comp565_fall2023_A1

1. (1%) Implement the efficient Bayes factor for each causal configurations: y = Xλ + ϵ, ϵ ∼ N (0 , σ 2 I n ) , λ ∼ N (0 , s 2 λ σ 2 ∆ γ ) where s 2 λ is user-defined prior variance in the unit of σ 2 , ∆ γ is the diagonal matrix with di- agonal equal to γ (causal configuration). You may assume that s 2 λ = 0 . 005 . Therefore, assuming there are k causal SNPs, then Σ CC = Ns 2 I k = 2 . 49 I k For the multivariate Gaussian density function, we may find many existing libraries. In R, mvtnorm . In Python, it can be found in scipy https://docs.scipy.org/doc/scipy-0.14. 0/reference/generated/scipy.stats.multivariate_normal.html 2. (1%) Implement the prior calculation for each configurations 3. (2%) Implement posterior inference over all possible configurations assuming at maxi- mum 3 causal SNPs (i.e., ( 100 1 ) + ( 100 2 ) + ( 100 3 ) = 100 + 4950 + 161700 = 166 , 750 possible configurations). Therefore, no stochastic sampling is required here (as opposed to the original FINEMAP). To obtain all possible configurations, we may also use existing libraries. In R, this is done by gtools:combinations . In Python, check out itertools.combinations https://docs. python.org/2/library/itertools.html Some configurations may result in non-finite multivariate Gaussian density. Discard those configurations. Visualize the configuration posteriors by ranking them in increasing order as shown in Figure 2. As we can see, the vast majority of the configurations have very small posterior probabilities. 4. (2%) Implement posterior inclusion probabilities (PIP) to calculate SNP-level posterior probabilities. Visualize the normalized inferred PIP aligned with GWAS marginal -log10 p-values in Fig- ure 3. It looks like we missed one of the 3 causal SNPs due to its nearly perfect LD with the other causal SNPs. But in general, we are able to pull down quite a few non-causal ones. That is, if we were going to experimentally validate the top SNPs, 2 out of 6 top SNPs based on PIP are true causal ones, whereas we would have got a lot more false positives if we were to follow the -log10 P-values instead. Similar to Question 1, submit your code in COMP565 A1 finemap.ipynb or COMP565 A1 finemap.Rmd via myCourses. Your code should generate the plots illustrated in Figure 2 and 3. There may be some difference due to the numerical implementations of various MVN libraries but it should not differ too much from the provided PIP values in SNP pip.csv.gz . This way the TA can run your notebook to validate its output. Do not submit the data provided to you as long as you 3