Sunday, September 6, 2009

Brainbreaker for a Sunday morning: Vilenkin's Principle of Mediocrity

Scientists like to believe that they deal in objective data, and that cultural biases play little part in their experiments and observations, even though sociologists studying scientists have known that this is not the case for some time. A prime example [and I cannot give a link here because nothing from the paper version of the 1993 study has made it onto the net as near as I can tell] is a study of biologists in Japan examining the mating habits of certain primates. The scientists, thankfully, filmed all of the primate interactions in their study, which made the following experiment possible:

The original team of biologists was all Japanese males, and, based on what they saw, they reached a set of ten major conclusions about primate sexual behavior. With their permission, the sociologists then gave the raw data [the tapes and other experimental notations] to a group that was 50% female, but held equivalent credentials. The second group, of course, was not given the first group's conclusions. What a surprise! The second group, because the female researchers brought different perspectives and asked different questions about gender/sexual relations, came up with an entirely different set of conclusions, including four that completely contradicted the original research. All from exactly the same evidence.

So who was right? Interesting question.

The validity of this question is not limited, however, either to subjective observations or to biology, but this sort of observational bias shows up in all sort of subtle ways.

For example, cosmologist Alexei Vilenkin has posited what he terms The Principle of Mediocrity to explain why other regions of space that we cannot see and will never be able to see are likely to be similar [especially in terms of natural laws] to those we can observe [Warning; Heavy sledding for a Sunday morning ahead]:

Quantum fluctuations in the course of eternal inflation ensure that all possible values of the constants are realized somewhere in the universe. As a result, remote regions of the universe may drastically differ in their properties from our observable region. The values of the constants in our vicinity are determined partly by chance and partly by how suitable they are for the evolution of life. The latter effect is called anthropic selection.

If some "constant" varies from one region of the universe to another, its value cannot be predicted with certainty, but we can still try to make a statistical prediction. Suppose, for example, I want to predict the height of the first man I am going to see when I walk out into the street. Having consulted the statistical data on the height of men in the United States, I find that the height distribution follows a bell curve with a median value at 1.77 meters. The first man I meet is not likely to be a giant or a dwarf, so I expect his height to be in the mid-range of the distribution. To make the prediction more quantitative, I can assume that he will not be among the tallest 2.5% or shortest 2.5% of men in the United States. The remaining 95% have heights between 1.63 and 1.90 meters. If I predict that the man I meet will be within this range of heights and then perform the experiment a large number of times, I can expect to be right 95% of the time. This is known as a prediction at 95% confidence level.

In order to make a 99% confidence level prediction, I would have to discard 0.5% at both ends of the distribution. As the confidence level is increased, my chances of being wrong get smaller, but the predicted range of heights gets wider and the prediction less interesting.

A similar technique can be used to make predictions for the constants of nature. Suppose the Statistical Bureau of the Universe collected and published the values of some constant X measured by observers in different parts of the universe. We could then discard 2.5% at both ends of the resulting distribution and predict the value of X at a 95% confidence level.

What would be the meaning of such a prediction? If we randomly picked observers in the universe, their observed values of X would be in the predicted interval 95% of the time. Unfortunately, we cannot perform this experiment, because all regions with different values of X are beyond our horizon. We can only measure X in our local region. What we can do, though, is to think of ourselves as having been randomly picked. We are just one in the multitude of civilizations scattered throughout the universe. We have no reason to believe a priori that the value of X in our region is very large or small, or otherwise very special compared with the values measured by other observers. Hence, we can predict, at 95% confidence level, that our measurement will yield a value in the specified range. The assumption of being unexceptional is important in this approach; I called it "the principle of mediocrity".

First, understand that I understand that in using the example of human heights, Vilenkin is employing an explanatory metaphor rather than a rigorous mathematical proof. Nonetheless, metaphors are probably more important than those proofs in some ways, because human beings tend to reason and imagine metaphorically, or--to put it another way--the metaphor will more often lead to thinking about the proof than the proof will lead to thinking about the metaphor.

That said, Vilenkin's observations about the height of human beings is both understandable and correct, but recent research suggests that it means almost exactly the opposite of what he thinks it does.

Human beings, it turns out, have so little statistical variations in height [especially among populations] that they are not unexceptional--within the animal kingdom they are quite exceptional.

From Wired:

Crunch the numbers on the animal kingdom’s sizes and shapes, and humans differ from each other far less than most species. The reason why is a mystery.

“We don’t have an answer. We have this interesting observation, but the explanation is an open hypothesis,” said evolutionary biologist Andrew Hendry of McGill University.

Hendry and Queens University biologist Ann McKellar combed through the scientific literature on body size and length in more than 200 species, from insects to fish to birds and, of course, humans.

In terms of sheer mass, humans variation was par for the animal course. So was the height difference between populations — between, say, the average Maasai man and the average Australian aborigine. But when it came to variation within a population, such as that Maasai or aboriginal village, humans had less variation than 95 percent of all the species studied. The results were published Tuesday in Public Library of Science ONE.

Through most of human history, it seems that evolution stretched or shrunk people to fit their local environments, then rigidly enforced the size limits. People were no taller or shorter than their neighbors.

So Vilenkin's statement, The first man I meet is not likely to be a giant or a dwarf, so I expect his height to be in the mid-range of the distribution. To make the prediction more quantitative, I can assume that he will not be among the tallest 2.5% or shortest 2.5% of men in the United States, is--in fact--not a global observation, but one specifically limited by his residential environment. In this case it is the United States

But what would happen to his example if he were told that he had been dropped in some random, unknown location in Africa, which contains far higher proportions (the Massai on one end and !Kung on the other) of peoples at the extreme ranges of human heights? The average height would be much the same, but the variability would be much higher. Or, to put it in terms of his theory, his own average height would have become significantly less unexceptional, less mediocre as a result.

Vilenkin's Principle of Mediocrity has always bothered me since I read it. It is an understandable attempt to speculate on unknown conditions from a single data point, but the validity of the theory is heavily dependent upon the choice of examples that one uses for comparison. It is just as easy to select classes of items for comparison wherein the statistical mean is much less helpful in predicting with any confidence what the next, randomly selected example of the set will look like.

To suggest but one example, what if--instead of height--I am trying to predict whether the next person I randomly meet will have an innie or an outie belly button? I have no real idea what the normal distribution of these characteristics is, but I know that there will be far more difficulty in making a prediction about height with 95% confidence. Why? Because I am selecting based on a single, effectively binary characteristic, whereas height is the summation of multiple characteristics all blended together, and actually tells me nothing much useful about bone strength, nutrition, muscle density, or anything else except ... average height.

Vilenkin's Principle of Mediocrity is ultimately an exceptionally important example of an unexceptional reality: even scientists tend to understand the world and form their hypotheses in ways that are consistent rather than at variance with their cultural assumptions and their very human insistence that the area and the people around them are representational rather than exceptional.


Miko said...

Interestingly, Vilenkin's discussion of anthropic selection shows that he's wrong and that (if he were to juxtapose the first paragraph with some of the latter ones) he should know that he's wrong.

On this subject, I always recommend Feynman's "Cargo Cult Science."

Tyler Nixon said...

Of course, none of this takes into account the most exceptionally diverse yet unquanitfiable distinction of our species that makes the physical variations an essential nullity.

It's called The Spirit.

Being as we can never truly know the mind, the "consciousness" of any other species except by empirical correlation to our own, we can never know what variations or level of variation, if any, exist in that realm.

Perhaps every fly is an individual expression of existential brilliance...but observably they do little more than eat shit and propagate themselves.

I guess the finer points of existence for any species are universal, and thus perhaps the metaphysics of 'mind' are the same.

Remember all this next time you murder a fly.