Chapter 2: Bayesian hierarchical models in geographical genetics

1. Chapter 2: Bayesian hierarchical models in geographical genetics Manda Sayler

2. Geographical genetics is the field of population genetics that focuses on describing the distribution of genetic variation within and among populations and understanding the processes that produce those patterns. • Statistical sampling uncertainty arises from the process of constructing allele frequency estimates from population samples. • Genetic sampling uncertainty arises from the underlying stochastic evolutionary process that gave rise to the population we sampled. • Note: increasing the sample size of alleles with each population reduces statistical uncertainty, but it cannot reduce the magnitude of genetic uncertainty. • Weir and Cockerham approach is the most widely used approach for analysis of genetic diversity in hierarchically structured populations. • Bayesian approach provides a model-based approach to inference that is enormously powerful and flexible. • Hierarchical Bayesian models provide a natural approach to inference in geographical genetics.

3. Weir and Cockerham Approach • To illustrate the formalism, consider a set of populations segregating for 2 alleles, A1 and A2 at a single locus • pk frequency of allele at A1 • Xij,k frequency of genotype AiAj in the kth population k=1,…,K • Variance • Fst can be interpreted as the fraction of genetic diversity due to differences in allele frequencies among populations . where and

4. Hierarchical Bayesian Models • A hierarchical Bayesian model uses the full power of the data for simultaneous estimators of the parameters while accounting for both statistical and genetic uncertainty. • To account for statistical uncertainty assume that alleles are sampled independently within populations. • Also assume the samples are drawn independently across loci and population. • Likelihood of the sample from a single population is binomial.

5. To account for genetic uncertainty we must assume a parametric form for the among-population allele frequency distribution. • It is natural to assume that population allele frequencies follow a Beta distribution, where E(pik) = πand Var(pik) = θπ(1 - π). • Thus, θ is equivalent to Fst. • The posterior distribution for the parameters is where P(πi) and P(θ) are the prior distributions for πi and θ, respectively.

6. A fully hierarchical model • To estimate the correlation of allele frequencies within loci, we need to add an additional level to the hierarchy that describes the distribution of mean allele frequencies across loci P(πi| π,θy). • Regard the loci in the sample as a sample from a larger universe of loci from which we might have sampled. • Regard the populations in our sample as a sample from a larger universe of populations from which we might have sampled. • The likelihood is unchanged. The posterior becomes where is the Beta distribution for θx, and is the Beta distribution for θy.

7. Developing an MCMC sampler • The process begins by picking an initial value for p, called p0, then p0 is updated until we have a large sample of values pt using either • Metropolis-Hastings algorithm (Figure 2.2) • Slice algorithm (Figure 2.3) • Estimate any property of the posterior to an arbitrary degree of accuracy. • Ensure that the MC has converged the values from an initial burn-in period are discarded. • Values retained from the following sample period represent the full posterior distribution and summary statistics are calculated directly from this sample. • Reduce the autocorrelation of values in the sample, it is sometimes useful to thin the sample.