Hi, So, to resume your statements, by using Bayesian/Max.Entr. we can distinguish between two distributions that can not be distinguished by maximum likelihood (least square)? Hard to swallow, once the restored peak profiles are "the same" inside the noise. What other information than the peak profile, instrumental profile and statistical noise we have that Bayes/Max.ent. can use and the least square cannot?
"prior distributions to be uniform" - if I understand correctly you refer to the distributions of "D0" and "sigma" of the lognormal (gamma) distribution from which the least square "chooses" the solution, not to the distribution itself (logn, gamm). Then, how is this prior distribution for Baye/Max.ent.? Best, Nick Popa > 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 > > 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 > > > > > > > > > > > > > > > > > > > > > -- > > > UTS CRICOS Provider Code: 00099F > > > DISCLAIMER: This email message and any accompanying attachments may > > contain > > > confidential information. If you are not the intended recipient, > > do not > > > read, use, disseminate, distribute or copy this message or > > attachments.If > > > you have received this message in error, please notify the sender > > immediately > > > and delete this message. Any views expressed in this message are > > those of > > the > > > individual sender, except where the sender expressly, and with > > authority,> states them to be the views the University of > > Technology Sydney. Before > > > opening any attachments, please check them for viruses and defects. > > > > > > > > > > > > -- > UTS CRICOS Provider Code: 00099F > DISCLAIMER: This email message and any accompanying attachments may contain > confidential information. If you are not the intended recipient, do not > read, use, disseminate, distribute or copy this message or attachments. If > you have received this message in error, please notify the sender immediately > and delete this message. Any views expressed in this message are those of the > individual sender, except where the sender expressly, and with authority, > states them to be the views the University of Technology Sydney. Before > opening any attachments, please check them for viruses and defects.
