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Rosenhouse and Discrete Hypercube Evolution

Photo credit: Foam, via Flickr (cropped).

I am reviewing Jason Rosenhouse’s new book, The Failures of Mathematical Anti-Evolutionism (Cambridge University Press), serially. For the full series so far, go here.

Because of the centrality of the “searching protein space” model to Jason Rosenhouse’s argument, it’s instructive to illustrate it with the full rigor of his track 2 (see my post from yesterday on that). Let me therefore lay out such a model in detail. Consider a 100-dimensional discrete hypercube of 100-tuples of the form (a_1, a_2, …, a_100), where the a_i’s are all natural numbers between 0 and 100. Consider now the following path in the hypercube starting at (0, 0, …, 0) and ending at (100,100, …, 100). New path elements are now defined by adding 1’s to each position of any existing path element, starting at the left and moving to the right, and then starting over at the left again. Thus the entire path takes the form

0:   (0, 0, …, 0)

1:   (1, 0, …, 0)

2:   (1, 1, …, 0)

100:   (1, 1, …, 1)

101:   (2, 1, …, 1)

102:   (2, 2, …, 1)

200:   (2, 2, …, 2)

300:   (3, 3, …, 3)

1,000:   (10, 10, …, 10)

2.000:   (20, 20, …, 20)

10,000:   (100, 100, …, 100)

The hypercube consists of 101^100, or about 2.7 x 10^200 elements, but the path itself has only 10,001 path elements and 10,000 implicit path edges connecting the elements. 

A Uniform Probability Distribution 

For simplicity, let’s put this discrete hypercube under a uniform probability distribution (we don’t have to, but it’s convenient for the purposes of illustration — Rosenhouse mistakenly claims that intelligent design mathematics automatically defaults to uniform or equiprobability, but that’s not the case, as we will see; but there are often good reasons to begin an analysis there). Given a uniform probability on the discrete hypercube, the path elements, all 10,001 of them considered together, have probability roughly 1 in 2.7 x 10^196 (10,001 divided by the total number of elements making up the hypercube). That’s very small, indeed smaller than the probability of winning 23 Powerball jackpots in a row (the probability of winning one Powerball jackpot is 1 in 292,201,338). 

Each path element in the hypercube has 200 immediate neighbors. Note that in one dimension there would be two neighbors, left and right; in two dimensions there would be four neighbors, left and right as well as up and down; in three dimensions there would be six neighbors, left and right, up and down, forward and backward; etc. Note also for path elements on the boundary of the hypercube, we can simply extend the hypercube into the ambient discrete hyperspace, finding there neighbors that never actually end up getting used (alternatively, the boundaries can be treated as reflecting barriers, a device commonly used by probabilists). 

Next, let’s define a fitness function F that is zero off the path and assigns to path elements of the form (a_1, a_2, …, a_100) the sum a_1 + a_2 + … + a_100. The starting point (0, 0, …, 0) then has minimal fitness and the end point (100, 100, …, 100) then has maximal fitness. Moreover, each successive path element, as illustrated above, has higher fitness, by 1, than its immediate predecessor. If we now stay with a uniform probability, and thus sample uniformly from the adjoining 200 neighbors, then the probability p of getting to the next element on the path, as judged by the fitness function F, is 1 in 200 for any given sample query, which we can think of and describe as a mutational step

Waiting Times

The underlying probability distribution for moving between adjacent path elements is the geometric distribution. Traversing the entire path from starting point to end point can thus be represented by a sum of independent and identically distributed (with geometric distribution) random variables. Thus, on average, it takes 200 evolutionary sample queries, or mutational steps, to move from one path element to the next, and it therefore takes on average 2,000,000 (= 200 x 10,000) evolutionary sample queries, or mutational steps, to move from the starting to the end point. Probabilists call these numbers waiting times. Thus, the waiting time for getting from one path element to the next is, on average, 200; and for getting from the starting to the end point is, on average, 2,000,000. 

As it is, the geometric distribution is easy to work with and illustrates nicely Rosenhouse’s point about evolution not depending on brute improbability. But suppose I didn’t see that I was dealing with a geometric distribution or suppose the problem was much more difficult probabilistically, allowing no closed-form solution as here. In that case, I could have written a simulation to estimate the waiting times: just evolve across the path from all zeros to all one-hundreds over and over on a computer and see what it averages to. Would it be veering from Rosenhouse’s track 2 to do a simulation to estimate the probabilities and waiting times? Throughout his book, he insists on an exact and explicit identification of the probability space, its geometry, and the relevant probability distributions. But that’s unnecessary and excessive. 

Poker and Biology

In many practical situations, we have no way of assigning exact theoretical probabilities. Instead, we must estimate them by sampling real physical systems or by running computer simulations of them. Even in poker, where all the moving parts are clearly identified, the probabilities can get so out of hand that only simulations can give us a grasp of the underlying probabilities. And what’s true for poker is even more true for biology. The level of specificity I’ve given in this hypercube example is way more than Rosenhouse gives in his “searching protein space” example. The hypercube makes explicit what he leaves implicit, namely, it distinguishes mathematically the entire search space from the evolutionary paths through it from the neighborhoods around points on the path. It thus captures a necessary feature of Darwinian evolution. But it does so at the cost of vast oversimplification, rendering the connection between Darwinian and real-world evolution tenuous at best.

Why have I just gone through this exercise with the 100-dimensional discrete hypercube, giving it the full track 2 monty? Two reasons. One, it is to rebut Rosenhouse’s insistence on Darwinian gradualism in the face of intelligent design (more on this later in this review series). Two, it is to show Darwinist critics like Rosenhouse that we in the intelligent design community know exactly what they are talking about when they stress that rather than brute improbability, the real issue for evolvability is the improbability of traversing evolutionary pathways subject to fitness. I’ve known this for decades, as have my intelligent design colleagues Mike Behe, Steve Meyer, and Doug Axe. Rosenhouse continually suggests that my colleagues and I are probabilistically naïve, failing to appreciate the nuances and subtleties of Darwinism. We’re not. I’ll be returning to the hypercube example because it also illustrates why Rosenhouse’s Darwinism is so implacably committed to sequential mutations and must disallow simultaneous mutations at all costs. But first …

Next, “Rosenhouse and ‘Mathematical Proof.’”

Editor’s note: This review is cross-posted with permission of the author from BillDembski.com.