The method of mixtures and the posterior mode plug-in

The method of mixtures and the plug-in method are two standard ways to build test supermartingales from e-variables. We ask when a mixture supermartingale has the same asymptotic growth rate as a plug-in supermartingale derived from the corresponding posterior. For finite families of e-variables, and for some infinite families, the growth rates of the mixture supermartingale and the posterior mode plug-in supermartingale both converge to the essential supremum of e-power, almost surely and in expectation. We investigate the behavior of the method of mixtures and the posterior mode plug-in method in several examples and then conclude by highlighting some open questions.