An earthquakes occurs when the forces acting on a fault overcome its intrinsic strength and cause it to slip abruptly. Understanding more specifically why earthquakes occur at particular locations and times is complicated because in many cases we do not know what these forces actually are, or indeed what processes ultimately trigger slip. The goal of this study is to develop, test, and implement a Bayesian method of reliably determining tectonic stresses using the most abundant stress gauges available — earthquakes themselves.Existing algorithms produce reasonable estimates of the principal stress directions, but yield unreliable error bounds as a consequence of the generally weak constraint on stress imposed by any single earthquake, observational errors, and an unavoidable ambiguity between the fault normal and the slip vector.A statistical treatment of the problem can take into account observational errors, combine data from multiple earthquakes in a consistent manner, and provide realistic error bound...