Today I talked to a doctor about the design of a randomized clinical trial that would use a Bayesian monitoring rule. The probability of response on each arm would be modeled as a binomial with a beta prior. Simple conjugate model. The historical response rate in this disease is only 5%, and so the doctor had chosen a beta(0.1, 1.9) prior so that the prior mean matched the historical response rate.
For beta distributions, the sum of the two parameters is called the effective sample size. There is a simple and natural explanation for why a beta(a, b) distribution is said to contain as much information as a+b data observations. By this criterion, the beta(0.1, 1.9) distribution is not very informative: it only has as much influence as two observations. However, viewed in another light, a beta(0.1, 1.9) distribution is highly informative.
This trial was designed to stop when the posterior probability is more than 0.999 that one treatment is more effective than the other. That’s an unusually high standard of evidence for stopping a trial — a cutoff of 0.99 or smaller would be much more common — and yet a trial could stop after only six patients. If X is the probability of response on one arm and Y is the probability of response on the other, after three failures on the first treatment and three successes on the other, Pr(Y > X) > 0.999.
The explanation for how the trial could stop so early is that the prior distribution is, in an odd sense, highly informative. The trial starts with a strong assumption that each treatment is ineffective. This assumption is somewhat justified by of experience, and yet a beta(0.1, 1.9) distribution doesn’t fully capture the investigator’s prior belief.
(Once at least one response has been observed, the beta(0.1, 1.9) prior becomes essentially uninformative. But until then, in this context, the prior is informative.)
A problem with a beta prior is that there is no way to specify the mean at 0.05 without also placing a large proportion of the probability mass below 0.05. The beta prior places too little probability on better outcomes that might reasonably happen. I imagine a more diffuse prior with mode 0.05 rather than mean 0.05 would better describe the prior beliefs regarding the treatments.
The beta prior is convenient because Bayes’ theorem takes a very simple form in this case: starting from a beta(a, b) prior and observing s successes and f failures, the posterior distribution is beta(a+s, b+f). But a prior less convenient algebraically could be more robust and better adept at representing prior information.
A more basic observation is that “informative” and “uninformative” depend on context. This is part of what motivated Jeffreys to look for prior distributions that were equally (un)informative under a set of transformations. But Jeffreys’ approach isn’t the final answer. As far as I know, there’s no universally acceptable resolution to this dilemma.