Bayesianness is next to Godliness

Is there evidence for God?

I watched a debate recently between William Lane Craig and Lawrence Krauss on the topic "Is there evidence for God?"  Dr. Craig argued the affirmative; Dr. Krauss the negative.  Since their names are monosyllabic, Germanic, and alliterative, it might be hard to keep them straight.  I will call them Affirmative and Negative.

Affirmative spoke first and offered the following definition of evidence: "To say that there is evidence for a hypothesis is just to say that the hypothesis is more probable given certain facts than it would have been without them."  He explained that if E is a set of facts and H is a hypothesis, then E is evidence for H if

P(H | E & B) > P(H | B),

where B represents background information independent of E.  To simplify I will make B implicit and write

P(H | E) > P(H).

In Bayesian terms, P(H) is the prior probability, that is, prior to taking E into account;  P(H | E) is the posterior probability.

This definition of evidence sounds reasonable to me, and I think it is uncontroversial.  But Negative chose to refute it: "I will get to the fact that [Affirmative's] claim for what evidence is is not at all what we use in science nowadays.  It doesn’t relate at all to what we use in science."

I think I have a pretty good idea what we use in science nowadays, and I have no idea what Negative is talking about.  He never explained.

In his rebuttal, Affirmative replied, "This is the standard definition of 'is evidence for,' used in probability theory.  And I’m astonished to hear [Negative] attacking logic in Bayesian probability theory as the basis for his argument."

So am I.  On this point I think Affirmative is basically right, but I do have one reservation.  People often assume, incorrectly, that E is evidence for H if E is consistent with H, but in fact, that's not enough.

Oliver's blood!

To see why not, let's look at an exercise from one of my favorite books, David MacKay's Information Theory, Inference, and Learning Algorithms:

MacKay's blood type problem
Two people have left traces of their own blood at the scene of a crime.  A suspect, Oliver, is tested and found to have type O blood.  The blood groups of the two traces are found to be of type O (a common type in the local population, having frequency 60%) and of type AB (a rare type, with frequency 1%).  Do these data (the blood types found at the scene) give evidence in favour of the proposition that Oliver was one of the two people whose blood was found at the scene?

Because the findings at the scene are consistent with the hypothesis that Oliver was at the scene, it is tempting to say that they are evidence for the hypothesis.  In fact, they are not.

Let's apply the test proposed by Affirmative.  I use G for "guilty" to represent the hypothesis that Oliver left blood at the scene.  I assume that the the prior probability P(G) is some unspecified value p; it will turn out not to matter.  Then according to Bayes's theorem the posterior probability is

P(G | E) = p * P(E | G) / P(E)

The term P(E | G) is the likelihood of the evidence if the hypothesis is true. This is easy to evaluate.  If Oliver left blood at the scene, that accounts for the type O trace.  So P(E | G) is the probability that the other sample, from an unknown person, is type AB, which is 1%.

The denominator, called the normalizing constant, is the probability of seeing E under any circumstances at all.  It is not always easy to formulate P(E).  But in this case we can, by defining NG, which is the hypothesis that Oliver did not leave blood at the scene.  Applying Bayes's theorem again, the posterior probability of NG is

P(NG | E) = (1-p) * P(E | NG) / P(E)

P(E | NG) is the likelihood of the evidence under the hypothesis that Oliver is not one of the people who left blood at the scene.  In that case, the blood was left by two unknown people.  If we choose two people at random, the chance of finding types O and AB is 2 * 60% * 1% = 1.2%.

Now we can get rid of P(E) by computing the ratio of the posterior probabilities:

P(G | E) / P(NG | E) = p / (1-p) * P(E | G) / P(E | NG)

In words, the posterior ratio is the product of the prior ratio and the likelihood ratio.  That means that the probability of G will increase if

P(E | G) / P(E | NG) > 1;

that is, if the evidence is more likely under G than under NG.  In this example the ratio is 1/1.2, which means that the posterior probability of G is slightly lower.  So the evidence is mildly exculpatory.

Surprised?  So was I.  To understand why, think about the evidence.  In the local population, AB blood is rare, so the AB sample is harder to explain.  If Oliver accounts for the type O sample, there is only one candidate to account for the AB sample.  If neither sample is accounted for, we get two bites at the apple.  And in this example, that makes the difference.

The moral of this story is that evidence consistent with your hypothesis does not necessarily support your hypothesis.  The evidence has to be more likely under your hypothesis than under the alternative.

Evidence for God?

In the previous section, I started with the definition that E is evidence for H if

P(H | E) > P(H),

and used Bayes's theorem to derive the equivalent requirement

P(E | H) > P(E | NH),

where NH is the complement of H.  I think this way of expressing the requirement is clearer and less prone to abuse.

To demonstrate, I will analyze one of the points of evidence proposed by Affirmative, the existence of contingent beings.  He explains, "By a contingent being I mean a being which exists but which might not have existed.  Examples: mountains, planets, galaxies, you and me.  Such things might not have existed."

He then offers a version of Aquinas's argument from First Cause.  You can listen to it here (between 2:30 and 4:30) and read about it here.  Based on this argument, he concludes "The most plausible explanation of the universe is God.  Hence the existence of contingent beings makes God's existence more probable than it would have been without them."

But I think he skipped a few steps.  If E is the existence of contingent beings and G is the hypothesis that God exists, we have to evaluate both P(E | G) and P(E | NG).

If Aquinas's First Cause is Yahweh as described in the Torah, then P(E | G) = 1, and it's hard to do better than that.  But one of the standard objections to Aquinas's argument is that it provides no reason to identify the First Cause as Yahweh.

Because we don't know anything about what the First Cause is (or even whether the concept makes sense), P(E | G) is just P(E); that is, the hypothesis has no bearing on the probability of the evidence.

How about under NG?  What is the probability of the universe if there is no god?

I have no idea.

And that brings me to...

The long-awaited point of this article

Bayes's theorem is a computational tool that is only useful if the right side of the equation is easier to evaluate than the left.  In the case of Oliver's Blood, it provides a classic divide-and-conquer algorithm.  But for the God Hypothesis, all it does is replace one imponderable with another.

So I applaud Dr. Craig (Affirmative) for breaking out Bayesian probability, but I don't think it helps his case.  As for Dr. Krauss (Negative), I am still waiting to hear "what we use in science nowadays."