Sharing data is the foundation of open science, helping efficiency and reproducibility of research, and a fertile ground for a vibrant new data-driven economy. Yet, whatever the value created by sharing data, we cannot ignore the risks that it presents to privacy protection. The best known data protection measure, differential privacy, is so complex that neither data scientists, lawmakers, nor data subjects can appreciate the guarantees it gives! In contrast, probability is a commonly understood measure of risk. Can we use the probability of re-identification of a subject by an attacker to measure anonymity? We will answer this question by building Bayesian models for a range of standard statistical procedures and anonymization algorithms. We will then test the effectiveness of existing Bayesian inference algorithms for assessing the reidentification risk. If successful, we will be able to initiate building new foundations for data sharing in science