A prevailing viewpoint in paleoclimate science is that a single paleoclimate record contains insufficient information to discriminate between typical competing explanatory models. Here we show that by using SMC² (sequential Monte Carlo squared) combined with novel Brownian bridge type proposals for the state trajectories, it is possible to estimate Bayes factors to sufficient accuracy to be able to select between competing models, even with relatively short time series. The results show that Monte Carlo methodology and computer power have now advanced to the point where a full Bayesian analysis for a wide class of conceptual climate models is now possible. The results also highlight a problem with estimating the chronology of the climate record prior to further statistical analysis, a practice which is common in paleoclimate science. Using two datasets based on the same record but with different estimated chronologies, results in conflicting conclusions about the importance of the astronomical forcing on the glacial cycle, and about the internal dynamics generating the glacial cycle, even though the difference between the two estimated chronologies is consistent with dating uncertainty. This highlights a need for chronology estimation and other inferential questions to be addressed in a joint statistical procedure.