- Feb 2019
g likelihood or Bayesian probabilistic phylogene
If you have a molecular data partition, you can just use total evidence approach and the standard 1-parameter Markov model.
Potential synapomorphies will be compatible with the molecular tree and considered not likely to change. Potential homoiologies and symplesiomorphies are partly ("semi-")compatible with the molecular tree and, hence, considered less likely to change than highly homoplastic traits with (random) convergence.
Just try out a couple of datasets, and infer the (Bio)NJ and ML trees and then compare the result with the strict consensus network (not tree) of all equally parsimonious trees and the Bayesian tree sample.
Note that if you apply TNT's iterative character weighting procedure, what you effectively do is sorting the random convergences from parallelisms/ characters that are more compatible with the preferred tree.
In principle, I do sympathize with the general idea, but the laid out approach will have little use.
The main drawback is that you can only define homoiologies using an external data set (e.g. the molecular "gold" tree). But when you have a reliable molecular tree, you can just go for total evidence approaches to select a more likely, in a mathematical and general sense, alternative without the need to make any prior destinction between your characters. Homoiologies will be inferred, like synapmorphies or symplesiomorphies or shared apomorphies (non-stochastically distributed convergences) on the fly.
If you define the homoiologies on a inferred (e.g. parsimony) tree only based on a morphological data matrix (e.g. for an extinct group of organisms), you will inevitably misinterpret some characters, because your clades are not necessarily monophyletic. Homoiologies like symplesiomorphies may appear as (pseudo-)synapomorphies.
The only application left would be that the molecular tree cannot resolve certain relationships, and we use more tree-compatible morphological characters to discern between alternatives. However, the first choice would then be to maximise the number of synapomorphies. Only if that would be the same for all alternatives, one could count the number of symplesiomorphies and homoiologies (as the distinction between both via a tree-inference is very tricky; and their are often just two side of the same evolutionary process).
However, one could also just directly change to a network-analysis framework, which will pretty much solve all these problems at once.
For further details see my (upcoming, March 4th) post at Genealogical World of Phylogenetic Networks