Mathematical Models of the Genetic Architecture in Complex Human Disorders

Slides: https://bayesgroup.github.io/bmml_sem/2019/Frei_Genetics.pdf

Oleksandr Frei, Researcher at NORMENT (Norwegian Centre for Mental Disorders Research), University of Oslo, Norway

Modern studies on genetics of complex human disorders collect large samples, often exceeding N=10^6 individuals and M=10^7 genetic variants, posing challenging mathematical problems, such as solving a system of linear equations with huge NxM design matrix. In this presentation we will describe the Gaussian Mixture model (MiXeR [1], [2]) and three approaches for estimating its probability density function using (1) random sampling, (2) Fourier convolution, and (3) moment-preserving approximations. Further, we discuss our optimization protocol, based on direct maximization of the likelihood function using differential evolution and Nelder-Mead algorithms. Finally, we derive posterior estimates for some quantities of interest. If time allows we may also discuss related work [3] based on Mixed Linear Models, REML (Restricted Maximum Likelihood) and Variational iteration for Bayesian linear regression with Gaussian mixture prior.

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