## Sunday, March 25, 2012

### How large a boost should the priors of simpler laws receive?

Simpler laws should have higher prior probabilities. Otherwise, the curve-fitting problem will kill scientific theorizing, since any set of data can be fitted with infinitely many curves. If, however, we give higher probabilities to simpler laws, then the curves describable have a hope of winning out, as they should.

So simpler formulae need to get a boost? How much of a boost? Sometimes a really big one.

Here's a case. Let's grant that Newton was justified in accepting that the force of gravitation was F=Gmm'/r2. But now consider the uncountably many force laws of the form F=Gmm'/ra, where a is a real number. Now in order for Newton to come to be justified on Bayesian grounds in thinking that the right value is a=2, he would have to have a non-zero prior probability for a=2. For the only way you're going to get out of a zero prior probability of a hypothesis would be if you had evidence with zero prior probability. And it doesn't seem that Newton did.

So Newton needed a positive prior for the hypothesis that a=2. But a finitely additive probability function can only assign a positive probability to countably many incompatible hypotheses. Thus, if Newton obeyed the probability axioms, he could only have positive priors for countably many values of a. Thus for the vast majority of the uncountably many possible positive values of a, Newton had to assign a zero probability.

Thus, Newton's prior for a=2 had to be infinitely greater than his prior for most of the other values of a. So the simplicity boost can be quite large.

Presumably, the way this is going to work is that Newton will have to have non-zero priors for all the "neat" values of a, like 2, 1, π, 1/13, etc. Maybe even for all the ones that can be described in finite terms. And then zero for all the "messy" values.

Moreover, Newton needs to assign a significantly larger prior to a=2 than to the disjunction of all the uncountably many other values of a in the narrow interval between 2−10−100 and 2+10−100. For every value in that interval generates exactly the same experimental predictions within the measurement precision available to Newton. So, all the other "neat" values in that narrow interval will need to receive much smaller priors, so much small that when they're all summed up, the sum will still be significantly smaller than the prior for a=2.

One interesting question here is what justifies such an assignment of priors. The theist at least can cite God's love of order, which makes "neater" laws much more likely.