Maximum likehood unleashed!

[Michal Frackowiak]

On Wed, 2002-06-05 at 13:08, Boud Roukema wrote:
> Thanks Michał,
> 
> On 3 Jun 2002, Michal Frackowiak wrote:
> 
> > I am sending a short theory behind max. likehood method and its
> > adaptation in comparing curves with given errors. I believe this will be
> > helpfull - it seems to me as the best way that does the trick:
> > - it takes particular errors into account
> > - lets you estimate errors of the fit
> > - as well as contours of confidence when making a grid.
> 
> It's a nice explanation.
> 
> > This is well tested in my own soft so I hope should work in this case as
> > well.
> > In case of questions - I would be even more than glad to help.
> 
> Well, although it's clear the method can give a result, for it to
> give a correct result, the different  r  values would need to be
> independent from one another.
> 
> So I see three problems, 2 easily solvable, 1 more fundamental:
> 
> - solvable:
>  (1) If we combine all three: L_{12} L_{23} L_{13} then one of these
> three is dependent on the other two. So it seems to me that we have to
> (arbitrarily) remove one of the three, even though it's clear that
> this is an arbitrary choice.

Not exactly. If you remove one of them, imagine the situation: you have
2 curves (A and B) almost identical and 1 (C) very different. Now if you
calculate L wita A-B and A-C you get nothing from the first pair - they
are identical - but much from B-C. BUT if you chose to comare A-C (much
difference) and B-C (also much difference) you get useless info. That is
why you have to compare each witch each.

> 
>  (2) There is some smoothing in the curves output by DEplotcorrnall.
> This can be removed (just set ismoo=0 on line 283 of DEplotcorrnall in
> DE-V0.04, I think this should be OK), but then my worry is that
> the result will be extremely noisy. An alternative solution would
> be to to only test one out of every  (ismoo+1)  values of  r  .
> 

In eqs. (4) and (5) we should then replace the sum with an integral -
that would be of course more natural. I have ommited it.

> - fundamental:
>  (3) Different bins in a correlation function depend on one another.
> A single quasar is a member of many pairs, and different pairs fall
> into different bins. So the different  r  values in a single function
> zeta(r)  depend on one another.
> 
> So for this reason I find it hard to believe that  L  (rescaled) would
> be a true probability density function.
> 
I do not think it willd be a problem - since you rescale it properly. I
will think about it, but the fact that the values are somehow corelated
should not affect the method - you can estimate errors for any r as I
understand and the method is just to compare curves! 

> It's certainly a good idea, so I'll put it as one of the DEplot_cf
> tests, but I don't think it'll give true error bars.
> 
I think it will. For error bars for the best fit you can calculate the
second derivative matrix for l({parameters}), than inverse it - and you
get covariance matrix! From that point elements \sqrt{c_{ii}} will give
you errors of 1-sigma for i-th parameter. That's a method for estimating
errors of minimising method. It works for me. And agrees with plotted
contours.

regards
Michal



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