Hi
Sorry for the delay. The Bayesian results showed that the lognormal was more
probable. Yes, the problem is ill-condition which why you need to use the
Bayesian/Maximum entropy method. This method takes into account the
ill-conditioning of the problem. The idea being it determines the most probable
solutions from the set of solutions. This solution can be shown to be the most
consistent solution or the solution with the least assumptions given the
experimental data, noise, instrument effects etc (see Skilling & Bryan 1983;
Skilling 1990; Sivia 1996). This is the role of entropy function. There are
many mathemaitcal proofs for this (see Jaynes' recent book). The Bayesian
analysis maps out the solution/model spaces.
Also the least squares solution is simple a special case of a class of
deconvolution problems. This s well established result. It is not the least
ill-posed, since it assumes the prior distributions to be uniform (in a
Bayesian case. See Sivia and reference therein). In fact it's likely to be the
worst solution since it assumes a most ignorant state knowledge (ie. uniform
proir) and doesn't always take into consideration the surrounding information.
Moreover, it doesn't account for the underlying physics/mathematics, that the
probability distributions/line profiles are positive & additive distributions
(Skilling 1990; Sivia 1996).
Best wishes, Nick
Dr Nicholas Armstrong
NIST-UTS Research Fellow
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----- Original Message -----
From: Nicolae Popa <[EMAIL PROTECTED]>
Date: Saturday, March 26, 2005 9:10 pm
> Hi, once again,
> Fine, I'm sure you did. And what is the most plausible, lognormal
> or gamma?
> From the tests specific for least square (pattern fitting) they are
> equallyplausible. And take a combination of the type w*Log+(1-
> w)*Gam, that will be
> equally plausible.
> On the other hand, why should believe that the Baesian
> deconvolution (or any
> other sophisticated deconvolution method that can imagine) give the
> answermuch precisely? Both, the least square and deconvolution are
> ill-posed
> problems, but the least square is less ill-posed than the
> deconvolution. At
> least that say the manuals for statistical mathematics.
>
> Best wishes,
> Nicolae Popa
>
>
>
>
>
> > Hi,
> > I pointed out that each model needs to be tested and their
> plausibilitydetermined. This can be achieved by employing Bayesian
> analysis, which
> takes into account the diffraction data and underlying physics.
> >
> > I have carried out exactly same analysis for the round robin CeO2
> samplefor both size distributions using lognormal and gamma
> distributionfunctions, and similarly for dislocations: screw, edge
> and mixed. The
> plausibility of each model was quantified using Bayesian analysis,
> where the
> probability of each model was determined, from which the model with
> thegreatest probability was selected. This approach takes into
> account the
> assumptions of each model, parameters, uncertainties, instrumental
> andnoise effects etc. See Sivia (1996)Data Analysis: A Bayesian
> Tutorial(Oxford Science Publications).
> >
> > Best wishes,
> > Nick
> >
> > Dr Nicholas Armstrong
>
> >
>
> >
> > > Hi,
> > > But the diffraction alone cannot determine uniquely the physical
> > > model. An
> > > example at hand: the CeO2 pattern from round-robin can be
> equally well
> > > described by two different size distributions, lognormal and gamma
> > > and by
> > > any linear combinations of these two distributions. Is the
> situation> > different with the strain profile caused by different
> types of
> > > dislocations,possible mixed?
> > >
> > > Best wishes,
> > > Nicolae Popa
> > >
> > >
> > >
> > > > Best approach is to develop physical models for the line profile
> > > broadening and test them for their plausibility i.e. model
> selection.> > >
> > > > Good luck.
> > > >
> > > > Best Regards, Nick
> > > >
> > > >
> > > > Dr Nicholas Armstrong
> > >
> > >
> > >
> >
> >
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