Natural Selection By Proxy

Here, I'll give a short summary of one of my favourite studies of recent decades: John Endler's ingenious field and laboratory experiments on small tropical fish¹, which in my (distinctly non-expert) opinion constitute one of the most compelling and 'slam-dunk' proofs available of the theory biological evolution by natural selection. After I've done that, in an act of unadulterated vanity, I'll suggest an extension to these experiments that I feel would considerably boost the information content of their results. That will be what I have dubbed 'selection by proxy'.

Don't get me wrong, Endler's experiments are brilliant. I first read about them in Richard Dawkins' delightful book, 'The Greatest Show on Earth,' and they captured my imagination, which is why I'm writing about them now.

Endler worked on guppies, small tropical fish, the males of which are decorated with coloured spots of varying hues and sizes. Different populations of guppies in the wild were found to exhibit different tendencies with regard to these spot patterns. Some populations show predominantly bright colours, while others prefer more subtle pigments. Some have large spots, while other have small ones. Its easy to contemplate the possibility that these differences in appearance are adaptive under different conditions. Two competing factors capable of contributing a great deal to the fitness of a male guppy are (1) ability to avoid getting eaten by predatory fish, and (2) ability to attract female guppies for baby making. Vivid colourful spots might contribute much to (2), but could be a distinct disadvantage where (1) is a major problem, and if coloration is determined by natural selection, then we would expect different degrees of visibility to be manifested in environments with different levels of predation. And so colour differences might be accounted for.

Furthermore, the idea suggests itself to the insightful observer that in the gravel-bottomed streams in which guppies often live, a range of spot sizes that's matched to the predominant particle size of the gravel in the stream bed would help a guppy to avoid being eaten, and that the tendency for particle and spot sizes to match will be greater where predators are more of a menace, and greater crypsis is an advantage. 

These considerations lead to several testable predictions concerning the likely outcomes if populations of guppies are transplanted to environments with different degrees of predation and different pebble sizes in their stream beds. These predicted outcomes are extremely unlikely under the hypothesis that natural selection is false. Such transplantations, both into carefully crafted laboratory environments, and into natural streams with no pre-existing guppy populations, constituted the punch line of Endler's experiments, and the observed results matched the predictions extraordinarily closely, after only a few months of naturally selected breeding. 

Its the high degree of preparatory groundwork and the many careful controls in these experiments, however, that result in both the high likelihood, P(D_p | H I), for the predicted outcome under natural selection, and the very low likelihood, P(D_p | H' I), under the natural-selection-false hypothesis. These likelihoods, under almost any prior, lead to only one possible logical outcome, when plugged into Bayes' theorem, and make the results conclusive.

The established fact that the patterning of male guppies is genetically controlled served both causes. Of course, natural selection can not act in a constructive way if the selected traits are not passed on to next generation, so the likelihood under H goes up with this knowledge. At the same time, alternate ways to account for any observed evolution of guppy appearance, such as developmental polymorphisms or phenotypic plasticity (such as the colour variability of chameleons, to take an extreme example), are ruled out, hitting P(D_p | H' I) quite hard.

Observations of wild populations had established the types of spot pattern frequent in areas with known levels of predation - there was no need to guess what kind of patterns would be easy and difficult for predators to see, if natural selection is the underlying cause. The expected outcome under this kind of selection could be forecast quite precisely, again enhancing the likelihood function under natural selection.

Selection between genotypes obviously requires the presence of different genotypes to select from, and in the laboratory experiments, this was ensured by several measures leading to broad genetic diversity within the breeding population. This, yet again, increased P(D_p | H I). (Genetic diversity in the wild is often ensured by the tendency for individuals to occasionally get washed downstream to areas with different selective pressures, which is one of the factors that made these fish such a fertile topic for research.)

The experiment employed a 3 × 2 factorial design. Three predation levels (strong, weak, and none) were combined with 2 gravel sizes, giving 6 different types of selection. The production of results appropriate for each of these selection types constitutes a very well defined prediction and would certainly be hard to credit under any alternate hypothesis, and P(D_p | H' I) suffers further at the hands of the expected (and realized) data.

Finally, additional blows were dealt to the likelihood under H', by prudent controls eliminating the possibility of effects due to population density and body size variations under differing predation conditions.

With this clever design and extensive controls, the data that Endler's guppies have yielded offer totally compelling evidence for the role of natural selection. Stronger predation led unmistakably to guppies with less vivid coloration, and greater ability to blend inconspicuously with their environment, after a relatively small number of generations.

I first read about these experiments with great enjoyment, but there was another thing that came to my mind: what the data did not say. It is quite inescapable from the results that natural selection of genetic differences was responsible for observed phenotypic changes arising in populations placed in different environments, but the data say nothing about the mechanism leading to those genetic differences. This, of course, is something that is central to the theory of natural selection. Indeed, we might consider the full name of this theory to be 'biological evolution by natural selection of random genetic mutations.' For the sake of completeness, we would like to have data that speak not only of the natural selection part, but also of the random basis for the genetic transformation.

I'm not saying that there is any serious doubt about this, but neither was there serious doubt about natural selection prior to Endler's result. (In fact, there is some legitimate uncertainty about the relative importance of natural selection v's other processes, such as genetic drift - uncertainty that work of Endler's kind can alleviate.) The theory of biological evolution, though, is a wonderful and extremely important theory. It stands out for a special reason: every other scientific theory we have is ultimately guaranteed to be wrong (though the degree of wrongness is often very small). Evolution by natural selection is the only theory I can think of that in principle could be strictly correct (and with great probability is), and so deserves to have all its major components tested as harshly as we reasonably can. This is how science honours a really great idea.

To test the randomness of genetic mutation, we need to consider alternative hypotheses. I can think of only one with non-vanishing plausibility: that at the molecular level, biology is adaptive in some goal-seeking way. That the cellular machinery strives, somehow, to generate mutations that make their future lineages more suitably adapted to their environment. I'll admit the prior probability is quite low, but I (as an amateur in the field) think its not impossible to imagine a world in which this happens, and as the only remotely credible contender, we should perhaps test it.

We could perform such a test by arranging for natural selection by proxy. That is, an experiment much like Endler's, but with a twist: at each generation, the individuals to breed are not the ones that were selected (e.g. by mates or (passively) by predators), but their genetically identical clones. At each generation, pairs of clones are produced, one of which is added to the experimental population, inhabiting the selective environment. The other clone is kept in selection-free surroundings, and is therefore never exposed to any of the influences that might make goal-seeking mutations work. Any goal-seeking mechanism can only plausibly be based on feedback from the environment, so if we eliminate that feedback and observe no difference in the tendency for phenotypes to adapt (compared to a control experiment executed with the original method), then we have the bonus of having verified all the major components of the theory. And if, against all expectation, there turned out to be a significant difference between the direct and proxy experiments, it would be the discovery of the century, which for its own sake might be worth the gamble. Just a thought.

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[1]	Natural Selection on Color Patterns in Poecilia reticulata, Endler, J.A., Evolution, 1980, Vol. 34, Pages 76-91 (Downloadable here)

Total Bayesianism

If you've read even a small sample of the material I've posted so far, you'll recognize that one of my main points concerns the central importance of Bayes' theorem. You might think, though, that the most basic statement of this importance is something like "Bayes' theorem is the most logical method for all data analysis." This, for me though, falls far short of capturing the most general importance of Bayes' rule.

Bayes' theorem is more than just a method of data analysis, a means of crunching the numbers. It represents the rational basis for every aspect of scientific method. And since science is simply the methodical application of common sense, Bayes' theorem can be seen to be (together with decision theory) a good model for all rational behaviour. Indeed, it may be more appropriate to invert that, and say that your brain is a superbly adapted mechanism, evolved for the purpose of simulating the results of Bayes' theorem. 

Because I equate scientific method with all rational behaviour, I am no doubt opening myself up to the accusation of scientism¹, but my honest response is: so what? If I am more explicit than some about the necessary and universal validity of science, this is only because reason has led me in this direction. For example, P.Z. Myers, author of the Pharyngula blog (vastly more well known than mine, but you probably knew that already), is one of the great contemporary advocates of scientific method - clear headed and craftsmanlike in the way he constructs his arguments - but in my evidently extreme view, even he can fall short, on occasion, of recognizing the full potential and scope of science. In one instance I recall, when the league of nitwits farted in Myers' general direction, and he himself stood accused of scientism, he deflected the accusation, claiming it was a mistake. My first thought, though, is "hold on, there's no mistake." Myers wrote: 

The charge of scientism is a common one, but it’s not right: show us a different, better path to knowledge and we’ll embrace it.

But how is one to show a better path to knowledge? In principle, it can not be done. If Mr. X claims that he can predict the future accurately by banging his head with a stone until visions appear, does that suffice as showing? Of course not, a rigorous scientific test is required. Now, if under the best possible tests, X's predictions appear to be perfectly accurate, any further inferences based on them are only rational to the extent that science is capable of furnishing us (formally, or informally) with a robust probability estimate that his statements represent the truth. Sure, we can use X's weird methodology, but we can only do so rationally, if we do so scientifically. X's head smashing trick will never be better than science (a sentence I did not anticipate writing).

To put it another way, X may yield true statements, but if we have no confidence in their truth, then they might as well be random. Science is the engine generating that confidence.

So, above I claimed that all scientific activity is ultimately driven by Bayes' theorem. Lets look at it again in all its glory:

P(H | DI)   =    P(H | I) × P(D | H I) P(H | I) × P(D | H I)   +   P(H' | I) × P(D | H' I)

(1)

(As usual, H is a hypothesis we want to evaluate, D is some data, I is the background information, and H' means "H is not true.")

The goal of science, whether one accepts it or not, is to calculate the term on the left hand side of equation (1). Now, most, if not all, accepted elements of experimental design are actually adapted to manipulate the terms on the right hand side of this equation, in order to enhance the result. I'll illustrate with a few examples.

Firstly, and most obviously, the equation calls for data, D. We have to look at the world, in order to learn about it. We must perform experiments to probe nature's secrets. We can not make inferences about the real world by thought alone. (Some may appear to do this, but no living brain is completely devoid of stored experiences - the best philosophers are simply very efficient at applying Bayes' theorem (usually without knowing it) to produce powerful inferences from mundane and not very well controlled data. This is why philosophy should never be seen as lying outside empirical science.)

Secondly, the equation captures perfectly what we recognize as the rational course of action when evaluating a theory - we have to ask 'what should I expect to see if this theory is true? - what are its testable hypotheses?' In other words, what data can I make use of in order to calculate P(D | HI)?

Once we've figured out what kind of data we need, the next question is how much data? Bayes' rule informs us: we need P(D | HI) to be as high as possible if true, and as low as possible if false. Lets look at a numerical example:

Suppose I know, on average, how tall some species of flower gets, when I grow the plants in my home. Suppose I suspect that picking off the aphids that live on these flowers will make the plants more healthy, and cause them to grow higher. My crude hypothesis is that the relative frequency with which these specially treated flowers exceed the average height is more than 50 %. My crude data set results from growing N flowers, applying the special treatment to all of them, and recording the number, x, that exceed the known average height.

To check whether P(D | HI) is high when H is true and low when H is false, we'll take the ratio

P(D | H I) P(D | H' I)

(2)

If H_f says that the frequency with which the flowers exceed their average height is f, then P(D | H_fI) (where D is the number of tall flowers, x, and the total number grown N) is given by the binomial distribution. But our real hypothesis, H, asserts that f is in the range 0.5 < f ≤ 1. This means we're going to have to sum up a whole load of P(D | H_fI)s. We could do the integral exactly, but to avoid the algebra, lets treat the smoothly varying function like a staircase, and split the f-space into 50 parts: f = 0.51, 0.52, ...,0.99, 1.0. To calculate P(D | H'I), we'll do the same, with f = 0.01, 0.02, ...., 0.50.

What we want, e. g. for the P(D | HI), is P(D | [H_0.51 + H_0.52 + ....] I).

Generally, where all hypotheses involved are mutually exclusive, it can be shown (see appendix below) that,

P(D | [H1 + H2 + .....] I)   =    P(H1 | I) P(D | H1 I)  +  P(H2 | I) P(D | H2 I)  +  ..... P(H1 | I)  +  P(H2 | I)  +  .....

(3)

But we're starting from ignorance, so we'll take all the priors, P(H_f | I), to be the same. We'll also have the same number of them, 50, in both numerator and denominator, so when we take the desired ratio, all the priors will cancel out (as will the width, Δf = 0.01, of each of the intervals on our grid), and all we need to do is sum up P(D | H_f1I) + P(D | H_f2I) + ....., for each relevant range. Each term will come straight from the binomial distribution:

P(x | N, f)  =   N!    xf   (N - x)1-f x!   (N - x)!

(4)

If we do that for say 10 test plants, with seven flowers growing beyond average height, then ratio (2) is 7.4. If we increase the number of trials, keeping the ratio of N to x constant, what will happen?

If we try N = 20, x = 14, not too surprisingly, ratio (2) improves. The result now is 22.2, an increase of 14.8. Furthermore if we try N = 30, x = 21, ratio (2) increases again, but this time more quickly: now the ratio is 58.3, and further increase of 36.1.

So, to maximize the contrast between the hypotheses under test, H and H', what we should do is take as many measurements as practically possible. Something every scientist knows already, but something nonetheless demanded by Bayes' theorem.

How is our experimental design working out, then? Well, not that great so far, actually. Presumably the point of the experiment was to decide if removing the parasites from the flowers provided a mechanism enabling them to grow bigger, but all we have really shown is that they did grow bigger. We can show this by resolving e.g. H' into a set of mutually exclusive and exhaustive (within some limited model) sub-hypothesis:

H' = H'A1 + H'A2 + H'A3 + ......

(5)

where H' is, as before, 'removing aphids did not improve growth,' and some of the A's represent alternative causal agencies capable of affecting a change in growth. For example, A₁ is the possibility that a difference in ambient temperature tended to make the plants grow differently. Lets look again at equation (3). This time instead of H_f's, we have all the H'A_i, but the principle is the same. Previously, the priors were all the same, but this time, we can exploit the fact that they need not be. We need to manipulate those priors so that the P(D | H'I) term in the denominator of Bayes' theorem, is always low if the number of tall plants in the experiment is large. We can do this by reducing the priors for some of the A_i corresponding to the alternate causal mechanisms. To achieve this, we'll introduce a radical improvement to our methodology: control.

Instead of relying on past data for plants not treated by having their aphids removed, we'll grow 2 sets of plants, treated identically in all respects, except the one that we are investigating with our study. The temperature will be the same for both groups of plants, so P(A₁ | I) will be zero - there will be no difference in temperature to possibly affect the result. The same will happen to all (if we have really controlled for all confounding variables) the other A_i that corresponded to additional agencies offering explanations for taller plants.

This process of increasing the degree of control can, of course, undergo numerous improvements. Suppose, for example, that after a number of experiments, I begin to wonder if its not actually removing the aphids that affects the plants, but simply the rubbing of the leaves with my fingers that I perform in order to squish the little parasites. So as part of my control procedure, I devise a way to rub the leaves of the plants in the untreated group, while carefully avoiding those villainous arthropods. Not a very plausible scenario, I suppose, but if we give a tentative name to this putative phenomenon, we can appreciate how analogous processes might be very important in other fields. For the sake of argument, lets call it a placebo effect.

Next I begin to worry that I might be subconsciously influencing the outcome of my experiments. Because I'm keen on the hypothesis I'm testing, (think of the agricultural benefits such knowledge could offer!) I worry that I am inadvertently biasing my seed selection, so that healthier looking seeds go into the treatment group, more than into the control group. I can fix this, however, by randomly allocating which group each seed goes into, thereby setting the prior for yet another alternate mechanism to zero. The vital nature of randomization, when available, in collecting good quality scientific data is something we noted already, when looking at Simpson's paradox, and is something that has been well appreciated for at least a hundred years.

Randomization isn't only for alleviating experimenter biases, either. Suppose that my flower pots are filled with soil by somebody else, with no interest in or knowledge of my experimental program. I might be tempted to use every second pot for the control group, but suppose my helper is also filling the pots in pairs, using one hand for each. Suppose also that the pots filled with his left hand receive inadvertently less soil than those filled with his right hand. Unexpected periodicities such as these are also taken care of by proper randomization.

Making real-world observations, and lots of them; control groups; placebo controls; and randomization: some exceedingly obvious measures, some less so, but all contained in that beautiful little theorem. Add these to our Bayesian formalization of Ockham's razor, and its extension, resulting in an explanation for the principle of falsifiability, and we can not avoid noticing that science is a thoroughly Bayesian affair.

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Appendix

You might like to look again at the 3 basic rules of probability theory, if your memory needs refreshing.

To derive equation (3), above, we can write down Bayes' theorem in a slightly strange way:

P(D | [H1 + H2 + ....], I)   =    P(D | I) × P([H1 + H2 + ....] | D I) P([H1 + H2 + ....] | I)

(A1)

This might look a bit backward, but thinking about it a little abstractly, before any particular meaning is attached to the symbols, we see that it is perfectly valid. If you're not used to Boolean algebra, or anything similar, let me reassure you that its perfectly fine for a combination of propositions, such as A + B + C, (where the + sign means 'or') to be treated as a proposition in its own right. If equation (A1) looks too much, just replace everything in the square brackets with another symbol, X.

As long as all the various sub-hypotheses, H_i, are mutually exclusive, then when we apply the extended sum rule above and below the line, the cross terms vanish, and (A1) becomes:

P(D | [H1 + H2 + ....], I)   =    P(D | I) × [ P(H1 | D I)  +  P(H2 | D I)  +  ..... ] P(H1 | I)   +   P(H2 | I)   + .....

(A2)

We can multiply out the top line, and also make note that for each hypothesis, H_i, we can make two separate applications of the product rule to the expression P(H_i D | I), to show that

P(D | I)   =    P(Hi | I) P(D | Hi I) P(Hi | D I)

(A3)

(This is actually exactly the technique by which Bayes' theorem itself can be derived.)

Substituting (A3) into (A2), we see that

P(D | [H1 + H2 + .....] I)   =    P(H1 | I)  P(D | H1 I)  +  P(H2 | I)  P(D | H2 I)  +  ..... P(H1 | I)  +  P(H2 | I)  +  .....

(A4)

which is the result we wanted.

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[1]  From Wikipedia:

Scientism is a term used, usually pejoratively, to refer to belief in the universal applicability of the scientific method and approach, and the view that empirical science constitutes the most authoritative worldview or most valuable part of human learning to the exclusion of other viewpoints.