Re: Anyone found an Elementary Bayesian Book yet?

Reef Fish wrote:
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)


No. Beta(r+1+alpha, n-r+1+beta) as a posterior density would contain
p to the power of (r+alpha), not (r+alpha-1) as required.
Beta(alpha+r, beta+n-r) is the answer given in cookbooks.


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


No Beta(1,1) (uniform) gives (a scaled version of) the likelihood
function for the posterior. As noted by others, this doesn't
necessarily mean that the uniform can be counted as non-informative
.... one of my books says that even Bayes had doubts about making such
a claim.



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.

No, Bayesian theory copes adequately with a non-informative
experiment, which is what the case n=0 amounts to. If n=0, the outcome
R can reasonably be assigned as R=0, with Pr(R=0)=1 (so that the
non-experiment provides no information about p) ... and this is what
the usual interpretation of the formula for the binomial distribution
gives for the case n=0. If the prior is proper, then the posterior
density is the same as the prior. If the prior is improper, then the
posterior is improper (ie. no density) but should logically be the
same as the prior. Thus both cases are covered by treating the
likelihood function as L=1, or L=any constant. Obviously Bayesian
theory also works for the case where an experiment is informative
about some parameters but not others.

David Jones


.

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