Postprocessing of Genealogical Trees

Abstract

We consider inference for demographic models and parameters based upon postprocessing the output of an MCMC method that generates samples of genealogical trees (from the posterior distribution for a specific prior distribution of the genealogy). This approach has the advantage of taking account of the uncertainty in the inference for the tree when making inferences about the demographic model and can be computationally efficient in terms of reanalyzing data under a wide variety of models. We consider a (simulation-consistent) estimate of the likelihood for variable population size models, which uses importance sampling, and propose two new approximate likelihoods, one for migration models and one for continuous spatial models. . Often this approach is applied informally, with the qualitative features of the inferred tree being used to suggest plausible demographic histories for the sample (e.g., Shen et al. 2000).The second approach involves joint inference of the genealogical tree and the parameters. In many cases the genealogical tree is a nuisance parameter, and calculation of the likelihood for the parameters involves integrating out the unknown tree, for example, in inference about various demographic models under a coalescent prior, including variable population sizes (Griffiths and Tavaré 1994a;Kuhner et al. 1998;Drummond et al. 2005) and population structure (Bahlo and Griffiths 1998;Beerli and Felsenstein 1999), inference for selection (Coop and Griffiths 2004), dispersal of a population (Brooks et al. 2007), and inference for recombination rates (Griffiths and Marjoram 1996;Kuhner et al. 2000;Fearnhead and Donnelly 2002). (In the latter case the genealogical information is contained in a graph and not in a tree.)The advantage of the second approach is that, assuming the model for the genealogical tree is reasonable, the uncertainty in this genealogy is correctly incorporated into the inference about the parameters of interest. This is particularly important for data where there is considerable uncertainty in the genealogy (which is common for many data sets). The first approach of conditioning on a single estimate of the genealogy can sometimes lead to biases in estimates and, more generally, to underestimates of the uncertainty in the parameters. These problems often mean that analysis conditional on the tree is often used primarily to test hypotheses (Templeton et al. 1987;French et al. 2005), rather than for estimating parameters of appropriate models.However, implementing the second approach is considerably more challenging and generally requires the use of modern computationally intensive statistical methods (Stephens and Donnelly 2000). In particular, this often requires the development of customized programs to analyze the data under the specific model or models of interest, and the application of this approach can be limited by the availability of suitable software.In this article we consider a new approach, which lies between these two approaches. The basic idea is (i) to perform infer...

Likelihood-Based Inference in Isolation-by-Distance Models Using the Spatial Distribution of Low-Frequency Alleles

“…A further consideration is that, in practice, the extent of density regulation may vary across species. For example, Meligkotsidou and Fearnhead (2007) found the performance of their estimator improved in data derived from populations that have been expanding, and we would expect a similar pattern to hold for our estimator.…”

“…The observation of upwardly biased estimates of dispersal in a CIBD model is not unique to our study. Meligkotsidou and Fearnhead (2007) observe a similar bias, and both our methods use the same approximation; namely, the probability distribution for events in the genealogical process occurs independently of the locations of the sampled lineages. This approximation implicitly ignores population density regulation, which is a key feature of the lattice model.…”

“…Furthermore, we make the approximation that the intraallelic genealogy is independent of the geographical configuration of the lineages. This approximation is also used by Neigel et al (1991) and Meligkotsidou and Fearnhead (2007) and implicitly assumes weak population density regulation. For our purposes, we note that, even in density‐regulated populations, the approximation may be more accurate for low‐frequency alleles.…”

“…More recently, efforts have been made to derive maximum‐likelihood estimators of dispersal parameters in CIBD models (Rousset and Leblois 2007; Meligkotsidou and Fearnhead 2007). Maximum‐likelihood estimators have the advantage of making full use of the data but the potential disadvantage of being difficult to implement.…”

Inference of Homologous Recombination in Bacteria Using Whole-Genome Sequences

“…Such extensions would introduce new parameters that would appear in both the prior for the clonal genealogy (Equation 1) and the prior for recombinant edges (Equation 7) and would therefore make the decomposition into three inference steps problematic. An interesting alternative would be to leave inference as it is and investigate extensions of the model in a postprocessing step using importance sampling (Meligkotsidou and Fearnhead 2007).…”