2. Pedigree structure and gene identity by descent
The segregation of genes in pedigrees, including the case of
multiple linked loci, and a genomic continuum. The concept of gene
identity by descent which underlies the joint probabilities of
observable data on individuals. Probabilities of patterns of gene
identity by descent among relatives.
3. Recursive computation of gene identity, descent, and ancestry
Coefficients of kinship and inbreeding, and their generalizations to
multiple loci and/or multiple individuals. Recursive equations for
computing these coefficients, and implications for inference of
relationship, and inference of gene ancestry.
4. Paradigms for likelihood computations on pedigree structures
A likelihood on a pedigree as a latent variable problem. The classical
view of genotypes as latent variables. The modern view of meiosis
indicators (inheritance vectors) as the latent variables. The advantages
of each paradigm for different forms of likelihood computation on a
pedigree; the implementation of ``peeling'' algorithms under each paradigm.
5. Markov chain Monte Carlo (MCMC) on pedigree structures
The use of Monte Carlo to estimate expectations. The use of MCMC to
estimate conditional expectations. What can be realized, conditional
upon data, and how these realizations can be used. Estimation of
genotype probabilities, meiotic events, and genome sharing.
6. Monte Carlo EM and Monte Carlo likelihood
The framework for genetics models, and EM as a method for parameter
estimation in fitting models. The E-step, using MCMC. Likelihood
ratios as conditional expectations; ideas of Monte Carlo likelihood,
in the context of genetic data on pedigrees.
7. Monte Carlo likelihood for genetic models, and linkage analysis
Applications of Monte Carlo likelihood in genetic modeling and gene mapping, including multilocus linkage analyses, and mapping of
quantitative trait loci.
8. Monte Carlo likelihood for inference of relationships, from genomic data
Likelihoods for relationship inference, from discrete markers, and from genomic data. Monte Carlo likelihoods and Bayesian posterior in the
inference of relationship from genetic data.
9. Monte Carlo likelihoods for gene ancestry and fine-scale genetic
mapping
Coalescents, and ancestry of alleles. The realization of the recombination history of a disease haplotype, and implications for Monte Carlo likelihoods for fine-scale genetic mapping.
10. Improved MCMC procedures in genetic analysis
Problems of mixing and irreducibility in MCMC on pedigrees. Methods for ensuring irreducibility, and improving Monte Carlo efficiency. These include simulated tempering, regenerative samplers, and mixture likelihoods.