Friday, July 13, 2007

Probability and the anthropic principle

In the comments to his review of Bede Rundle's "Why there is Something rather than Nothing", [Update: Law has pulled the examples into their own post] Stephen Law attempts to undermine the anthropic principle with a number of counterexamples:
Here's a Swinburne type illustration. Suppose I am asked to guess each one of 52 cards, one by one. If I ever get one wrong, my brains will be blown out.

I start guessing, and amazingly, I get all 52 cards correct. Now you may say, "What's so improbably about that? After all, the probability of you getting them all right is 1, as you wouldn't be here otherwise would you?"

But of course, there's a sense in which something deeply improbably has happened. So improbable, in fact, that it would be reasonable for me to suspect this result wasn't just a matter of chance.

As a condemned spy, you are put before a firing squad of twenty expert marksmen, who load aim, and fire at your heart from close range.

Amazingly, they all miss. You feign death, and survive.

Pure luck that they all missed? Possibly. But highly unlikely.Far more likely that the miss was deliberately arranged.

It won't do to now say "But their all missing is not amazing at all. It's wholly unremarkable. After all, had they not all missed, I would not be here to ponder my luck!"
From the context, I suspect that Law is not really trying to argue against the weak anthropic principle itself, but arguing rather against naive conceptions which appear to undermine the Rundle's probabilistic argument. I think Rundle's argument fails for other reasons (which I might address later), but the arguments Law reproduces are good counterexamples for the weak anthropic principle and deserve rebuttal on their own right.

The Weak Anthropic Principle[1] states:
WAP: We observe X because if not-X, then we would not be here to observe it.
The weak anthropic principle is the strongest response to the Fine Tuning Argument for the existence of God, but, as Law notes, it can be discussed in more prosaic contexts.

It is dangerous and misleading for anyone, even a professional statistician (as I have been told by more than one professional statistician), to trust one's own superficial intuition about probability. We have apparently evolved and learned probabilistic intuitions that, while useful in the special circumstances of ordinary life, are wildly off-base in other contexts.

In order to understand any probabilistic argument, it is, in my opinion, always necessary to rigorously quantify the underlying numbers to be explicit about precisely what one asserts some probability. It is more important to be rigorous than realistic: a rigorous quantification over fictions or counter-factuals such as possible worlds or experiments not performed is preferable to no quantification at all. If the quantification is rigorous, I can at least apprehend the meaning of the probability; any metaphysical argument over the unreality of the quantification can at least be an argument about something well-defined.

In the Frequentist interpretation of probability[2], a probability is the ratio between the count of some subset of a population and the count of the entire population. A frequentist probability is usually estimated by physically computing the the ration between count of a subset of a sample and a count of the entire sample. For instance, if I want to estimate the probability that a white male 18-35 has watched episode CABF08 of the Simpsons, I can randomly select some white males 18-35 and ask them if they watched that episode. In this case,
  • The population is all the white males 18-35
  • The probability is the number of white males 18-38 who watched that episode divided by the total number of all white males 18-35
  • The sample is all the white males 18-35 who I actually asked
  • The estimate of the probability is the number of white males 18-35 who actually told me they watched that episode divided by the number of white males 18-35 in the sample
All of these items are strictly quantifiable: there is an rigorously definable number for each item. It's not always necessary to estimate the probability of a population by means of a sample: In many cases—such as the probability of various poker hands—the relevant numbers can be quantified directly from the population.

It's critically important to note that the population is defined by how you choose the sample. In this example, I might call people at random, ask them their race, sex and age, and accept them only if they say they are white, male, and 18-35; otherwise I will not include them in the sample. Since race, sex and age are criteria for choosing the sample, they define the population.

However, it is not just explicit criteria that define the sample, and thus the population; implicit and unconscious criteria also define the sample. For instance, if I were to ask some of my friends who are white males 18-35, then my population would be white males 18-35 who are friends of Larry. If I call people at random from the phone book, then my sample, and thus my population, are white males 18-35 who live in my city. Whenever the definition of sample—implicit or explicit—does not match the definition of the population, a statistical argument becomes vulnerable to a counterargument of selection bias. (As an exercise, examine the selection criteria of Testing Major Evolutionary Hypotheses about Religion with a Random Sample for sampling biases; I've detected two. If Wilson were actually computing statistics, would these biases affect his results?)

I say "vulnerable to" instead of "guilty of" because, of course, every sample has an inherent selection bias: One could reasonably say that the "population" is every white male 18-35 included in the sample. One must make the positive argument that the criterion included in the sample is uncorrelated to the statistic being measured. (I'll discuss in a further essay how to make such an argument.)

The weak anthropic principle, whether applied to the fine tuning argument or the more prosaic examples above is best understood not as a positive argument in itself but rather as a rebuttal to a statistical argument on the grounds of selection bias. Thus, to understand whether or not the weak anthropic principle undermines some argument, it is necessary to quantify the probabilities and determine whether survival or existence is biasing the sample. We then have a basis for evaluating the validity of the weak anthropic principle with regard to Law's examples.

In the first example
Here's a Swinburne type illustration. Suppose I am asked to guess each one of 52 cards, one by one. If I ever get one wrong, my brains will be blown out.

I start guessing, and amazingly, I get all 52 cards correct. Now you may say, "What's so improbably about that? After all, the probability of you getting them all right is 1, as you wouldn't be here otherwise would you?"
there are two quantifiable populations, about which we can discuss two distinct probabilities.

The first probability is the probability of surviving the exercise. In this case, there are several ways of quantifying the population: all the possible worlds in which the guesser guesses accurately or inaccurately, the population of arrangements of cards and guesses, the number of counter-factual statements about guesses, etc. They all give the same numbers; you pick whatever suits your metaphysical preferences. It's important to note that survival is not a criterion for defining the population: The population includes possible worlds/arrangements of cards/counter-factual statements in which the guesser dies. So we can meaningfully say that the probability of guessing 52 cards correctly is very low, without regard to the fact that the guesser dies if he guesses wrong, since we're including cases where the guesser guesses wrong (and dies) in the population.

However, we can ask a different question, with a a different population: Given this game exists, what is the probability that I would speak to (and, more importantly, receive an oral response from!) a guesser who has survived? Having survived the game is part of my selection criteria; my population is those people who have played the game and survived. I may be legitimately astonished that he survived the game, but, I'm not statistically entitled to be astonished that I'm talking to a survivor.

There is, of course, the observation that we shouldn't be talking to a survivor at all. But even this observation depends on a rigorous quantification: There have been only a countable number human beings ever alive, and the probability that anyone would have survived is very low, and we know this number independently of the details of the game. We can use a possible worlds or other abstract or fictional quantification to evaluate the probability of anyone surviving.

The weak anthropic principle does not itself argue for a chance rather than causal explanation for some low-probability event. It argues only that we're not entitled to compute some probabilities about some populations, because our sample is biased, and does not represent the population we're trying to draw conclusions about.

[1] The strong anthropic principle states, in contrast, that X is true because we are here to observe it, e.g. that human intelligence caused the physical universe to come into existence or to have its particular properties. This is not as science-fictional a principle as it might first appear: The strong anthropic principle has been offered as a serious explanation to measurement problem and the role of the observer in quantum mechanics.

[2] It's not my intention in this essay to deny the Bayesian interpretation of probability. Even the staunchest Bayesian admits that the frequentist interpretation has validity and meaning; she denies that only frequentist probabilities have meaning.

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