### A Simple Proof: Occam's Razor

How do you know that I'm not a robot? How do you know we're not living in the matrix?

The usual resolution is some form of Occam's razor: sure, it's possible that I'm a robot, but the simpler explanation is that I'm human, and simpler explanations are preferable.1

This just pushes the question back: why are simpler explanations better?

There is a straightforward proof that comes from Computer Science, of all places, which I hope to explain here.

Suppose I enter the world as a blank slate - I have a "bag" of hypotheses about how things work, and I consider them all equally probable. As I perform experiments, I disprove some of my hypotheses, while others remain. As time goes on, my bag of plausible hypotheses gets smaller and smaller.

If I eventually reach a point at which I only have two hypotheses remaining and I randomly choose one to believe, I'm 50% certain that I've got the right one. But if I randomly believe one out of a hundred possible hypotheses, I've almost certainly chosen wrong (i.e. I've probably selected a hypothesis that by luck happened to fit with all the observed data, even though it's in fact wrong).

Believe it or not, this concludes the proof.

If I have a simple hypothesis ("fire is hot") there's really only one other hypothesis that could be in my bag ("fire is not hot"), so I can rapidly determine which is the right one. If my hypothesis is complicated ("fire is hot, provided it's the first full moon of a year with zodiac symbol ...") there are tons of equally complex hypotheses, and some of them are bound to fit the data, so I'm unlikely to have chosen the right one.

In my job, I spend some time in the back rooms at medical offices, which means I hear nurses complain about doctors, and doctors complain about patients. One conversation I had with a dietition sticks into my memory: she was complaining about patients who expect the faddish, complicated dietary advice you hear on TV - "good" carbs, antioxidants etc. - but all she does is give people a calorie target, and recommend eating more fresh fruits and vegetables.

I told her to give her patients a brochure on Occam's razor. I doubt they've implemented my suggestion.

Postscript: This proof is a vague mishmash of the motivation for Bonferroni correction and VC theory. Any book on computational learning theory will have a better one, but you can see de Wolf's thesis for an explicit application of PAC learning to Occam's razor. You might also like my post why you will never see an eight-sided snowflake.

1. That's not true. The usual resolution is to ignore the problem.