Re: [MLUG - DISCUSSION] statistical inference

[Mike Miller <EMAIL:PROTECTED>]

On Mon, 5 Jun 2006, Stephen Montgomery-Smith wrote:

> So consider this problem.  You have a machine that delivers heads or 
> tails. (I won't say "coin" because that gives you too much info.)

If we cannot assume independence, we really can't proceed.


> The uninformed prior distribution for the parameter p that describes the 
> probability that the machien will give you a head, is 1/p(1-p) which is 
> definitely not proper.

For p, why isn't the uninformative prior just 1 (uniform)?  The prior you 
are using is called Jaynes' prior, I believe.  Jeffreys' prior, 
p^½(1-p)^½, is yet another possible choice.

Your prior is a little unusual in that it isn't quite in the beta 
distribution family, is it?  The weirdness is that, at least according to 
this page:

http://mathworld.wolfram.com/BetaDistribution.html

The alpha and beta parameters must exceed zero, but yours both equal zero.


> So do the following experiment - do precisely one trial.  (I.e. toss the 
> coin one time.)
>
> If you get a head, and you normalize the resulting distribution for p, 
> you conclude that p=1 with certainty.  Similarly if you get a tail, you 
> conclude that p=0 with certainty.  Clearly this is not the right answer.

With one success you update by multiplying your 1/p(1-p) by p and you get 
a density with kernel 1/(1-p).  My first thought was "That has infinite 
density at 1, but it is not a point mass at 1, so what's the problem." 
But I guess the problem is that you can't integrate it to find an 
appropriate constant.  It ends up that the density has to go to zero 
everywhere except at 1.  That's interesting, in a way, but doesn't it mean 
that the Jaynes' prior is a little wacky?  Isn't it best to stick to the 
beta family?  What was wrong with the uniform prior?

I'll get back to the rest of it later, but now is a good time to go to 
sleep!

Mike

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