Tuesday, May 29, 2012

The theorem that never was: Diversionary “erratum” from Dembski and Marks

In my last post, I showed that the research program of William A. Dembski and Robert J. Marks II depends on making performance gain in so-called search look like information gain, and calling it active information. It was clear that the misrepresentation arose from severe misunderstanding of the "no free lunch" theorems, not an intent to deceive. I would call it an honest error, if it were not the result of intrinsically dishonest activities — apologetics and culture war.*

Now, unfortunately, I indict both the math and the men. When things like Marks' approval of a master's thesis plagiarizing his and Dembski's publications present themselves to me, I feel honor-bound to report on them. Also, people who know of Marks' considerable technical achievements need to know also that he has, in late career, begun using his reputation and connections to pursue socio-poltical ends. They may want to give him the benefit of the doubt, as I did when he began collaborating with Dembski. This post shows why that's not such a good idea. (And so do this, this, and this.)

There never was a Horizontal No Free Lunch Theorem

In the abstract of "The Search for a Search: Measuring the Information Cost of Higher Level Search," Dembski and Marks write:

We prove two results: (1) The Horizontal No Free Lunch Theorem, which shows that average relative performance of searches never exceeds unassisted or blind searches, and (2) The Vertical No Free Lunch Theorem, which shows that the difficulty of searching for a successful search....

That has the ring of a grand contribution. They've extended the famous "no free lunch" results both horizontally and vertically. Problem is, they knew long before publication of the paper that the statement of the first "theorem" was semantically ill-formed (not even a proposition, let alone a theorem).

How can I say that they knew? The paper was originally scheduled for publication in 2008, in an obscure Polish journal that shut down after two years of operation. Dembski and Marks disseminated the paper themselves, however, after it was accepted, and soon heard of several errors from the mathematically talented D. Eben. I saw not only Eben's detailed criticisms on the Web (e.g., here and here), but also some of the email he sent to Marks. The version of the paper subsequently published in July 2010, in a Japanese open-access journal, includes revisions of arcane mathematics, and acknowledges Eben. But it leaves in place the "theorem" made into gobbledy-gook by an error in sophomore-level discrete math.

Was Marks too dumb to see the mistake, or was he unwilling to dismantle the lovely "horizontal-and-vertical" rhetoric, perhaps believing that he could produce the theorem later? I have never suggested that Marks is anything less than a very bright man.

Dembski and Marks release a diversionary “erratum”

Now, almost two years later, Dembski and Marks have added an erratum to the end of the paper. (Here are Eben's announcement and subsequent discussion of it, along with some of my comments.) They begin by acknowledging the mistake. Then they give a theorem and a corollary that seem, at first blush, to correct those in the original paper. However, they've pulled the old switcheroo. The theorem is actually a lemma, and the corollary is the main result. "Search" is reduced to the utterly uninteresting case of a single draw from the sample space. And, hilariously, the average "active information" in the corollary is undefined. The cause of this is a) correct consideration of insoluble problems, for which expected performance is 0, and b) logarithmic transformation of expected performance to make it into faux information. That is, $\log 0 = -\infty$. Oops.

Dembski and Marks can predicate magical knowledge that the problem has a solution, and make it so that the average "active information" is not always undefined. But there is no way for them to arrange for it always to be defined. As discussed in more detail on Mr. Eben's blog, the "erratum" predicates a condition in which average performance does not depend on the choice of "search" (sampler). Yet the average "active information" varies from 0 to $-\infty$, depending on the "search." This contradicts claims in the paper like this:

If no information about a search exists, so that the underlying measure is uniform, then, on average, any other assumed measure will result in negative active information, thereby rendering the search performance worse than random search.

Yet the "erratum" does nothing to identify and correct this crucial error. Dembski and Marks never had a Horizontal No Free Lunch Theorem, could not possibly have believed that they published one, evidently know that they cannot produce one, and clearly seek to admit to absolutely as little as possible.

* To get an idea of how Dembski will exploit what he and Marks have published in the next judicial test of public-school instruction in "intelligent design" creationism, see Expert Witness Report: The Scientific Status of Intelligent Design and Rebuttal to Reports by Opposing Expert Witnesses, which he prepared for Kitzmiller v. Dover Area School District (2005). After the trial, Dembski, who had backed out of testifying, dubbed a Flash animation, now heavily redacted, with the judge in the case "represented as a pull-to-speak doll spouting snippets of his decision in a high-pitched voice with added farting noises, and various pro-science advocates... represented as pulling the string."

Marks makes no secret of how he feels about Kitzmiller in the following bait-and-switch guest lecture he gave in "Introduction to Engineering" at Baylor University (Fall 2011). He reads [1:00:00] from the transcript of the trial, supplying a dopey voice for Robert Pennock, who testified on the artificial-life system Avida. Ever so coincidentally, this is followed by a dubbed "OK" in a high-pitched voice.

I hope to devote an entire post to this video: "Academic Freedom Does Not Entail Instructional Freedom." My guess is that program review teams from the Accreditation Board for Engineering and Technology would frown on "the universe is not old enough nor big enough to allow the evolution of complex life" [49:55] indoctrination of freshmen.