Re: different priors (flat, uniform, etc)


Anon. wrote:
John Uebersax wrote:
Saludos Ruth!

I have a probability on several parameters and one of them is
nuisance one. I want to margizalize over it by simply
integrating the probability along the whole range of values
that such nuisance parameter is allowed to take.

Okay good. You want to "integrate out a nuisance parameter."

Doing it that way is what people call using a flat prior?

I think possibly you're combining two different issues here.
A "prior distribution" is a Bayesian term, and means, basically,
what you think the distribution of a parameter is before
(prior to) some additional data or evidence leads you to
modify your prior beliefs (i.e., produce an updated, or
posterior estimate of the distribution).

So I think we should just dispense with the word "prior"
here. It seems like all you really want to know is: given
that I don't know the shape of my nuisance parameter
distribution, what's a good guess?

Shouldn't that just come from the likelihood? i.e. it should be in the
model already.
--
Bob O'Hara

You are ALMOST correct. That's why I said Uebersax is NOT a
Bayesian. We already know Bob O'Hara isn't one. :-)

The posterior distribution is the likelihood function if the prior is
"diffuse" (which is NOT the same as a "uniform" or "flat" prior).

For Bayesian Inference on the parameter p of a Binomial distribution
or a Bernoulli Process, the beta distribution is a member of the
conjugate prior family -- meaning both the prior AND posterior
belongs to the same distribution family -- Beta.

The uniform distribution on (0,1) is a Beta distribution with
parameters (1,1) and is an INFORMATIVE prior.

Beta(1/2, 3) is reverse J-shaped.
Beta(1/2, 1/2) is U-shaped, symmetric around 1/2.
Beta(2, 2) is symmetric unimodal, so is Beta(2,3).
Beta(2,1) is the triangular distribution on (0,1)
Beta(3,2) is unimodal, skewed to the left.
Beta(3,1) is J-shaped, so is Beta(2. 1/3).

As you can see, the Beta family CAN represent a wide
variety of opinion about p and hence is a reasonably
good APPROXIMATE prior distribution for one to choose
that best-reflects one's opinion, without having to do any
work on integration of the product of the prior and likelihood,
because the posterior distribution form can be written
IMMEDIATELY given the sample information.

That's the usefulness of a CONJUGATE prior on certain
problems (for Bayesians). However, even the family of
conjugate priors are grossly inadequate for a true Bayesian
for expressing his opinion about a particular p of a
Binomial. That's why Robert Schlaiffer had spent a large
amount of time providing numerical assessment software
and numerical integration software for just that ONE
problem (and other uniariate parameter problems) and
had written a book about it.

-- Reef Fish Bob.

.

Relevant Pages

  • Re: different priors (flat, uniform, etc)
    ... integrating the probability along the whole range of values ... that such nuisance parameter is allowed to take. ... A "prior distribution" is a Bayesian term, and means, basically, ...
    (sci.stat.math)
  • Re: Whats a random sequence for Bayesians?
    ... That Bayesian would reason as in your number 3. ... with far more "prior" precision than any empirical data. ... >then you should say what kind of departures from that distribution you ... between prior weights; there is no symmetry. ...
    (sci.stat.math)
  • Re: different priors (flat, uniform, etc)
    ... as is probably obvious should you look at his history ... John Uebersax PhD ... JU> So I think we should just dispense with the word "prior" ... RF> simply ASSUME anything has a normal distribution or any other ...
    (sci.stat.math)
  • Re: different priors (flat, uniform, etc)
    ... that such nuisance parameter is allowed to take. ... A "prior distribution" is a Bayesian term, and means, basically, ... We already know Bob O'Hara isn't one. ...
    (sci.stat.math)
  • Re: different priors (flat, uniform, etc)
    ... integrating the probability along the whole range of values ... that such nuisance parameter is allowed to take. ... A "prior distribution" is a Bayesian term, and means, basically, ...
    (sci.stat.math)
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