Ockham's Razor: A Probabilistic Justification

Lately, I've been spending a lot of time with probability and statistics, partly because we're about to make some significant decisions about our boy's treatment, and partly because I've been reading The Drunkard's Walk: How Randomness Rules our Lives, which, despite its overdone name, is really about -- you guessed it -- probability and statistics.

Anyhow, I was lying in bed the other night, and for some reason I was thinking about Ockham's Razor, which despite Wikipedia's protests to the contrary, is usefully summarized as "the simpler explanation is more likely the correct one." So if you imagine an explanation that rests on 5 propositions, and an explanation that rests on 10 propositions, you should give preference to the simpler argument. Sometimes this is justified in terms of "falsifiability": the longer argument is "more easily falsifiable" than the shorter argument, because it has more places it can break.

But that night, I was thinking that a slightly more interesting way of justifying the principle is in terms of probability.

Suppose you know nothing about the content of the propositions, or the quality of the argument. If it helps, think about this as being a situation where the content of the propositions is so arcane, you have no idea how to evaluate whether or not they are true. But even this little bit of information -- the number of propositions -- can help you choose an explanation, if you suppose that each proposition has an equal chance of being correct. And what else can you do, since you don't understand the argument? If that is the case, then the chance of all propositions being correct is the combination of the chances of each proposition being correct. You don't know exactly how to combine the chances, because you don't know how the propositions are related, but what you do know is that every proposition you add decreases the chances of all the propositions being correct.

To illustrate, let's look at our 5- and 10-proposition arguments and plug in some numbers. If we suppose that each proposition has a 50% chance of being correct, independently of any of the other propositions, we get:

10-proposition argument
1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/1024

5-proposition argument
1/2 * 1/2 * 1/2 * 1/2 * 1/2 = 1/32

So there you go. Ockham's Razor is a good rule because every additional proposition decreases the chances of all the propositions being true.

Obviously, though, an actual analysis of the propositions is preferable to simply applying Ockham's Razor and taking that as proof. Right?

Well, probably. While this probability justification has the benefit of applying when we ourselves are too ignorant to analyze the argument, it also applies if we suppose that people in general are ignorant, and as likely to fail as to succeed when either composing or analyzing the truth of propositions. In that case, well, our best bet is just to guess ... and Ockham's Razor helps us make a better guess. I mean, unless I'm confused about the truth of that proposition. =)