Fitness surfaces and searches: Dembski, Ewert, and Marks's search for design

This post is by Joe Felsenstein and Tom English

Back in October, one of us (JF) commented at Panda’s Thumb on William Dembski’s seminar presentation at the University of Chicago, Conservation of Information in Evolutionary Search. In his reply at the Discovery Institute’s Evolution News and Views blog, Dembski pointed out that he had referred to three of his own papers, and that Joe had mentioned only two. He generously characterized Joe’s post as an “argument by misdirection”, the sort of thing magicians do when they are deliberately trying to fool you. (Thanks, how kind).

Dembski is right that Joe did not cite his most recent paper, and that he should have. The paper, “A General Theory of Information Cost Incurred by Successful Search”, by Dembski, Winston Ewert, and Robert J. Marks II (henceforth DEM), defines search differently than do the other papers. However, it does not jibe with the “Seven Components of Search” slide of the presentation (details here). One of us (TE) asked Dembski for technical clarification. He responded only that he simplified for the talk, and stands by the approach of DEM.

Whatever our skills at prestidigitation, we will not try to untangle the differences between the talk and the DEM paper. Rather than guess how Dembski simplified, we will regard the DEM paper as his authoritative source. Studying that paper, we found that:

  1. They address “search” in a space of points. To make this less abstract, and to have an example for discussing evolution, we assume a space of possible genotypes. For example, we may have a stretch of 1000 bases of DNA in a haploid organism, so that the points in the space are all 41000 possible sequences.
  2. A “search” generates a sequence of genotypes, and then chooses one of them as the final result. The process is random to some degree, so each genotype has a probability of being the outcome. DEM ultimately describe the search in terms of its results, as a probability distribution on the space of genotypes.
  3. A set of genotypes is designated the “target”. A “search” is said to succeed when its outcome is in the target. Because the outcome is random, the search has some probability of success.
  4. DEM assume that there is a baseline “search” that does not favor any particular “target”. For our space of genotypes, the baseline search generates all outcomes with equal probability. DEM in fact note that on average over all possible searches, the probability of success is the same as if we simply drew randomly (uniformly) from the space of genotypes.
  5. They calculate the “active information” of a “search” by taking the ratio of its probability of success to that of the baseline search, and then taking the logarithm of the ratio. The logarithm is not essential to their argument.
  6. Contrary to what Joe said in his previous post, DEM do not explicitly consider all possible fitness surfaces. He was certainly wrong about that. But as we will show, the situation is even worse than he thought. There are “searches” that go downhill on the fitness surface, ones that go sideways, and ones that pay no attention at all to fitnesses.
  7. If we make a simplified model of a “greedy” uphill-climbing algorithm that looks at the neighboring genotypes in the space, and which prefers to move to a nearby genotype if that genotype has higher fitness than the current one, its search will do a lot better than the baseline search, and thus a lot better than the average over all possible searches. Such processes will be in an extremely small fraction of all of DEM’s possible searches, the small fraction that does a lot better than picking a genotype at random.
  8. So just by having genotypes that have different fitnesses, evolutionary processes will do considerably better than random choice, and will be considered by DEM to use substantial values of Active Information. That is simply a result of having fitnesses, and does not require that a Designer choose the fitness surface. This shows that even a search which is evolution on a white-noise fitness surface is very special by DEM’s standards.
  9. Searches that are like real evolutionary processes do have fitness surfaces. Furthermore, these fitness surfaces are smoother than white-noise surfaces “because physics”. That too increases the probability of success, and by a large amount.
  10. Arguing whether a Designer has acted by setting up the laws of physics themselves is an argument one should have with cosmologists, not with biologists. Evolutionary biologists are concerned with how an evolving system will behave in our present universe, with the laws of physics that we have now. These predispose to fitness surfaces substantially smoother than white-noise surfaces.
  11. Although moving uphill on a fitness surface is helpful to the organism, evolution is not actually a search for a particular small set of target genotypes; it is not only successful when it finds the absolutely most-fit genotypes in the space. We almost certainly do not reach optimal genotypes or phenotypes, and that’s OK. Evolution may not have made us optimal, but it has at least made us fit enough to survive and flourish, and smart enough to be capable of evaluating DEM’s arguments, and seeing that they do not make a case that evolution is a search actively chosen by a Designer.

This is the essence of our argument. It is a lot to consider, so let's explain this in more detail below:

As usual I will pa-troll the comments, and send off-topic stuff by our usual trolls and replies to their off-topic stuff to the Bathroom Wall

The target

DEM have a "target" for which the search is searching. Except that they don't actually require that the "search" actually search for something that makes sense. The target can be any set of points. If each point is a genotype and each of them has a fitness, the target can be genotypes with unusually high fitnesses, with unusually low fitnesses, mediocre fitnesses, or any mixture of them. They do not have to be points that are "specified" by fitness or by any other criterion. DEM do not require that the "search" even consider the fitnesses. They calculate the fraction of all _M_ points that are in the target. If |_T_| is the size of the target, for this fraction If we divide that by the number of points in the space, _N_, we get _p_ = |_T_|/|_N_|. This of course is also the probability that a random point drawn uniformly from the space hits the target.

Searches as distributions on the space of points

DEM consider the probability distribution of all outcomes of a search. Different instances of the search can find different results, either because they choose different starting points, or because of random processes later during the search. They assume very little about the machinery of the search -- they simply identify the search with the distribution of results that it gets. Suppose that two searches lead to the same distribution of outcomes, say a probability 0.6 of coming up with point _x1_, probability 0.4 of being coming up with _x12_, and probability 0 of everything else. They consider these two processes to be the same identical search. They don't consider what intermediate steps the searches go through. Correspondingly, two searches that lead to different probability distributions of outcomes are considered to be different searches. All distributions that you can make can apparently be found by one or another of DEM's search processes. From this point on they talk about the set of possible distributions, which to them represent the set of possible searches.

Note that this means that they are including "searches" that might either fail to be influenced by the fitnesses of the genotypes, and even ones that deliberately move away from highly fit genotypes, and seek out worse ones. Anything that gets results is a "search", no matter how badly it performs.

Are “searches” search algorithms?

Mathematicians and computer scientists working on optimization are accustomed to investigating the properties of algorithms that try to maximize a function. Once an algorithm is given, its behavior on different functions can be studied mathematically or numerically. DEM do not make this separation between the algorithm and the function. Their definition of a "search" includes both the algorithm and the function it encounters. As an evolutionary algorithm may have different results on different fitness surfaces, in their argument the same evolutionary model can be two different "searches" if it encounters two different fitness surfaces. As we have noted, even "searches" that do not try to maximize the fitness are included in their space.

DEM’s “Search For a Search”

A probability distribution on a set of _N_ points simply assigns probabilities to each of them. These probabilities can be positive or zero, but not negative, and they must add up to 1. So DEM consider the _N_ probabilities _a1_, _a2_, ..., _aN_. The conditions that they be nonnegative and add up to 1 forces them to lie in a region of _N_-dimensional space called a simplex. For example, if _N_ is 3, the numbers must lie in an equilateral triangle in a 3-dimensional space of points (_x_,_y_,_z_), where _x_+_y_+_z_ = 1, with its corners on the points (1,0,0), (0,1,0), and (0,0,1). For that small case, each probability distribution would have three probabilities, and be a point in the triangle such as (0.2344, 0.6815, 0.0841).

Now DEM discuss the distribution of searches -- that is, the distribution of probability distributions. Since each probability distribution is a point in the simplex, the distribution of searches is a distribution on that simplex. This is the probability distribution from which the search is said to be chosen. They go to a fair amount of effort, in this paper and in earlier papers by Dembski and Marks and by Dembski, to argue that a uniform distribution of searches on the simplex is a natural starting point from which the searches can be regarded as chosen. They also consider, in the DEM paper, initial distributions that are nonuniform. That does not make much difference for the argument made here. We're not going to argue with the details of their mathematics, but instead concentrate on what in evolutionary biology corresponds to such a choice of a search.

Their theorem

When one draws a probability distribution, which is one of the points in the simplex, one might get one that assigns a higher probability to the target, or one that assigns a lower probability of the target. On average, they argue, one gets one that has the probability _p_ of hitting the target. DEM show that, in the original uniform distribution of searches, at most a fraction _p/q_ of them will have a probability of finding the target as large as, or larger than _q_.

They then calculate a quantity that they call "active information" by taking the negative logarithm of this ratio and conclude that this is the amount of information that is built in by the choice of that search. In their argument it is implied that the improved success is due to some Designer having made choices that built that information in.

Mostly not using the fitness.

In Joe's earlier post, he argued that Dembski and Marks were examining the choice of a fitness surfaces from among all possible fitness surfaces. _He was wrong._ In fact, most of the searches in their distribution of searches cannot involve going uphill on any fitness surface. One is already in a very small portion of their distribution of searches as soon as the process is doing that. In that case one has an evolutionary search, and that is drawn from a very small fraction of all of their searches. Here is how we can see that.

A simple “greedy” search algorithm

Evolutionary processes occur in populations of organisms that have genotypes and fitnesses. Will a situation like that do as badly as a randomly-chosen search, where the probability of hitting the target is the same as it would be for random draws from the space? We can make a simple model, which easily shows that it is not the same.

Consider a space of DNA sequences, say all possible sequences of a stretch of 1000 nucleotides. The organism has one of these DNA sequences. In each generation it looks at all of its neighbor DNA sequences that have just one of these 1000 bases changed from the present sequence. There are 3000 of these, since each of the 1000 bases has one of the four bases A, C, G, and T and this means that there are 3 others possible at that site. Each DNA sequence has a fitness. Let's assume that the organism has just one DNA sequence, so it is located at one point in the genotype space. If the most fit of these 3000 neighbors has a higher fitness than the present DNA sequence, let's assume that the organism changes its DNA sequence to that DNA sequence. Otherwise it stays the same. It goes through _m-1_ generations of this.

This of course is a very simpleminded model of an evolving population, one that looks only at the neighbors of one genotype, but which also responds perfectly to any fitness differences. The question is not whether this is fully realistic, but whether this simple biasing by natural selection has a major effect on the probability of hitting the target. Let's call this beast a Greedy Uphill Climber "bug". We introduce it because it is easy to see what it will do.

Searching for a small target

To make the case even simpler, let's assume that all the genotypes have different fitness values -- there are no ties. There is then only one genotype that has the highest fitness. For our test case, let's define that one as the target _T_. In DEM's argument, the target can be defined in any way you want. It could even be a set of genotypes of unusually low fitness. But as the issue for evolution is whether natural selection can find highly-fit adaptations, it does not make sense to have a target that has unusually low fitness, especially since natural selection will actively move away from it.

Let's also simplify things by choosing the starting genotype at random from among all possibilities. Our GUC Bug then makes _m_ steps, each time to the most fit of the 3001 sequences that consist of its own genotype, plus the genotypes of its 3000 current neighbors.

Probability of the GUC Bug finding the target

Remember that if we drew at random from a distribution (a "search") which itself was randomly chosen from the simplex of all probability distributions, we would have only a probability _p_ of hitting the target. That is the same as if we just drew the outcome randomly from the set of possible DNA sequences. In the case of our GUC Bug, we start out with a randomly sampled genotype, and if that were all we did, we would have that small probability of hitting the target.

But if we let the bug do just one more step, so _m = 2_, it will move to the fittest of the 3001 immediate neighbors. This mimics the effect of natural selection, and that makes us much more likely to hit the target. The GUC Bug will find the target if it starts with the genotype which is the target, or if it starts with any genotype that is an immediate neighbor of the target. As there are 3000 neighbors of each of these DNA sequences, the probability of hitting the target will be about 3001 times greater than _p_.

If we take more steps, it is not clear how much larger is the set of starting points that will allow us to arrive at the target. It depends on how smooth the fitness surface is. At its smoothest, the fitness surface has no local peaks. For each genotype outside the target, there is a best neighbor of higher fitness, so the GUC Bug will move to that neighbor. If _m = 50_, there will be a great many neighbor genotypes that are less than 50 steps away from the target. In fact, there will be 1.211×10107 of those neighbors in all. That's a lot. All of those genotypes are starting points that will lead to _T_ in 49 steps or less. So the probability of a GUC Bug reaching the target is not just _p_, in the most favorable case it is vastly larger than that.

Behavior on a “white noise” fitness surface

One of us (TE) has carried out computer simulations of this case. He considered 1000-base nucleotide sequences and a GUC Bug started at a random sequence. Running the bug until it reached a local peak of the fitness surface, where no immediate neighbor is more fit, he found that these peaks were typically higher than 99.98% of all points. So even on one of the worst possible fitness surfaces, a GUC Bug does far better than choosing a DNA sequence at random.

Can DEM’s “searches” all be carried out by a greedy search bug?

This immediately establishes that most of the searches in DEM's space of searches are much worse at finding the target _T_ than any search that has a GUC Bug and a fitness surface. In our case the average chance of success of one of their searches is only _p_, which is more than 3000 times lower than the average for a GUC Bug that looks at neighbors on a fitness surface once. So a GUC Bug moving on a fitness surface must be far more successful than a random one of DEM's searches. This is true no matter what the fitness surface is. Simply by having a process that moves to more fit neighbors, we immediately narrow down DEM's searches to a tiny fraction of all possible searches.

But what about more realistic models of evolution?

These have the same property. In the GUC Bug model, we had only one DNA sequence in the species. If instead there is a population of sequences, then the genotypes of the species have multiple DNA sequences, and by multiple mutations and recombination parts of the space further afield can be reached. On the other hand the GUC Bug is more efficient in moving uphill to more fit genotypes than actual evolutionary processes are. So more realistic models of evolution might be either better or worse at climbing the fitness surface. But all of them move to the target from some reasonably large set of points in the neighborhood of the target. All such models will end up at the target far more often that a blind search will, and that immediately signals that these processes are far different from most of the searches in DEM's space of searches.

What causes smooth fitness surfaces?

We can see that evolutionary processes are not typical members of DEM's space of searches, because all of them, no matter what the shape of their fitness surface, do much better than blind search. Within the class of evolutionary processes those that have smoother fitness surfaces do better yet -- enormously better. DEM acknowledge this but do not discuss what makes fitness surfaces smooth. As one of us (JF) argued in his previous posts ([here](, [here](, [here](, and [here](, the ordinary laws of physics, with their weakness of long-range interactions, lead to fitness surfaces much smoother than white-noise fitness surfaces.

In the white-noise surfaces, changing one base in the DNA brings us to a fitness that is in effect randomly chosen from all possible fitnesses. In fact, it brings us to a fitness that is just as bad as if all bases in the DNA were changed simultaneously. That is not like actual biology. Furthermore in a white-noise fitness surface interactions among changes in different sites in the DNA are ubiquitous and incredibly strong. Changing one base leads to a randomly-different fitness. So does changing another. Changing both of those leads to a fitness that is also randomly-chosen, without regard to what the effects of the two earlier changes were. Combining two deleterious changes will then make no prediction that the result will be even more deleterious. Similarly, combining two advantageous changes will make no prediction that the result will be even more advantageous. But with real physics, those predictions can often be made.

Thus we can see that simply having genotypes with different fitnesses leads to results much better than most of the searches in DEM's space. Considering that "because physics" the fitness surfaces will be nonrandomly smooth brings us to an even tinier fraction of all possible searches, ones that are even more successful. Dembski and Marks would consider these smooth fitness surfaces to have large amounts of "active information", because they lead to much greater success at reaching any target which includes the genotypes of highest fitness. So these two effects do not require any intervention of a Designer, just the presence of genotypes that have fitnesses, and the action of ordinary laws of physics. Some, quite possibly all, of Dembski and Marks's "active information" is present as soon as we have genotypes that have different fitnesses, and genotypes whose phenotypes are determined using the ordinary laws of physics.

Is evolution a search?

The modeling of evolutionary processes as searches is of limited help. It is generally not best to regard evolutionary processes as carrying out a search for a target which is an optimal organism.

Evolution does not withhold its approval until it sees whether the single most-fit possible phenotype is found. Whether a species goes extinct depends on its fitnesses along the way, and a species can be quite successful without ever finding the most-fit genotypes. It is almost certain that we are not as fit as the best organism possible anywhere in in our space of genotypes. Requiring that evolution find that optimum result is unreasonable; we may always be stuck in some isolated region of genome space, and all of our wonderful adaptations may be the ones found there. But that is good enough for us to have developed remarkable abilities, including being capable of analyzing arguments about the evolutionary process, and seeing whether they imply the existence of the intervention of a Designer in the evolutionary process. Or whether they do not.