Revisiting the Paleomagnetic Reversal Test: A Bayesian Hypothesis Testing Framework for a Common Mean Direction

For over 60 years a suite of field tests has provided paleomagnetists with a means to evaluate the timing of magnetic remanence acquisition and subsequent remanence stability. These tests are crucial for detecting potential overprinting by secondary remanences and for assessing the fidelity of the geological information carried by paleomagnetic signals. The reversal test was developed to detect secondary remanent magnetizations that could potentially bias paleomagnetic reconstructions. More recently, the reversal test has been applied to a broader set of problems, which require statistical assessment of common or antipodal paleomagnetic directions. From a statistical standpoint, the reversal test must distinguish whether two sets of paleomagnetic directions originate from populations with a common mean. However, earlier work has demonstrated that the reversal test may be ambiguous for small numbers of observations because insufficient information is available to reject the null hypothesis of a common mean direction. Here we develop a Bayesian framework to estimate directly the probability that two Fisher-distributed sets of directions originate from populations with a common mean. This framework can be used to consider data sets with common or different precisions and, thus, provides a fully probabilistic version of the parametric reversal test. Additionally, adoption of a Bayesian framework means that ambiguity associated with the lack of information provided by small numbers of observations is incorporated into the final probability estimate in a natural way. Our new Bayesian test is demonstrated with numerical examples and case studies.