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.1) Comprehensive Cancer Center of the University of Puerto Rico. 2) MERCK 3) NIH