Hi,
Nick Armstrong has advised me he will in non-email-land for a week or so.
I'm sure he'll resume this discussion when he returns...
Jim
At 03:45 PM 3/28/2005 +0400, you wrote:
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
<snip>
James P. Cline [EMAIL PROTECTED]
Ceramics Division Voice (301) 975 5793
National Institute of Standards and Technology FAX (301) 975 5334
100 Bureau Dr. stop 8520
Gaithersburg, MD 20899-8523 USA
