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      2.3. From parsimony-based to semiparametric approaches

Event-based methods were based on explicit process models and therefore represented a considerable advance over cladistic methods in terms of statistical power. However, they still have several limitations. The cost of events, for example, cannot be estimated from the data but must be fixed a priori using ad hoc criteria such as the phylogenetic conservation of ancestral ranges. As explained above, through dispersal and extinction, the descendants come to occupy a different range than the ancestor, so these two processes are underestimated (“minimized”) in event-based reconstructions (Sanmartín 2012). Another constraint imposed by the use of parsimony is that biogeographic reconstructions are inferred on cladograms with no branch lengths or time information. This makes it difficult to discriminate between shared distribution history and biogeographic “pseudocongruence”, that is, shared patterns that are spatially but not temporally congruent (Page 1993; Donoghue and Moore 2003) and between competing dispersal and vicariance explanations: in vicariance, the appearance of the barrier within the ancestral range is the trigger for allopatric speciation, while in dispersal, the barrier predates geographic change (Chapter 1). Finally, parsimony-based inference ignores two sources of error associated with reconstructing trait evolution: “phylogenetic uncertainty” – ancestral areas are reconstructed over a single topology, the most parsimonious tree, assuming there is no uncertainty in phylogenetic relationships – and “reconstruction uncertainty”, where only the minimum-cost (most parsimonious) reconstructions are evaluated, even though alternative reconstructions could be almost as likely, or even more likely if additional information is considered (Ronquist 2004).

      Incorporating phylogenetic uncertainty in EBMs is relatively straightforward: run the analysis over a distribution of trees that represent the level of clade support in the phylogeny; this distribution can be obtained from bootstrap pseudoreplication or a Bayesian posterior probability distribution. Non-bifurcating nodes (polytomies) and nodes with low clade support can then be associated with low support for ancestral ranges. In Nylander et al.’s (2008) Bayes-DIVA method, DIVA parsimony-based reconstructions are averaged over a distribution of trees representing the posterior probability obtained from a Bayesian phylogenetic analysis. Figure 2.4 gives an example. Node “X” is a highly supported clade including three species distributed in areas C, B and A. Phylogenetic relationships in the rest of the tree are uncertain and vary over the Bayesian sample of trees, including the potential sister-group. For example, the stem or parent node of X (“Y” in Figure 2.4) has as the other descendant: “D” in tree 1, “E” in tree 2 and “F(ED)” in tree 3. Each of these tree topologies has a different posterior probability (PP) value. Because in Bayesian inference (BI), the frequency with which a tree is sampled in the analysis is proportional to its posterior probability (Ronquist 2004), the nodal ancestral area reconstructions in Bayes-DIVA can be interpreted as “marginal probabilities” (i.e. the different wedges in the pie chart in Figure 2.4). In other words, averaging DIVA reconstructions over a Bayesian sample of trees gives us estimates of ancestral ranges at nodes that are marginalized over the variation in the remainder tree topology. Notice that DIVA does not integrate branch length information, so the only parameter that is marginalized is the tree topology; in this sense, Bayes-DIVA can be considered an empirical Bayesian method (Nylander et al. 2008). It is also a semiparametric model since it contains a parametric (Bayesian phylogenetic inference) and a nonparametric (parsimony biogeographic inference) component. Another important distinction is given by tree 4: Pagel et al.’s (2004) definition of a “floating node”. Trees containing different definitions (bipartitions) of node X (tree 4, PP = 0.10) are excluded from the marginal estimations for that node in Bayes-DIVA: that is, the wedges in the pie chart sum to 0.90 (Figure 2.4). In other words, Bayes-DIVA uses a node-to-node approach in accounting for phylogenetic uncertainty: only those trees containing the node of interest (X) will be used in the estimation of ancestral-range marginal probabilities.