Intelligent Design: Frequentist, Bayesian, Physics, Biology
I had a good dinner conversation about intelligent design a couple of nights ago which inspired me to think about it in terms of frequentist (classical) versus Bayesian statistics and to contrast its application to physics with its application to biology. My notes follow.
(1) How do we do a classical (or frequentist) test here?
Null: Randomness, according to some distribution.
Alternative: Non-Random,
Test: Are the parameters in confidence region R?
(2) A simple example Let possible values of X be in {0,1,2,…1000}. Only X=1000 permits life.
Null: Randomness, X is U(0,1000).
Alternative: Non Random, X =1000 with probability 1.
Test: Reject if X is in R=[951,1000].
Probability(mistaken rejection) = 5%.
Probability(mistaken nonrejection if the Alternative is true) =0. (This test has very high– maximal– power.)
(3) Or, make Nonrandomness the Null.
Null: Nonrandomness, X =1000 with probability 1.
Alternative: Randomness, X is U(0,1000).
Test: Reject if X is in R=[0,999].
Probability(mistaken rejection) = 0%.
Probability(mistaken nonrejection if the Alternative is true) =1%. (This test has very high power too.)
(4) Now, condition on anyone being around to make the tests. The probability of mistaken rejection falls to 0 for both tests, and the probability of mistaken nonrejection rises to 1–minimal power.
This is a tricky argument, though. It has broad application. Here is one that might or might not be true in its premises, but which would work for other examples if not this one: Einstein’s theory of general relativity. If it is true, that would lead Einstein to come up with the theory, and suggest a test, and the test would verify it. If it is false, then Einstein would not come up with the theory, so the test would not be performed. So does the test tell us anything?
(5) Suppose we don’t do that existence-conditioning, though, and reject randomness (or accept Nonrandomness). For the significance level of the test, we don’t need to specify an alternative hypothesis. We just reject the Null and that’s that. But “it takes a theory to kill a theory”. We want to be able to Accept something, not just Reject something or fail to rejection something. We are talking about two particular theories, and we want to actually choose one. We need decision theory, which means we need Bayesian analysis.
Randomness is not really a theory, but it is very attractive as a null anyway, and should be, since it is simple. Now let the two possible theories be
Theory 1, Chance: Randomness, X is U(0,1000).
Theory 2, Intelligent Design: Nonrandomness, X = 1000, because there is a God and He wished to create a world.
Which theory is more probable, given the data we observe? That depends on our prior that Intelligent Design is true, which we will denote by P. Bayes Rule says:
Probability (Intelligent Designis true given X=1000) = Probability(X=1000 given Intelligent Design is true)* Probability (Intelligent Design is true)/Probability(X=1000)
Probability(Intelligent Design is true given X=1000) =
(1)*P/[(1*P + .001*(1-P)]= P/[.999P+.001]
P= .5 if .5 =P/[.999P+.001], true if .4985P+ .0005 = P, true if .0005=.5015P, true if P = 00025075.
Thus, if your prior, the probability there is a God and He wished to create a world, is greater than .025075 percent, then you would Accept Intelligent Design after seeing that X=1000.
(6) There is a flaw in that reasoning. I assumed that there were only two theories. For the case of the special values of constants in physics, there is a third theory:
Theory 3, New Law: Nonrandomness, X = 1000, because of an underlying law of physics that we just haven’t figured out yet, which would unify all the other laws.
For most scientists, I expect there is a higher prior on New Law than on Intelligent Design, which means it would have a higher posterior, and New Law is what they would believe in the end. We don’t prove Intelligent Design.
Note, however, that all this nonetheless is bad news for most people who oppose Intelligent Design, because most of them seem to hold to the Chance theory, which we have rejected. New Law has very different implications. Chance implies that we should shrug our shoulders at seeing the coincidences in the constants of physics. New Law implies that we should eagerly think about those coincidences, and there is a Nobel Prize to to be won.
People who reject the kind of reasoning I’ve presented are unlikely to win Nobel Prizes.
(7) Intelligent Design as applied to Evolution is different. There, the coincidences become coincidental mutations. What matters about that is that our prior on New Law falls drastically. From history, we know that anomalies in physics often are resolved by new laws; a single new law has often explained a puzzling observation. . Here we have just one more puzzling observation. In biology, though, everyone thinks of mutations as random, unconnected over time. We have not just one anomaly, like the constants being linked, but many anomalies, in the many creatures whose evolution is hard to explain. Thus, New Law isn’t as attractive if we reject Chance.