This probability is the sum of two independent probabilities, corresponding to the two types of compatibility, namely if the characters were compabilble due to a disjoint mapping or if they were compatible due to a nested mapping. The model of randomness proposed is that all characters that are nesting equivalent are equally likely. There exists a modified version of the compatibility analysis algorithm that computes these probabilities. Although one must still assess whether such improbable cliques result from true phylogenetic history or from potentially other non-random effects, such as those recently discussed by Naylor in one of the more recent issues of Systematic Biology, the method is useful in assessing the relative importance of characters to particular phylogenetic hypotheses. The magnitude of the probabilities may not be as significant to phylogenetic reconstruction as the finding that characters may differ widely in what we can expect them to tell us about other characters.
Compatibility attempts to measure the probability that a character is random, viz.
Of course, others suggest that a low probability of being random suggests that the probability of the character being random is low but nothing more. This characterization of Meacham's probability model is not factually correct. The Meacham probability model measures the probability that two or more characters will be compatible by chance. This probability is the sum of two independent probabilities, corresponding to the two types of compatibility, namely if the characters were compabilble due to a disjoint mapping or if they were compatible due to a nested mapping. The model of randomness proposed is that all characters that are nesting equivalent are equally likely.