Re: OMEGA HAS A NONZERO BLOCK

From: "Lewis B. Sheiner" Subject:Re: [NMusers] OMEGA HAS A NONZERO BLOCK Date: Fri, 04 Oct 2002 13:48:04 -0700 I, too have been following this discussion, and am puzzled by the fact that we are covering very old ground. This is a solved problem, although we may not much like the 'solution'. The problem we are discussing is called the 'ill-posed inverse problem'. Peter's and Serge's results confirm the nature of the problem: ill-posed inverse problems are typified by instability of estimates. Indeed, not only is this well-known, but it is also well known that if the problem can be solved at all (that is if the model is a priori identifiable) one can ONLY achieve stability by adding more information; that contained in the current data is insufficient. Moreover, there is really no longer any serious argument about where this information should come from, or even how to add it once we have it! Regarding the former, the principled answer is 'from science' . Regarding the latter, the only principled way I know of is to adopt a Bayesian perspective, and to use Bayesian methods. So, returning to the current interchange, consider the 'how to' alternatives being discussed: fix parameters to some value, or use full prior distributions on them (in either case, the first principle says that the specific values chosen must be justified on the basis of science). First, note that simplifying a model (that is, by adopting a simpler sub-model than the original one) is formally equivalent to fixing certain parameters (of the full model) to prior values. These two 'solutions', then, are the same solution, and I refer to them both when I say 'fixing parameters'. From the bayesian perspective, if you 'add knowledge' by fixing parameters, this is equivalent to asserting that you have perfect prior knowledge of certain parameter values. This is a hyper-subjective position with which any scientist should feel distinctly uncomfortable (except perhaps in the case that the parameter in question is Planck's constant, or the speed of light, or other such universal constant). Not only is this practice unrealistic, it is dangerous: Although, admittedly, you will fix only parameters to which predictions of the current data are insensitive, making the 'danger' unobvious on those data, there is no guarantee that predictions of futuredata, especially extrapolations to designs considerably different from those used with the current data, will be similarly insensitive. In contrast, by putting a formal, informative, but appropriately diffuse (i.e. correctly representing the current state of scientific uncertainty) prior on (all) the parameters, one is being MORE objective than one is when fixing parameters: uncertainty is recognized and correctly factored in. This, in more words, is one of the points Ken was making. Where he and I part company is that he thinks (to mangle Wittgenstein) that 'whereof the data do not speak, thereof we should be silent', and I claim that that is not possible: choosing 'not to speak' is always equivalent to a sharp prior restriction on some model parameters. Moreover, I contend, when one is doing science, one is never completely ignorant -- there is ALWAYS some relevant past data which do speak, however softly, and those data define an informative prior distribution. In my view, then, Ken and Nick differ only on the sharp prior value to which to fix the pesky correlation. Ken, the empiricist, says fix it to unity because the data do not contradict that value, and Nick, the theoretician, complains that a correlation of unity is unscientific (whereas one of .5 presumably is not). Their common meeting ground, in my view, would be an informative prior distribution, bounded away from unity (to satisfy Nick), but with enough dispersion so that if the data did say anything about the value, they would be heard (satisfying Ken's empiricism). To return to the first point--where does the extra information come from?--the implication of the appeal to science here is deep: There CANNOT BE an automatic (algorithmic) GENERAL procedure for making ill-posed inverse problems well-posed (such as setting all near-boundary correlations to boundary values), as the only principled source of extra information is prior DOMAIN-SPECIFIC scientific knowledge. Sermon over. _/ _/ _/_/ _/_/_/ _/_/_/ Lewis B Sheiner, MD (lewis@c255.ucsf.edu) _/ _/ _/ _/ _/ Professor: Lab. Med., Biophmct. Sci. _/ _/ _/ _/_/_/ _/_/ Mail: Box 0626, UCSF, SF,CA,94143 _/ _/ _/ _/ _/ Courier: Rm C255, 521 Parnassus,SF,CA,94122 _/_/ _/_/ _/_/_/ _/ 415-476-1965 (v), 415-476-2796 (fax)