When the discovery of the Higgs Boson was announced there was controversy over the use of classical Hypothesis Testing and the lack of Bayes in particle physics. Some reactions, with useful links, can be found here, here, here, and here. In this post, I’ll point out what particle physics has to lose by ignoring Bayesians.
One goal of particle physics is to determine the mass of particles such as neutrinos. A couple of centuries back, Laplace faced the similar problem of estimating the mass of Saturn from some error prone, earthbound measurements of the mutual perturbations of Jupiter and Saturn. A prior can be obtained, in the words of Jaynes, from the “common sense observation that can’t be so small that Saturn would lose its rings; or so large that Saturn would disrupt the solar system”. Jaynes goes on to describe Laplace’s success:
Laplace reported that, from the data available up to the end of the 18th Century, Bayes’ theorem estimates to be (1/3512) of the solar mass, and gives a probability of .99991, or odds of 11,000:1, that lies within 1 percent of that value. Another 150 years accumulation of data has raised the estimate 0.63 percent.
This stands in contrast to use of confidence intervals in physics, which are well known to posses nothing like the coverage properties Frequentists claim are objectively guaranteed.
But this raises an interesting point. Prior information like “ can’t be so small that Saturn would lose its rings” is very difficult for Frequentist to use. The problem, as explained here, is that this information in no away physically affects the errors in the earthbound observatories, or causes us to modify their sampling distribution or model. It only affects the prior distribution!
Moreover, this kind of prior information is available for neutrinos. A discussion can be found here, but a flavor is given by the quote:
If the total energy of all three types of neutrinos exceeded an average of 50 eV per neutrino, there would be so much mass in the universe that it would collapse.
So that’s what Bayes has to offer particle physics.
This also puts lie to the belief sampling distributions are hard facts while priors are subjective, if not outright meaningless. While I’ve never met anyone who’s measured the errors given by any device in a laboratory or observatory to “check” the assumed sampling distribution, and Laplace certainly didn’t do so, almost everyone’s seen images like the one below verifying the prior.
UPDATE: The charming Dr. Mayo reposted a discussion on the use of background information to assign prior probabilities here. The choice quote is
There is no reason to suppose that the background required in order sensibly to generate, interpret, and draw inferences about H should—or even can—enter through prior probabilities for H itself!
I love that “or even can” line. Yep, there is no reason to suppose. The good Dr. Mayo has searched high and low, but can’t find a reason anywhere.