• Applying Bayes’ Theorem

Before I begin I need to lay down a few ground rules. First: giving an exact probability for some of the things I’m going to would be extremely difficult, if not impossible. I am not going to try. That does not mean we cannot use Bayes’ Theorem to judge evolution, though.

We may not know the exact prior probability for the hypothesis “Evil Gnomes stole my socks.” That does not prevent us from saying that its prior probability is certainly less than one percent, whatever it is. We could take that figure of one percent and assign it to the evil gnome hypothesis anyway, and with the result that Bayes’ Theorem churns out, we could take it and say “the final probability of the evil gnome hypothesis is less than the result given here, simply because we know we have overestimated its prior probability.” Richard Carrier has talked about this in a recent lecture and in his book Proving History.

I will also be making use of something called “The Principle of Indifference” to help find a few probabilities. The principle of indifference states that all possibilities ought to be treated as equally likely when there isn’t any evidence at all indicating that one is more likely than the other. I see this as being obviously correct. Example: if you fill out a 500 question multiple choice test (with each question having four possible answers) about a subject you knew nothing at all about (so that you could only guess at the answers) around 25% of your answers would be correct. For the sake of completeness, I’ll address an objection to the principle of indifference at the end of this post.

Anyway, I will be using the principle of indifference in ways such as this: if evolution absolutely predicts some piece of evidence, then we will say that such evidence is 100% likely under evolution. How likely is that same evidence under the theory that evolution did not occur? Throughout this series, we will see that if evolution did not occur, then there really isn’t any reason at all to think the evidence in question should exist. It might exist if evolution was false, or it might not. So, I will treat it such evidence as 50% likely under the theory that evolution did not happen. As we will see, when I do this we can usually be confident that the probability of the evidence is not even predicted that well under “No Evolution” hypothesis, but I am going to grant the hypothesis that much predictive power just so that we are being as generous as possible to it, and therefore, when we come up with our final calculation, we can say that the probability of “No evolution” is at least such-and-such, as with the evil gnome example given above.

The Principle of Indifference: An Objection

WARNING: The following is a dry discussion of philosophy and math, read only if interested. There’s a certain kind of paradox which has led many philosophers to reject the principle of indifference, and it goes like this: suppose that there is a factory that makes tiles with sides between one and three inches. The tiles therefore have an area between one and nine inches. Using the principle of indifference, what is the probability that the tile side is two inches? One third. In the case of two-inch tile sides, the area will be four inches, so it follows that there is a one out of three chance that the tiles have an area of four inches.

What if you did that calculation a little differently, and started out by figuring out the probability of a tile having an area of four inches? In that case, if there are 9 possibilities (the tile can be between one and nine inches in area) then there is only a one out of nine chance that the tile will have an area of four inches. There’s a contradiction between the two estimates, it seems, so the principle of indifference is false, or so the argument goes.

I think this is deeply mistaken. In the first example, each inch length (one inch, two inches, three inches) is treated as a single possibility, and there is no consideration of the fractions that lie between those lengths (one and a half inches, one and three quarters inches, and so on). If we adjust our thought experiment so that the length will always be an exact inch length and not a fraction more, then there will only be three possible area lengths; that is, if the sides of the tiles can only be exactly one, two or three inches and not “one and three quarters inches” or “two and sixth-sevenths inches” then it follows that there can only be three possible area lengths: one, four, and nine, in which case the probability estimate of a four inch area would remain one out of three. If we adjust our thought experiment so that the length can include a fraction, then we must consider whether the fraction can be infinitely small or not. We cannot use finite math to deal with infinities, and so the principle of indifference not being able to render a probability judgement in this case is no failing on its part. If, however, we allow fractions of one-half inch for the side of the tiles, then that means there will be six possibilities for the side of the tile and six possibilities for the area, and applying the principle of indifference to the possible sides and possible areas yield consistent answers. The same can be said about any fraction length you specify, as long as you restrict its variation to something finite, like tenths of inches or hundredths of inches, and not allow the length to vary infinitely.

I’ve found that the same thing applies to just about every objection people launch at the principle of indifference; it allows boils down to a mistake and usually of the kind that I mentioned.

Category: Bayes Theorem ### Article by: Nicholas Covington

I am an armchair philosopher with interests in Ethics, Epistemology (that's philosophy of knowledge), Philosophy of Religion, Politics and what I call "Optimal Lifestyle Habits."