A hypothesis is nested within a more general hypothesis when it is a special case of the more general hypothesis. Composite hypotheses consist of more than one component, and in many cases different composite hypotheses can share some but not all of these components and hence are overlapping. In statistics, coherent measures of fit of nested and overlapping composite hypotheses are technically those measures that are consistent with the constraints of formal logic. For example, the probability of the nested special case must be less than or equal to the probability of the general model within which the special case is nested. Any statistic that assigns greater probability to the special case is said to be incoherent. An example of incoherence is shown in human evolution, for which the approximate Bayesian computation (ABC) method assigned a probability to a model of human evolution that was a thousand-fold larger than a more general model within which the first model was fully nested. Possible causes of this incoherence are identified, and corrections and restrictions are suggested to make ABC and similar methods coherent. Another coalescent-based method, nested clade phylogeographic analysis, is coherent and also allows the testing of individual components of composite hypotheses, another attribute lacking in ABC and other coalescent-simulation approaches. Incoherence is a highly undesirable property because it means that the inference is mathematically incorrect and formally illogical, and the published incoherent inferences on human evolution that favor the out-of-Africa replacement hypothesis have no statistical or logical validity.approximate Bayesian computation | coalescence | logic | nested clade analysis | statistics C oherence is an important statistical property when comparing nested or composite models (1, 2). Coherence means that the statistics or probabilities used to measure the goodness of fit of the models obey the constraints imposed by formal logic. For example, consider comparing two models, A and B, such that model A is fully nested within model B, as shown by the Venn diagram in Fig. 1. From elementary probability theory and Boolean logic, the probabilities of A and B must satisfy the constraint that Probability(A) ≤ Probability(B) because all observations that support model A also support model B, A being is a special case of model B; however, some observations can support model B but not model A. Any goodness-of-fit statistic or posterior probabilities on models A and B that are consistent with this logical constraint are coherent, and any statistics or posterior probabilities that can violate this logical constraint are incoherent. Another example involves partially overlapping models. Let {M 1 ,. . .,M n } be a set of models such that at least one pair, say M j and M k , is overlapping; that is, Probability(M j and M k ) > 0. Then, the probability of at least one of the M i s being true is less than the sum of the probabilities of each M i . This situation arises because the probabiliti...