Can We Avoid “SIN” in the House of “No Common Mechanism”?

In 'no common mechanism' (NCM) models of character evolution, each charactercan evolve on a phylogenetic tree under a partially or totally separate process(e.g. with its own branch lengths). In such cases, the usual conditions that suffice to establish the statisticalconsistency of tree reconstruction by methods such as maximum likelihood (ML)break down, suggesting that such methods may be prone to statisticalinconsistency (SIN). In this paper we ask whether we can avoid SIN for treetopology reconstruction when adopting such models, either by using ML or anyother method that could be devised. We prove that it is possible to avoid SINfor certain NCM models, but not for others, and the results depend delicatelyon the tree reconstruction method employed. We also describe the biologicalrelevance of some recent mathematical results for the more usual 'commonmechanism' setting. Our results are not intended to justify NCM, rather to setin place a framework within which such questions can be formally addressed.