(X) will be used in the estimation of ancestral-range marginal probabilities.
Figure 2.4. Accounting for phylogenetic uncertainty: the Bayes-DIVA approach. Trees 1–3 represent different phylogenetic relationships among nine species distributed in areas A–I. These trees were obtained by Bayesian inference, so each of them is associated with a posterior probability (PP) value, that is, the frequency with which the tree appears in the Bayesian posterior distribution. In all trees (1–3), the species present in areas A–C form a well-supported clade, node “X”; however, the remainder of the tree topology, including the identity of the sister-group and the definition of the “parent node Y”, are uncertain. In Bayes-DIVA, a DIVA analysis is run on each of these alternative trees. Because each tree is “weighted” by its posterior probability in the Bayesian sample, the end result is marginal probabilities of ancestral ranges, that is, DIVA reconstructions are marginalized or integrated over the uncertainty in the tree topology; this is represented in the central pie chart, where range A receives the highest marginal probability, followed by B and AB. Notice that only those trees containing node X are included in the computation of marginal probabilities. Tree 4, in which C, B and A do not form a clade, will not be included in the computation of marginal probabilities. For a color version of this figure, see www.iste.co.uk/guilbert/biogeography.zip
Nylander et al. (2008) demonstrated that accounting for phylogenetic uncertainty may also reduce biogeographic uncertainty: that is, for a given node, some ancestral ranges that were equally optimal in DIVA will be associated with higher marginal probabilities in the Bayes-DIVA analysis; in Figure 2.4, the ancestral range for node X that receives the highest marginal probability is A.
Harris and Xiang (2009) subsequently developed S-DIVA, implemented in the software RASP (Yu et al. 2010), to introduce “reconstruction uncertainty” in a Bayes-DIVA analysis. In a DIVA reconstruction, there could be several equally parsimonious ancestral ranges optimized at a given node (e.g. A, B or AB in Figure 2.4), as well as for each of these equally parsimonious ancestral ranges, there could be multiple different pathways (combinations of dispersal and cladogenetic events) by which a given ancestral range (e.g. A) is optimized along the tree. In Nylander et al. (2008)’s Bayes-DIVA approach, biogeographic uncertainty was defined as the first option: equally parsimonious ancestral ranges were assigned an equal weight (1/N), where N is the number of most parsimonious (MP) alternative ranges for a given node. In Harris and Xiang’s S-DIVA approach, biogeographic uncertainty is defined as the frequency with which a particular ancestral range appears within the pool of most parsimonious biogeographic pathways: F(r) = i/Rt, where i is the number of times a range (r) occurs in the total number of MP scenarios (Rt) over the tree.
Nylander et al.’s interpretation of ancestral range uncertainty as marginal probabilities is consistent with Bayesian probabilistic inference – nodal ancestral ranges are marginalized over a parameter (tree topology), whose values are sampled according to a probability distribution (Huelsenbeck et al. 2000). However, the probabilistic interpretation of F(r) in S-DIVA is not that clear. This has to do with the distinction between joint and marginal reconstructions. In a likelihood context, a joint reconstruction is the single best reconstruction across all nodes in a tree, and the marginal reconstructions are the single best reconstruction for each node considered along and after integrating for all possible reconstructions in the rest of the nodes. MP pathways in DIVA are estimated jointly over the tree, whereas ancestral ranges are estimated node-by-node. Equating the output of Bayes-DIVA to marginal probabilities is straightforward: the probability for ancestral range A on node X is approximately equal to the product of the probability that node X exists (PP value) and the probability of state A on node X – assuming that the MP solution is the ML solution (Huelsenbeck et al. 2000). However, the output frequency scenarios from S-DIVA cannot be equated directly to the joint probabilities obtained from a model-based method (Pagel et al. 2004). This is because discrimination between DIVA reconstructions is based on the MP score (the number of dispersal events): optimal nodal ranges are “equally parsimonious” if they carry the same cost in terms of explanatory dispersal events. That they appear in a different frequency among the biogeographic reconstruction pathways (e.g. 0.3, 0.7) does not imply that the first has lower joint probability than the second. A simile can be found in parsimony phylogenetics, given three equally most parsimonious trees (with the same number of steps), if two of them include clade X and one includes clade Y, this does not imply that clade X has some “probability” proportional to 2/3 and clade Y, a “probability” proportional to 1/3. Therefore, I argue that the original Bayes-DIVA method, which is also implemented in RASP, is more appropriate than Harris and Xiang’s (2009) extension to integrate phylogenetic uncertainty in EBMs: the reported values for ancestral ranges in Nylander et al.’s (2008) approach have a probabilistic interpretation within an empirical Bayesian context, which S-DIVA lacks.
2.4. A new revolution: parametric approaches in biogeography
The last decade has witnessed the introduction of parametric approaches in biogeography without the biases inherent in the parsimony framework (Ronquist and Sanmartín 2011). A common feature of these methods is the use of statistical probabilistic models, whose variables or parameters are quantifiable biogeographic processes that are dependent on time. Thus, in addition to the tree topology and tip distributions, parametric models incorporate a third source of information: branch lengths – measured in numbers or units of time – provide direct evidence on the rate or probability of geographic evolution. Longer branches imply a higher probability of change in the geographic range than shorter branches do. As more time elapses since the divergence of the species from its ancestor, there is more opportunity for biogeographic change (by dispersal, extinction or range expansion) along the branch. Branch lengths also inform on the certainty or degree of error in biogeographic inference: a species subtended by a long branch would be associated with a higher uncertainty about its ancestral range than one subtended by a short branch (Sanmartín 2020).
Two parametric models with stochastic variables that are time-dependent are typically used to model range evolution: the Brownian Motion (BM) model and the continuous-time Markov Chain (CTMC) model. BM is used to model the random dispersal of individuals within a population in a continuous landscape. We will return to this model later. CTMC models are typically used to describe range evolution at the species level. A Markov chain is a stochastic, memory-less process that models transitions between discrete states over continuous or discrete-time; the probability of each transition event depends only on the state attained in the previous event. The difference between a discrete-time MC (DTMC) and a continuous-time MC (CTMC) is that in DTMC, the chain moves (transitions or changes state) in discrete intervals (Δt), whereas in continuous-time, the moves are in infinitesimal amounts of time (dt) so that it can effectively measure instantaneous change. CTMC models are used in historical inference disciplines, such as phylogenetics or biogeography, where the states observed in the present are the result of a stochastic process that evolves over time. Stochastic means that, unlike in a deterministic process, the outcome of the process cannot be predicted with certainty, that is, we cannot predict the result of evolution. At the same time, evolutionary processes are not entirely random or unpredictable: they can be described by probability distributions with parameters, whose values we infer from the data.
Figure 2.5 provides an example of the parametric approach. In biogeography, the states of the stochastic CTMC process that governs range evolution are the set of discrete geographic areas that form the distribution range of a taxon (e.g. A, AB, B). The rates of transition or change between states in the CTMC process (e.g. A to B), within an infinitesimal amount of time (dt), are governed by the so-called instantaneous rate matrix Q, which has as parameters biogeographic processes that determine the probability of range evolution as a function of time, for example, dispersal, extinction, speciation. Given a phylogeny with time-calibrated branch lengths, tip distributions coded as discrete entities
