Inverse problems in evolutionary biology

In this thesis, we explore three techniques which could be used to increase the efficiency of analyses in evolutionary genetics while still producing reasonably accurate results. The first of these methods improves the efficiency of analyses based on Markov chain Monte Carlo (MCMC) through the application of delayed acceptance sampling, an MCMC method with an additional proposal step in which an acceptance probability is computed from computationally less expensive approximate likelihoods. Rejection at the additional decision step should allow software like SNAPP (``SNP and AFLP Phylogenies") to avoid unnecessary computation of full likelihoods and, therefore, run more efficiently. The second method we discuss combines dynamic programming with classical numerical integration methods to compute likelihoods with respect to continuous trait models on trees. This method assumes explicitly known transition densities, but is efficient and has a relatively fast convergence rate. We apply the method to a threshold model which combines continuous traits with discrete observations. The third method we look at is another dynamic programming integration algorithm, except that this algorithm takes advantage of a basis function approximation of likelihood functions. This method allows for numerical solutions to PDEs to be applied directly and the use of Chebyshev polynomials as the basis functions make the method easy to implement. We apply this method to the computation of the likelihood given a genetic data set generated by diffusion processes