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.

Figure 1.—

ESS for analyzing data sets of size m = 10, 15, 20, 30, 40 simulated from the exponentially growing population size model with β = 0.7 and θ = 10, 20, 30.

Figure 2.—

The coalescent tree for a sample of m = 10 chromosomes from the constant population size model. The mutations are depicted by solid circles on the branches of the tree.

Figure 3.—

Histograms of the samples of the TMRCA for the coalescent tree analyzed under (a) the constant population size model, (b) the exponentially growing population size model, (c) the constant population size followed by exponential growth model, and (d) the bottleneck model. The true value of the TMRCA is indicated in each plot by a circle.

Figure 4.—

Contour plot of the mean log-likelihood surface of M₁₂, M₂₁ obtained from 100 simulated coalescent trees with sample size m = 10 under the migration model with D = 2 demes (each contour corresponds to 0.05 units of log-likelihood). The mutation rate used was θ = 30. Shown are the true parameter values, M₁₂ = 1.2 and M₂₁ = 2.8, and the values that maximize the surface, and .

Figure 5.—

Plots of the log-likelihood surface of σ for a range of parameter values, each obtained from 100 simulated data sets. Left-hand plot: θ = 15, β = 1, and m = 10 (blue); m = 20 (red); and m = 40 (green). Right-hand plot: m = 20, θ = 30, and β = 1 (black); θ = 30, β = 2 (blue); θ = 15, β = 1 (red); and θ = 15, β = 2 (green).

Figure 6.—

Probability–probability (PP) plots of a -distribution against the likelihood-ratio (LR) statistics for (red) our conditional likelihood method, (blue) analysis conditional on the maximum-likelihood estimate of the coalescence times, and (green) analysis conditional on the true coalescence times. a–c are based on 1000 data sets, with m = 2; β = 1; and (a) θ = 1, (b) θ = 2, and (c) θ = 4. d is based on 100 data sets with m = 5, θ = 2, and β = 1. We simulated all data sets conditional on there being no identical sequences in the data set.