P(T|O) = P(T) * P(O|T) / P(O) Bayes' theorem gives us a way to evaluate a theory given a series of observations. A difficulty, however, is that the probabilities are difficult to gauge. What is P(T), P(O|T) and P(O)? What we can do is use a conservative computation, giving evolution favorable treatment at every turn. For instance, let's assume we start out with a very high probability that evolution is true. Next, consider the ratio P(O|T) / P(O). If an evolutionist is certain that observation, O, will not be observed, then the numerator should be quite low, say one in a million or one in a thousand. If P(O) is 0.5 then the ratio would be 0.000002 or 0.002, respectively. But to be conservative, and give evolution favorable treatment, let's set the ratio to 0.2, orders of magnitude greater than is reflected in the evolutionists expectations. For our 14 falsified predictions, using these extremely conservative values, Bayes' theorem tells us that evolution is a one-in-a-billion shot (0.000000000164 to be exact).This is quite correct --- if Cornelius Hunter can calculate the overall probability
Cornelius Hunter: Telltale Traces of a Non-Evolutionary Theory Sighted
by Joe Felsenstein, http://evolution.gs.washington.edu/felsenstein.html
Cornelius Hunter collects phenomena that he argues represent “failed evolutionary
predictions”. He also argues that evolutionary biologists
are making a “religious presupposition” when they insist
that science must use methodological naturalism. When asked what
supernatural methodology he would have us use instead, he
admits that methodological naturalism is “generally a good way to do science” [in his comment on that post of January 29, 2010 11:05 PM], and says that he's only complaining that we're being unclear about the matter.
But his bottom line is that he wants to test evolutionary predictions without his ever putting forth any alternative scientific theory for comparison. Now, however, there are signs that he may have such a theory. In a post at his blog he invokes Bayes' Theorem for an observation (O) and a theory (T):
P(O) of the observation. He states it as 0.5 in
his hypothetical case. But there is simply no way to calculate a value like
0.5 unless you have a non-evolutionary theory as well as the evolutionary
theory. If Hunter thinks that such a calculation can be made, he must have
his own theory. Let me explain why.
The denominator P(O) is simply the sum of the numerator terms
P(O|T)P(T) over all possible theories T . If there are just two theories T_1 and T_2 , the denominator is
P(O) = P(O|T_1)P(T_1) + P(O|T_2)P(T_2)
In short, it's the weighted average of the probabilities of the
observation O given the different theories, each weighted by the prior
probability of that theory.
If you want to do the calculation with only one theory, it's possible to
do that, but the result is boring. P(O) is then just one term P(O|T)P(T) ,
and the ratio P(O|T)/P(O) is just 1/P(T) , and since we only have one theory
its prior probability P(T) = 1 , and the ratio P(O|T)/P(O) turns out to
always be 1. Which is reasonable if there is no alternative theory, but
very uninteresting.
The lesson is simple: if you want to use Bayes' Rule to calculate the
posterior probability of evolution, you need at least one alternative
scientific theory. As a sophisticated fellow, surely Hunter must have one
in mind. The world awaits it with bated breath (his prior probabilities
for the theories would be interesting too). Or else the world awaits Hunter's
admission that he didn't know what he was talking about when he invoked
Bayes' Theorem.
77 Comments
Joe Felsenstein · 13 April 2010
Doc Bill · 13 April 2010
perpetualstudent · 13 April 2010
Creationist misuses statistics. Film at 11. In other news, dog bites man and sky is blue.
Rilke's Granddaughter · 13 April 2010
That exchange is remarkably funny. Essentially everything Hunter said was nonsense, but rather than admit that he was simply making up his numbers, he kept changing and reversing his opinions.
Is Hunter normally that stupid?
Mike Elzinga · 13 April 2010
Rilke's Granddaughter · 13 April 2010
Joshua Zelinsky · 13 April 2010
One other thing to note is that many of Luskin's "falsified predictions" are either non-predictions, not falsified or otherwise so distorted as to be meaningless. The fact that he can't do basic probability calculations is simply another problem.
Mike Elzinga · 13 April 2010
There is another issue with Bayes Theorem in that the various theories, Ti, are usually assumed to be disjoint. It gets really messy if they are not.
In the case of T1 = E, and T2 = D, one could propose that E and D are not disjoint. That might produce some interesting – and to some ID/creationists – objectionable theology; to say nothing of confounded science observations.
But the problem actually goes deeper in that we very often encounter confounding factors in our observations.
There are other Ti’s, not necessarily related to the competing theories in question, affecting our observations, that we may or may not know about. We would not only have to know what such factors do to our observations, but we would also have to know how such influences are dependant on which theory(s) is/are true.
And the real world of experimentation and observation of complex systems is, in fact, quite messy. That is why real researchers spend so much time cross-checking and doing experiments and observations in a way that uncovers systematic and confounding issues.
This seems to be where ID/creationists fall flat on their faces. If research gets too hard for them, they throw up their hands and declare that goddidit, and then start writing books and giving speeches.
cgauthier · 13 April 2010
I think this whole thing is an elaborate Freudian slip. The guy's obviously feeming. Anybody got a spare J(O|I)=N(T). I'd invite him over for a B+(O/N)G=(L|O)-A/D, but I think he'd be a buzzkill.
Dandu · 13 April 2010
James F · 13 April 2010
It's comforting to see that creationists are as bad at math as they are at geology and biology.
Alex H · 13 April 2010
Joshua Zelinsky · 13 April 2010
Tom English · 13 April 2010
I recommend Peter Olofsson's technically accurate and highly readable articles on ID to everyone here.
Joe Felsenstein · 14 April 2010
What interests me is that Hunter wants to test evolution (and reject it) without stating any alternative hypothesis. We might imagine that a supernatural "poof" is his actual alternative. But then he seems to disavow this by embracing Methodological Naturalism after much criticism of scientists who invoke it. Is he having it both ways with MN, getting praise at Uncommon Descent from an audience that thinks he is slamming MN, while he actually admits that it is really OK? Similarly for arguing that he can reject evolution, without being clear in favor of what. (And of course as has been pointed out here, evolution and design are not even mutually exclusive).
Olofsson's corrections make clear that he cannot do any of this using a Bayesian formula.
DavidK · 14 April 2010
Steve P. · 14 April 2010
Mike Elzinga · 14 April 2010
TomS · 14 April 2010
What a letdown! Here I was expecting something like what an alternative would be, or at least that someone was interested in starting off in that direction.
Douglas Theobald · 14 April 2010
Both Joe and Peter have made some very good points, as there are many problems with Hunter's Bayesian post. I would like to point out one more. Hunter's analysis is a classic example of selection bias. Hunter is trying to calculate the posterior probability of evolution theory, but he is only using observations that (supposedly) contradict the theory (his "14 falsified predictions"). Even if he were correct that these 14 observations are low probability given that evolution is true, there are many, many other predictions that evolution has made that have been confirmed. You have to include those also. Of course you will end up with a low posterior probability for a hypothesis if you only include low probability outliers in the analysis. This is analogous to trying to find the average height of women in Boston by only looking at the centers on college women's basketball teams.
DS · 14 April 2010
So the probability of a theory being correct depends on how many other theories there are? Funny, I though that the evidence was the way to decide? So if creationists ever do come up with any untested hypothesis, that will somehow reduce the probability that evolution is correct? Really? Then why do they refuse to do so?
Douglas Theobald · 14 April 2010
DS -- Yes, in a Bayesian analysis, the probability of a theory depends on the other theories.
This brings up another point. Others have pointed out that you can't test evolutionary theory in isolation using Bayes theorem, because you can't calculate the normalizing constant P(O) without knowledge of alternative theories (if there actually are no alternatives then P(O) is trivial and the posterior probability of evolution P(E|O) is 1 -- absolutely certain).
But let's assume for the purposes of argument that Hunter can actually know P(O) without analyzing alternatives (perhaps God told him the correct value of P(O)). And let's also assume, for the purposes of argument, that Hunter is correct that the posterior probability of evolution P(E|O) is something like one in a billion. Without alternative theories to compare against, this posterior is useless and tells us nothing. Perhaps the next best theory has a posterior probability of one in a trillion. If so, even though evolution has a low posterior probability, evolution is still 1000 times more probable than the best competitor!
This is just another reason why we must have alternative theories to compare against in order to make any meaningful or valid conclusions.
TomS · 14 April 2010
Sherlock Holmes often said that when one eliminates all other explanations, the one that remains, no matter how improbable, must be the truth.
Given that there are no other explanations for many of the prominent features of the world of life, some explanation involving descent with modification, no matter how improbable, is the most likely.
One can make a parody of the argument that evolution is so improbable by doing a calculation of the probability that an intelligent designer without limits would have done it:
The probability of an event happening is a ratio of the number of cases in which the event happens divided by the number of cases which are consistent with the given explanation. In the extreme case, God can do anything, so the probability that God would choose to do such-and-such is something-or-other divided by infinity. That is, the probability that God would create a flagellum on a bacterium is zero. In any case, "intelligent designers" can "design" more things than "naturalism" can, so the probability of such-and-such, assuming ID, is always less than the probability, assuming naturalism.
Douglas Theobald · 14 April 2010
TomS -- The probabilistic analysis you just gave for ID is not a parody, it is correct. You've pinpointed the very problem with creationism trying to be a scientific hypothesis. In the absence of any knowledge of the Creator's intent (and ID-ists vigorously deny that we can know the Designer's intentions), we either (1) cannot assign probabilities to observations given the design hypothesis, or (2) we have to assign all possibilities equal probability (invoking the Bayesian "principle of indifference" or "principle of insufficient reason"), which amounts to all observations having a vanishing probability. Either way, the creation hypothesis loses -- either it is undefined and hence untestable, or it has a vanishing probability of being true.
raven · 14 April 2010
william e emba · 14 April 2010
Christophe Thill · 14 April 2010
Hang on...
What's Bayes got to do with this ?
From the point of view of philosophy of science: the guy is implying that Bayes' theorem enables him to topple a theory if it can't explain a single, specific observation... ?
From the point of view of probability: he's saying that he can (or could) compute the probability of something that is either true of false (ie "the theory of evolution is right") and find something different than 0 or 1?
Frank J · 14 April 2010
Frank J · 14 April 2010
I meant "big tenter" of course. Must be a "darwinian" slip given how I equate anti-evolution activists with panhandlers. :-)
Mike Elzinga · 14 April 2010
nanobot74 · 14 April 2010
the funny thing is Hunter not only doesn't understand statistics, he doesn't even understand the basics of evolution, the very theory he is convinced is irredeemably flawed. He had another post (now scrubbed) about a recent study showing that the body size of many bird species was decreasing, perhaps due to global warming. He said that the only way for this to happen would be for all of these bird species to have the same body-shrinking mutation at the same time,and that the odds of this happening was astronomically low. facepalm.
see here: http://darwins-god.blogspot.com/2010/03/global-warming-effect-disproves.html#comments
TomS · 14 April 2010
I can remember when The Creation Hypothesis: scientific evidence for an intelligent designer (InterVarsity Press, 1994) came out, I hurried to buy a copy, anticipating that at least there would be something about such a hypothesis (although even then, I was not expecting any evidence for it). That sort of marks the date for me when I last had such anticipations.
Mike Elzinga · 14 April 2010
Mike Elzinga · 14 April 2010
ed · 14 April 2010
Swimmy · 14 April 2010
Douglas Theobald · 14 April 2010
The difficulty in calculating P(O), including the difficulties that Elzinga mentions, can all be easily avoided. Many Bayesians think it is impossible or hopeless to try to calculate P(O) in the absence of omniscience. But if you have two specific hypotheses to compare, you can easily calculate an odds ratio (assuming you can specify the likelihood functions and priors):
P(H1|O)/P(H2|O) = [P(O|H1) P(H1)] / [P(O|H2) P(H2)]
The P(O) in the numerator and denominator cancels out, so you don't need to calculate it. The odds of two hypotheses is the important thing anyway, as I mentioned earlier -- the absolute posterior probability of a hypothesis means basically nothing unless it can be compared to the probability of something else.
The problem for Hunter, of course, is that he refuses to explicitly state a competing hypothesis so that it can be compared to evolution, head-to-head.
Mike Elzinga · 14 April 2010
Science Avenger · 14 April 2010
hoary puccoon · 14 April 2010
If I recall my applied statistics from a few decades ago, Bayes theorem can also be used to calculate which of two theories is preferred by using the cost if you choose the wrong theory times the reciprocal of the probability of that theory.
In a simple example, deciding whether to carry your raincoat on a cloudy day would compare the cost of getting drenched if you don't take a raincoat and it rains versus the cost of lugging your raincoat around if you do take it and the sun comes out.
Using that application of Bayes theorem, a scientist would have to factor in the damage caused by focusing on supernaturalism if it's wrong (a huge penalty- it's pretty much a career stopper) versus the damage caused by focusing on methodological naturalism if it's wrong (no penalty-- evolutionary biology is in a golden age, and methodological naturalism is the only game in town.)
So, based on an application of Bayes theorem that one could actually calculate, the rational scientist should ignore Hunter EVEN IF HE'S RIGHT.
So why is he wasting people's time with this nonsense?
Henry J · 14 April 2010
Mike Elzinga · 14 April 2010
Tom English · 14 April 2010
It's rather humorous that Corny seems to be entirely ignorant of Dembski's Design by Elimination vs. Design by Comparison (Chapter 33 of The Design Revolution). Dembksi recognizes that assignment of a likelihood to a design hypothesis is impossible. What's knee-slapping hilarious is that Dembski attempts not merely to argue that Bayesian hypothesis testing is inappropriate when "design" is one of the hypotheses, but instead attempts to discredit the approach entirely.
And then along comes Corny....
Mike Elzinga · 14 April 2010
Jimmy · 14 April 2010
amyc · 15 April 2010
I'm only in an elementary statistics class right now, but even I could see that he was just pulling numbers out of thin air. It's sad when an undergrad who hasn't even completed the beginning course can recognize the holes in his statistical "hypothesis" (i don't know what else to call it).
Swimmy · 15 April 2010
Frank J · 15 April 2010
Jesse · 15 April 2010
TomS · 15 April 2010
It's been quite a while since I last looked at "The Creation Hypothesis", but I'll mention the contributors to the volume, which should give an idea of the flavor: forward by Phillip E. Johnson, J. P. Moreland, Stephen C. Meyer, William A. Dembski, Hugh Ross, Walter L. Bradley, Charles B. Thaxton, Kurt P. Wise, John W. Oller, Jr., John L. Omdahl, appendix by John Ankerberg and John Weldon.
And I can understand how people can be distracted from the question of "what happened and when" when one of the essays puts out such bait as describing the negative real line as:
-(aleph sub 0) ... -3 -2 -1 0
Yes, there is a minus aleph null there! (Trying to point out that it's impossible to traverse an infinite number of steps after minus aleph null.) That is something that sticks in my mind after 15 years.
In my more generous moods, I imagine that the anti-evolution crowd doesn't spell out "what happened and when" because if they did take any steps in that direction it would be sufficiently obvious to even the casual observer that they all they could describe was so absurd that it would make omphalism look attractive by comparison.
eric · 15 April 2010
Frank J · 15 April 2010
eric · 15 April 2010
Mike Elzinga · 15 April 2010
eric · 15 April 2010
Ken B · 15 April 2010
Frank J · 16 April 2010
Mike Elzinga · 16 April 2010
Frank J · 16 April 2010
Henry J · 16 April 2010
Mike Elzinga · 16 April 2010
Mike Elzinga · 16 April 2010
I should add that in that “Great Debate” series, Hugh Ross clearly shows his skepticism of Jason Lisle’s glib tossed-off arguments.
Frank J · 17 April 2010
Mike Elzinga · 17 April 2010
TomS · 17 April 2010
I know very little about rhetoric, but I have understood that one of the oldest tricks in the book is to accuse your opponent of doing something just before you set out to do it yourself. This is a way of immunizing your argument from the accusation. If someone makes the accusation, it looks like they're engaging in a "tu quoque" ("you're one, too").
Mike Elzinga · 17 April 2010
Jesse · 17 April 2010
I personally think that it is long past time that we poison the well against the DI, AIG and all their ilk. It's a dirty political tactic, but after all, this whole debate is about religion and politics. They could change that by doing legitimate research. But then they'd become evil evolutionists.
We know the DI is a nest full of dirty liars. I'm not sure about Ken Ham. Is he nutzo? Is he a liar? Is he both? I personally lean towards the nutzo hypothesis, but that may just be me.
Mike Elzinga · 18 April 2010
Stanton · 18 April 2010
John Kwok · 18 April 2010
Alex H · 18 April 2010
John Kwok · 18 April 2010
John Kwok · 18 April 2010
John Kwok · 18 April 2010
Frank J · 19 April 2010
Mike Elzinga · 19 April 2010
DS · 19 April 2010
Frank J wrote:
"Whatever he personally believes regarding the whats, whens, wheres and hows of the origin of Earth, life, species, etc., he seems at least honest in the sense that he truly believes that the “masses” need to believe the YEC story, and that the only effective way to do that is to preach it outright. Which means occasionally criticizing OEC and ID claims (specific statements regarding the age of Earth, etc.) as well as their approach (what to say, what to leave out, etc.)."
I recently went to a debate where a YEC quoted Behe using the irreducible complexity of the bacterial flagellum as an argument that disproved evolution. When asked if Behe accepted that the earth was young the reply was that Behe is an evilutionist! Very interesting that someone would use the irreducible complexity argument to claim that evolution was disproven, then admit that the guy who came up with the idea reached no such conclusion. This is a contradiction that should be exposed at every opportunity.
Of course this guy also claimed that the "Cambrian explosion" included vertebrates, so hopefully everyone realized that he was completely ignorant and lying through his teeth already. Or maybe not.