More thoughts about Dembski and Marks’ project:

- The fact that the No Free Lunch theorem does not hold for a continuous search domain may have some consequences for the project pursued by Dembski and Marks. (That project, by the way, seems to have inspired at least two blogs devoted to criticizing it, DIEBLOG and Bounded Science. Some interesting points are raised there.) Their idea of studying search algorithms/heuristics for a target, then metaheuristics to select such heuristics, then meta-meta….-heuristics, and so on quickly generates uncountably infinite search domains for the meta-…-heuristics. Even if the search domain containing the target is finite, the set of all probabilistic search heuristics is already uncountably infinite, and it doesn’t get better in the space of metaheuristics. Dembski and Marks don’t define their targets in terms of a fitness function, and therefore aren’t
*directly*affected by failure of the No Free Lunch theorem for uncountably infinite search domains. However, they have left it unspecified how the target is to be selected. A very natural way to fill in those details would be in fact let the target by defined in terms of a fitness function; so indirectly I think their project is undermined. - In their new article,
*Bernoulli’s Principle of Insufficient Reason and Conservation of Information in Computer Search*, they restrict their endorsal of the Principle of Insufficient Reason to finite sample spaces. This is a bit interesting: (i) It allows them to avoid the most damning criticism against that principle and shows some awareness of the serious problems with it, though it isn’t exactly unproblematic for finite spaces either. Furthermore, the conditions under which the Principle of Insufficient Reason are supposed to apply are actually never satisfied in practice, as was noted by some commenters at one of the author’s blog. (ii) It undermines their previous motivation for using uniform probability measures over search heuristics, metaheuristics, etc. (iii) It pretty much commits them to a Bayesian view of what probabilities are. Nothing wrong with that—some of my best friends are Bayesians—but the first author has previously argued against such views, arguing that prior probabilities are hard to justify, at least in the context of “design inferences” (but it is hard to see why design inferences would be in any way special in this regard):

“As we’ve already seen, for the Bayesian approach to work requires prior probabilities. Yet prior probabilities are often impossible to justify. Unlike the example of the urn and two coins discussed earlier, in which drawing a ball from an urn neatly determines the prior probabilities regarding which coin will be tossed, for most design inferences, especially the interesting ones like whether there is design in biological systems, we have no handle on the prior probability of a design hypothesis, or that prior probability is fiercely disputed (theists, for instance, might regard the prior probability as high whereas atheists would regard it as low).” (Dembski, 2004)

(iv) The endorsement of the more general the Principle of Maximum Entropy puts them in danger of committing themselves to a Bayesian methodology as Bayesian conditionalization is closely linked to this principle. A further generalization, to the Principle of Minimal Discrimination Information (or the Principle of Minimum Relative Entropy) contains both conditionalization and Maximum Entropy as special cases. Again, nothing wrong with this, except that the first author has in the past rejected a Bayesian methodology.

**References**

William A. Dembski and Robert J. Marks II. Bernoulli’s Principle of Insufficient Reason and Conservation of Information in Computer Search. Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics. San Antonio, TX, USA – 2009 [Available at the second author’s homepage, here]

William A. Dembski. The Design Revolution: Answering the Toughest Questions About Intelligent Design. InterVarsity Press (2004) [Chapter 33, the source of the above quote, is available here]