Consider the problem of finding the dose that is as high as possible subject to having a controlled rate of toxicity. The problem is commonplace in oncology Phase I clinical trials. Such a dose is often called the maximum tolerated dose (MTD) since it represents a necessary trade-off between efficacy and toxicity. The continual reassessment method (CRM) is an improvement over traditional up-and-down schemes for estimating the MTD. It is based on a Bayesian approach and on the assumption that the dose-toxicity relationship follows a specific response curve, e.g., the logistic or power curve. The purpose of this paper is to illustrate how the assumption of a specific curve used in the CRM is not necessary and can actually hinder the efficient use of prior inputs. An alternative curve-free method in which the probabilities of toxicity are modeled directly as an unknown multidimensional parameter is presented. To that purpose, a product-of-beta prior (PBP) is introduced and shown to bring about logical improvements. Practical improvements are illustrated by simulation results.