Bayesian analysis is frequently confused with conjugate Bayesian analysis. This is particularly the case in the analysis of clinical trial data. Even though conjugate analysis is perceived to be simpler computationally (but see below, Berger's prior), the price to be paid is high: such analysis is not robust with respect to the prior, i.e. changing the prior may affect the conclusions without bound. Furthermore, conjugate Bayesian analysis is blind with respect to the potential conflict between the prior and the data. Robust priors, however, have bounded influence. The prior is discounted automatically when there are conflicts between prior information and data. In other words, conjugate priors may lead to a dogmatic analysis while robust priors promote self-criticism since prior and sample information are not on equal footing. The original proposal of robust priors was made by de-Finetti in the 1960's. However, the practice has not taken hold in important areas where the Bayesian approach is making definite advances such as in clinical trials where conjugate priors are ubiquitous.
We show here how the Bayesian analysis for simple binary binomial data, expressed in its exponential family form, is improved by employing Cauchy priors. This requires no undue computational cost, given the advances in computation and analytical approximations. Moreover, we introduce in the analysis of clinical trials a robust prior originally developed by J.O. Berger that we call Berger's prior. We implement specific choices of prior hyperparameters that give closed-form results when coupled with a normal log-odds likelihood. Berger's prior yields a robust analysis with no added computational complication compared to the conjugate analysis. We illustrate the results with famous textbook examples and also with a real data set and a prior obtained from a previous trial. On the formal side, we present a general and novel theorem, the "Polynomial Tails Comparison Theorem." This theorem establishes the analytical behavior of any likelihood function with tails bounded by a polynomial when used with priors with polynomial tails, such as Cauchy or Student's $t$. The advantages of the theorem are that the likelihood does not have to be a location family nor exponential family distribution and that the conditions are easily verifiable. The binomial likelihood can be handled as a direct corollary of the result. Next, we proceed to prove a striking result: the intrinsic prior to test a normal mean, obtained as an objective prior for hypothesis testing, is a limit of Berger's robust prior. This result is useful for assessments and for MCMC computations. We then generalize the theorem to prove that Berger's prior and intrinsic priors are robust with normal likelihoods. Finally, we apply the results to a large clinical trial that took place in Venezuela, using prior information based on a previous clinical trial conducted in Finland.
Our main conclusion is that introducing the existing prior information in the form of a robust prior is more justifiable simultaneously for federal agencies, researchers, and other constituents because the prior information is coherently discarded when in conflict with the sample information.
"A case for robust Bayesian priors with applications to clinical trials." Bayesian Anal. 4 (4) 817 - 846, December 2009. https://doi.org/10.1214/09-BA431