In this work, we consider the problem of parameter estimation in a Bayesian setting. We propose a new approach to lower-bounding the Bayesian risk, based on Sibson's alpha-Mutual Information. The results are then applied to specific settings of interest. As an example, we provide a lower-bound on the risk of the so-called "Hide-and-Seek" problem. Generalisations of the results and alternative directions are also briefly presented.