Re: [Boost-users] [distributions]: Inverse Gamma

Subject: Re: [Boost-users] [distributions]: Inverse Gamma
From: Paul A. Bristow (pbristow_at_[hidden])
Date: 2010-08-05 04:41:49

> -----Original Message-----
> From: boost-users-bounces_at_[hidden] [mailto:boost-users-bounces_at_[hidden]] On Behalf Of John Maddock
> Sent: Wednesday, August 04, 2010 4:32 PM
> To: boost-users_at_[hidden]
> Subject: Re: [Boost-users] [distributions]: Inverse Gamma
> 
> >> Yes I truly have one ;) The inverse gamma distribution and its special
> >> case, the scaled inverse chi-square distribution, is the conjugate prior
> >> to the normal distribution variance parameter in Bayesian statistics.
> >> Pretty much as uncommon and unheard of as it is outside Bayes world [to
> >> the best of my knowledge], it's very much central to Bayesian stats and
> >> appears in every textbook right after the introduction chapter ;)
> >>
> >> http://en.wikipedia.org/wiki/Scaled_inverse_chi-square_distribution
> >> http://en.wikipedia.org/wiki/Conjugate_prior
> >>
> >> Hence I wonder it has not been requested so far - but being a Bayesian
> >> C++ / booster I definitely want / need it :).
> >>
> >> @John: Yes it is a transformation deviate of the gamma, and an easy so.
> >> And it should be fairly easy to implement IMHO.
> >
> > The pdf and pdf etc looks fairly straightforward (only uses exp, pow and
> > gamma?) so I might be persuaded to do these.
> > But the inverses (qhantiles) may prove more troublesome if have to be done
> > by brute force numerically
> >
> > R library does it numerically
> > http://rss.acs.unt.edu/Rdoc/library/pscl/html/igamma.html
> >
> > - or are there analytic expressions for these?
> >
> > Or can it use the inverse of the gamma distribution?
> 
> Implementation would be the same as the gamma distro, all the information
> needed is on Wikipedia - cdf is simply:
> 
> gamma_q(alpha, beta/x);
> 
> so gamma_q_inv can be used for the quantile.  PDF is just a call to
> gamma_p_derivative.
> 
> >> Is contribution on my side expected (can be done just notice I am a
> >> [heavy !] user of the stats library only, not familiar with code /
> >> numerical stability issues).
> >
> > I'm not mathematician enough to deal with this - but I can deal with the
> > obfuscated code (by templating and policies) if
> > you can provide the equations.
> >
> > (And there is the question of testing - some parameters and value
> > combinations (preferably exact) are needed for sanity
> > and accuracy checks).
> 
> Paul, I don't mind filling in the implementation details if you could take
> care of docs and tests - if that helps?  Do we need the
> scaled-inverse-chi-squared as well?

John - I was just about to start to do the inverse gamma implementation - since it doesn't appear to need anything fancy
like root finding, it might just within my skill_set/pay_grade ;-)

I'd need more guidance on the scaled-inverse-chi-squared  (since we are doing an upgrade, it might be worth doing this
too).

But I'll certainly do the grunt work on docs and testing anyway.

Thomas - could you provide an *example* of using this distribution(s) for our docs?

(IMO this is especially important because the presentation of the distributions is unlike what your average FORTRAN or R
user would expect).

Paul

---
Paul A. Bristow
Prizet Farmhouse
Kendal, UK   LA8 8AB
[phone], mobile [phone]
pbristow_at_[hidden]