior BETA, from the Beta Prior
(alpha, beta) will be Beta (r+1+ alpha, n-r+1+beta),
which is why alpha and beta need to be both ZERO
for the posterior to be

Beta(r+1, n-r+1) which is the original likelihood function
p^r (1 - p)^(n-r)

prior : Beta(alpha,beta)

posterior:
Sorry, this should have been Beta(r+1+alpha, n-r+1+beta)

Well, no. Consider only the powers of p. The likelihood function that
comes originally from the Binomial(r,n), equivalently Beta(r+1,n-r+1),

That's what I said in the post you cited (see above)!

contributes has p to the power r. The prior distribution that is
Beta(alpha,beta), has p to the power (alpha-1). So the posterior has p
to the power (r+alpha-1), which is the term that would arise from a
Beta(alpha+r, beta+n-r) distribution.

I think you were still reading a part where I had a TYPO which I
corrected (see above). The posterior is

Beta(r+1+alpha, n-r+1+beta)

and that's why B(0,0) will give the likelihood function for the
posterior.


Another check is to see what happens when there is no data from the
experiment, so that the posterior is the same as the prior. "No
experiment" would be equivalent to n=r=0

The fly in the ointment is that with no data, you DON'T have a
likelihood function and you don't have a posterior density.


I also should have used the notation I had used elsewhere in
using ro as the r in the Beta density and r as the same ro in the
likelihood function. (Similarly for no as n+1 in the Beta, and
n in the likelihood). That would have eliminated not only
the ambiguity but the propensity for typos.

Sorry about that, but I am VERY GLAD that you found the
typo errors to make the same error Bob O'Hara did, so that
I can correct you. :-) At the same time, you would NOT
have made the error (and probably shouldn't even with my
typo) had I not made the typo error in the Beta index,
because the ESSENSE of the "cook book recipe" is that
you add the indices of the BETA in the likilihood and the
BETA in the prior to get the BETA in the posterior.


Actually, the books I see don't use the Beta distribution for the
likelihood, rather the Binomial, and you amy be thinking of adding the
indices of the Binomial and Beta in some sense. For the Beta's, you
need to add the indices and subtract one.

Which was what was done. But the RESULTS are the same.
Putting them all in Beta form has the mnemonic advantage of
remembering that Beta is the conjugate family -- and one simply
puts the updating result in the form Beta ^ Beta = Beta.

That's exactly the way the Normal conjugate family works too:
Normal ^ Normal = Normal.

David Jones

-- Reef Fish Bob.

.

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