**A digression on the scientific method**

I'm sure you've all seen this in some form, but I like laying it out clearly.

Suppose I have two coins. One of them is a two-headed coin; the other one is a normal coin. I choose one of them at random and flip it; it comes up heads. What is the chance that it is the two-headed coin? Well, this is a pretty simple exercise in using Bayes' Theorem. The total probability of getting heads is 3/4, and the total probability of picking the two-headed coin and getting heads is 1/2 x 1 = 1/2. So, by Bayes' Theorem, the probability of it being the two-headed coin given that it came up heads is P(2-headed given heads) = P(2-headed and heads) / P(heads) = (1/2)/(3/4) = 2/3. No problem.

All right, let's move on. I have a coin, and I flip it. It comes up heads. I say, "I have a theory: that this happened because this is a normal coin." My friend comes into the room and says, "Oh yeah? I have a better theory. Your coin came up heads because it was a two-headed coin. And look! You just proved that my theory has a 2/3 chance of being correct." Well, yes and no. If we assume that,

*a priori*, the coin was equally likely to be a regular coin and a two-headed coin, then yes, it's true that the coin has a 2/3 chance of being two-headed. But we have a better knowledge of the

*a priori*distribution than that -- we know that normal coins are far, far more common than two-headed coins, so the probability of it being a two-headed coin is still very small, even given this one piece of information.

But notice the key feature of this: the experiment (such as it is) can't give us a full picture of which hypothesis is more likely without some

*a priori*assumptions about which hypothesis is more likely in the first place.

So, I'll bet you can see where this is headed. I have a coin, and I flip it. It comes up heads. I say, "I have a theory: this happened because this is a normal coin." My other friend walks in, and says, "Well, I have a theory that this happened because invisible angels manipulated the trajectory of the coin to make it come up heads." The scientific method is completely helpless to resolve this dispute; you have to use your own

*a priori*knowledge to adjudicate it. This is where we like to talk about what makes a theory "scientific", but all that really means is setting out an objective framework for deciding what's likely in the first place and what's not. It's a big mess.

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