## A Case for Robust Bayesian Priors with applications to Clinical Trials

##### Abstract

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. On the other hand, robust priors 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 deFinetti 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, after expressing in its exponentially family form, is improved by employing Cauchy priors. This requires no undue computational cost, given the advances in computation and analytical approximations. Moreover, we
also 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 hyper-parmeters that give closed-form results when coupled
with a normal log-odds likelihood. Berger's prior yields the superior robust
analysis with no added computational complication compared to the conjugate analysis. We illustrate the results with famous textbook examples
and with a real data set and a prior obtained from a previous trial. On the
formal side, we use a general and novel theorem, called 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. For Berger's prior robustness can be established
directly since the exact expressions for posterior moments are known.