Re: Highest Posterior Density - AB is back...

Reef Fish wrote:
AB wrote:
Reef Fish ha scritto:

AB wrote:
Dear all,
by HPD I mean "high posterior density";
by "high posterior density" (e.g. at 95%) I mean an interval whose
probability under the posterior density is 95%,
That is indeed the definition given by Wikipedia.

It is the Bayesian CREDIBLE interval which may NOT even contain
the mode of the posterior distribution!

Just as a confidence interval is defined only by the coverage or
probability content, the same is true for Bayesian credible intervals.

If the posterior mode is a sharp spike far away from the bulk of
the high density region, such as,

x
x
x
x xxxxxxxxxx
x xxxxxxxxxx
x xxxxxxxxxxxx x x
_____________________________________
then the shortest credible interval of high percent will EXCLUDE
the m0de (the highest point of the posterior density).


Anyway, I'd try to distinguish between HPD and Credible Set (CR), since
CR should be "similar" to a confidence interval, as it leaves the same
probability to the right and to the left tail, while HPD is a sort of
"level set", including all the points having a density greater than a
certain value...(and so, it may sometimes not be an interval...)

Yes, I've seen the description of a still farther departure of HPD
from a Bayesian credible Interval to a Credible Set.

That is NOT even a STATISTICAL estimation concept (except in very
rare special circumstances), let alone a Bayesian concept. The usual
concept of a credible interval is (as in CI) to choose the SHORTEST
interval that has the Highest probability coverage of a prescribed
probability number, such as .95 or .9.

No, that's the concept behind the HPDI. A credible interval is simply an interval containing the correct amount of probability mass: obviously there are several ways of creating such an interval (e.g. see http://en.wikipedia.org/wiki/Credible_interval).

The idea of breaking such an
INTERVAL into many bits and pieces simply because of the height
of the density function is statistical perversion of no useful content,
that are dreamed up by NON-statisticians, in their own fields.

I'm interested (yes, genuinely) in why you think this is a "statistical perversion". Obviously reporting several intervals for a credible region is a bit of a pain, but it seems to be more in keeping with the philosophy of a HPDI, i.e. reporting the region with highest probability mass.

Bob

--
Bob O'Hara
Department of Mathematics and Statistics
P.O. Box 68 (Gustaf Hällströmin katu 2b)
FIN-00014 University of Helsinki
Finland

Telephone: +358-9-191 51479
Mobile: +358 50 599 0540
Fax: +358-9-191 51400
WWW: http://www.RNI.Helsinki.FI/~boh/
Journal of Negative Results - EEB: www.jnr-eeb.org
.

Relevant Pages

  • Re: Highest Posterior Density - AB is back...
    ... by "high posterior density" I mean an interval whose ... probability content, the same is true for Bayesian credible intervals. ... Anyway, I'd try to distinguish between HPD and Credible Set, since ...
    (sci.stat.math)
  • Re: Highest Posterior Density - AB is back...
    ... by "high posterior density" I mean an interval whose ... probability content, the same is true for Bayesian credible intervals. ...
    (sci.stat.math)
  • Re: Highest Posterior Density - AB is back...
    ... by "high posterior density" I mean an interval whose ... probability content, the same is true for Bayesian credible intervals. ...
    (sci.stat.math)
  • Re: Highest Posterior Density - AB is back...
    ... by "high posterior density" I mean an interval whose ... probability content, the same is true for Bayesian credible intervals. ... Anyway, I'd try to distinguish between HPD and Credible Set, since ...
    (sci.stat.math)
  • Re: Highest Posterior Density - AB is back...
    ... by "high posterior density" I mean an interval whose ... probability content, the same is true for Bayesian credible intervals. ... Anyway, I'd try to distinguish between HPD and Credible Set, since ...
    (sci.stat.math)