### Cutting Cake (with a Razor)

Suppose that a group of friends is trying to fairly divide a cake. By fair, I mean that you want each person to receive an equal portion. Let’s further suppose that this cake can be cut into arbitrarily small pieces while still retaining its cake-ness (so this is not a real cake, but a Real cake). If there are N people, then each gets 1/N of the cake.

But what happens when the group of friends is infinitely large? Let’s say that there is one person for each natural number – so we can talk about person 1, person 2, and so on. If we try to give each person 1/*k*th of the cake, then we’ll run out of cake after *k* slices, and the rest get nothing. So it’s not possible for everyone to get an equal, nonzero amount of cake. In other words, if we want everyone to get the same amount, then nobody gets anything.

But it *is* possible for everyone to get *some* cake. For example, we can give 1/2 of the cake to the 1st person, 1/4 to the 2nd person, 1/8th to the 3rd person, 1/16th to the 4th person, &c. So we can provide cake for infinitely many people – but only if it’s an *unfair* distribution.

Now suppose that instead of a set of people, we have a set of *hypotheses*, and instead of distributing cake, we are distributing *belief*. If we’re talking about a set of mutually-exclusive hypotheses, then we only have a fixed amount of belief to divide between them – let’s say we start with 1 unit of belief. If we don’t want to rule out any hypothesis outright, then each hypothesis must receive *some* belief, however small. So, for example, we could give a belief of 1/2 to the first hypothesis, 1/4 to the second, 1/8 to the third, and so on. Of course, this isn’t the only possible distribution.

The important thing here is that this is an infinite series which must sum to 1, and so the sequence must converge to zero, and *it doesn’t matter how you index the hypotheses*. No matter how you index the hypotheses, and no matter how you distribute your belief, there must exist some N such that you are 99% certain that the index of the correct hypothesis is less than N. Or 99.9% certain, or 99.99% certain. In particular, if you have some way of measuring *complexity*, and if there are only finitely-many hypotheses of a given complexity, then you can bound the complexity of the correct hypothesis with 99.99% (or arbitrarily high) confidence.

Now, this isn’t quite the same as Occam’s Razor. It doesn’t show that, given two hypotheses of differing complexity, the simpler one must be a-priori more probable. Also, this doesn’t show the expected complexity has to be bounded – it doesn’t. But it does show that as hypotheses become arbitrarily complex, your belief in them must become arbitrarily small.

Within the framework discussed above, there are many possible ways of distributing your belief between hypotheses. I’ll talk more about this later.