Priors vs Likelihoods

or why Bayes is not to be feared

In #articles

I was reading the article Why I am not a Bayesian where Greg Mayer complains about the Bayesian approach to inference, and espouses Maximum Likelihood methods. There is much to critique in the article, but let's consider one question he raises.

There are three ways round the problem of prior distributions. First, try really hard to find an objective way of portraying ignorance. This hasn’t worked yet, but some people are still trying. Second, note that the prior probabilities make little difference to the posterior probabilty as more and more data accumulate (i.e. as more experiments/observations provide more likelihoods), viz.

P(posterior) ∝ P(prior) × Likelihood × Likelihood × Likelihood × . . .

In the end, only the likelihoods make a difference; but this is less a defense of Bayesianism than a surrender to likelihood. Third, boldly embrace subjectivity. But then, since everyone has their own prior, the only thing we can agree upon are the likelihoods. So, why not just use the likelihoods?

The problem with Bayesianism is that it asks the wrong question. It asks, ‘How should I modify my current beliefs in the light of the data?’, rather than ‘Which hypotheses are best supported by the data?’. Bayesianism tells me (and me alone) what to believe, while likelihood tells us (all of us) what the data say.

I follow his question of "why not just use the likelihoods" with the following question: Given that Paul the Octopus predicted 12 out of 14 World Cup matches, does Greg Mayer accept that the octopus is psychic? Why or why not?

My guess is "no", but why? The likelihood (i.e. how well the hypothesis supports the data) is exceptionally high. Higher, I'd say, than nearly any other hypothesis. Actually, now that I think about it, the hypothesis "aliens that like the number 12 and influence octopi made it happen" is a better one. These are clearly ridiculous because they are implausible to start, and thus have a lower prior probability. In fact Bayesianism is asking the only interesting question, and it provides the framework to actually compare hypotheses, something that frequentist methods fail spectacularly at.

Greg Mayer is a Bayesian, he just fails to admit it.