[SciPy-user] computing Bayesian credible intervals

[Robert Kern]

On Tue, May 6, 2008 at 2:18 AM, Anne Archibald
<peridot.faceted at gmail.com> wrote:
>  Do you really need p(a)=p(b)? I mean, is this the right supplementary
>  condition to construct a credible interval? Would it be acceptable to
>  choose instead p(x<a)=p(x>b)? This will probably be easier to work
>  with, at least if you can get good numerical behaviour out of your
>  PDF.

It's one of the defining characteristics of the kind of credible
interval Johann is looking for. "Bayesian credible interval" is a
somewhat broad designation; it applies to pretty much any interval on
a Baysian posterior distribution as long as the interval is selected
according to some rule that statisticians agree has some kind of
meaning.

In practice, one of the most common such rules is to find the "Highest
Posterior Density" (HPD) interval, where p(a)=p(b) and P(b)-P(a)=0.95
or some such chosen credibility level. Imagine the PDF being flooded
with water up to its peak (let's assume unimodality for now). We
gradually lower the level of the water such that for both points a and
b where the water touches the PDF, p(a)=p(b). We lower the water until
the integral under the "dry" peak is equal to 0.95. Then [a,b] is the
HPD credible interval for that PDF at the 0.95 credibility level.

-- 
Robert Kern

"I have come to believe that the whole world is an enigma, a harmless
enigma that is made terrible by our own mad attempt to interpret it as
though it had an underlying truth."
  -- Umberto Eco