HI All,
It has been shown that when incorporating a priori information, Bayesian
approach is the logically the most consistent (Cox 1946), since it conserves
the probabilities and all information (Jaynes 2004). Least squares is a special
and limited case of Bayesian applications. There are many more mathematical
proof which demonstrate this. In addition, combining Bayes' theorem and entropy
function, produces the most consistent solution, from the set of solutions.
That is, the solution which has the least assumptions (Boltzman demonstrated
this almost 100 years ago!!). Again there are many other (general) mathematical
proofs which demonstrate this.
Hence, we have developed a method that ensures the information and
probabilities are conserved. Moreover, it has a firm mathematical and physical
basis. The use of a priori information, represents one's hypothesis, assumption
and/or belief. By using Bayes' theorem, hypothesis, assumption and/or beliefs
are quantified using probabilities relative to the experimental data and
noise/uncertainty. Please take note of the last point!
In addition, when applying Bayesian/MaxEnt model selection to size
distributions, we are *not* fitting the noise/instrument or secondary effects.
(I've been very care to check this. See above paragraph.) In the case of
profile function fitting, I strongly suggest seeing Sivia (1996). He points how
this can be done for line profile functions like Gaussian, Lorentzian (and
presume Voigts....).
About errors in the size distributions. The size of error-bars determined from
the Bayesian/MaxEnt method represent the qualtity of the data, how well the
background is estimated, and generally the poor conditioning of the size
distribution problem. This last point is critical. That is, the problem is
ill-conditioned and explains why many possible solutions can "fit" the data.
Another way to look at it is that the Bayesian/MaxEnt error-bars represent a
"slice" through the probability density function in the solution space, and
represent the "population" of solutions which can fit the data within the 68%
probability region. However, by using the entropy function we can always be
sure there exist one and only solution with a maximum entropy relative to the
experimental data etc... (again see the many mathematical proofs...). With a
least squares approach this is not always the case.
Best Regards, Nick
ps. I agree, it is inappropriate to use the email group to request Journal
articles.
Dr Nicholas Armstrong
NIST-UTS Research Fellow
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----- Original Message -----
From: alan coelho <[EMAIL PROTECTED]>
Date: Monday, April 18, 2005 7:03 pm
> This is by far the best topic on this list for a long time as
> opposed to
> requests for Journal papers which as pointed out by someone else is
> inappropriate in the first place and illegal in the second.
>
>
>
> Nicolae wrote:
>
> >(i) but a sum of two Lorentzians is not sharper than the sum of
> two pVs
> (Voigts)?
>
>
>
> This I know, it should not matter what is used as long as the
> mapping of
> the function to a distribution is done accurately. Whether it is
> lognomal, gamma or something else does does not matter.
>
>
>
> Every thing we are talking about is additive meaning that the sum of
> what ever in 2Th space translates to the sum of what ever
> distributions.From the resulting distribution you are free to
> extract what ever
> parameter you choose.
>
>
>
> The idea in Nick Armstrong's work of obtaining a distribution without
> knowing its functional form is a powerful one. But the Baysean
> approachwithout a functional form results in large errors bars in the
> distribution, see
> http://nvl.nist.gov/pub/nistpubs/jres/109/1/cnt109-1.htm
>
>
>
> What we should be looking for are cases where Voigt approximations are
> not possible. I have only ever seen one case where the actual Sinc
> typeripples are seen in a pattern. This was a pattern by Bob Cheary
> of gold
> columns. Other reports of ripples do exist in the literature (not
> available to me as I write). We must be careful not to include 2Th
> independent bumps produced by long narrow Soller slits inserted in the
> axial plane that limits horizontal divergence.
>
>
>
> When sample related ripples are seen then you can throw Voigt based
> approximations out the window. In the case of the gold columns we
> fittedthree Sinc functions added together. In other words the
> distribution was
> really a limited one.
>
>
>
> The work of Nick et al is sound and approaches the problems from a
> different perspective; it does not however negate the need to
> determinea priori information. The question that is open in my
> opinion is whether
> a priori information is more easily incorporated into a least squares
> process or a Bayesian Maxent approach.
>
>
>
> This discussion has reinvigorated my interest and like Bob, whom is
> nowlooking for an equation to approximate the log normal
> disribution, I
> will resurrect some code myself that I did a while back which
> calculates profiles from a arbitrary distribution for the Sinc
> function.Instead I will include the equation for spherical
> crystallites and of
> course a user defined one which can be hkl dependent.
>
>
>
> As a hint to those who write such code the calculation of a profile
> foran arbitraty distribution operates at around 5000 profiles per
> second as
> I noticed over the weekend - not much slower that a gaussian Nicolae.
> Maybe there's no need for a pseuod-Voigt / Lorentzian based
> approximations after all.
>
>
>
>
>
> all the best
>
> Alan
>
>
>
> -----Original Message-----
> From: Nicolae Popa [EMAIL PROTECTED]
> Sent: Sunday, April 17, 2005 9:00 AM
> To: rietveld_l@ill.fr
>
>
> Alan,
>
> (i) but a sum of two Lorentzians is not sharper than the sum of two
> pVs(Voigts)?
>
> (ii) We fitted the exact size profile caused by the lognormal
> distribution by a pV (for low lognormal dispersion) or by a sum of
> maximum 3 Lorenzians (for large lognormal dispersion).
> This is "cheaper" than the sum of 2 pVs. It involves the
> calculation of
> maximum 3 elementary functions with 4 independent parameters (3
> breadths+ 2 mixing parameters minus 1 constraint = 4)
> Sum of two pVs presumes 4 elementary function and 5 independent
> parameters (2 for one pV + 2 for the second one + a mixing parameter).
>
> Best wishes,
> Nicolae
>
>
>
>
>
> A pure peak fitting approach shows that two pV's (or two Voigts) when
> added with different FWHMs and integrated intensities but similar peak
> positions and eta values can almost exactly fit Pearsons II functions
> that are sharper that Lorentzians. This is not surprising as both
> profiles comprise 6 parameters.
>
>
>
> Thus from my observations two pVs added together can fit a bimodal
> distributions quite easily. In fact my guess is that two pVs can
> fit a
> large range of crystallite size distributions.
>
>
>
>
>
>
>
>
>
>
> all the best
>
> alan
>
>
>
>
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