  We investigate the utility to computational Bayesian analyses of a particular
family of recursive marginal likelihood estimators characterized by the
(equivalent) algorithms known as "biased sampling" or "reverse logistic
regression" in the statistics literature and "the density of states" in
physics. Through a pair of numerical examples (including mixture modeling of
the well-known galaxy data set) we highlight the remarkable diversity of
sampling schemes amenable to such recursive normalization, as well as the
notable efficiency of the resulting pseudo-mixture distributions for gauging
prior sensitivity in the Bayesian model selection context. Our key theoretical
contributions are to introduce a novel heuristic ("thermodynamic integration
via importance sampling") for qualifying the role of the bridging sequence in
this procedure and to reveal various connections between these recursive
estimators and the nested sampling technique.
