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Our aim is to give background and motivation for Scott's talk next week. Consider SNP association testing against a binary phenotype (disease vs. no disease). While linear regression enjoys very efficient inference, the simplest version is lacking due to:

• erroneous hard calls of variants (go with probabilities)
• multiple testing (go Bonferonni, FDR)
• confounding by ancestry, batch effects (go add PCs)
• cryptic relatedness (go full mixed model)
• binary phenotype (go logistic)
• overfitting (go Bayesian)
• nonlinear dependence of phenotype on covariates (go Gaussian process?)
• admixture (go topic model?)
• non-normal distribution of effect sizes (go GMM prior?)
• sparsity (go lasso?)
• epistatis (go neural net?)
• ascertainment bias (go do some research)
• high-dimensional phenotypes, both continuous and categorical (go do some modeling)

We will describe models addressing some of these points including Bayesian probit, logit, and mixed logit models, and time-permitting, some fancier models mixing continuous and discrete structure. Our emphasis will be on how exponential-family conjugacy makes inference easy via Gibbs sampling in certain cases, whereas its absence leads one toward despair (at least for six more days).