In this vignette, we demonstrate a procedure that helps SuSiE get out of local optimum.

We simulate phenotype using UK Biobank genotypes from 50,000 individuals. There are 200 SNPs. It is simulated to have exactly 2 non-zero effects at 34, 87.

```
library(susieR)
data('FinemappingConvergence')
b = FinemappingConvergence$true_coef
susie_plot(FinemappingConvergence$z, y = "z", b=b)
```

The strongest marginal association is a non-effect SNP.

Since the sample size is large, we use sufficient statistics (\(X^\intercal X, X^\intercal y, y^\intercal y\) and sample size \(n\)) to fit susie model. It identifies 2 Credible Sets, one of them is false positive. This is because `susieR`

get stuck around a local minimum.

```
fitted <- with(FinemappingConvergence,
susie_suff_stat(XtX = XtX, Xty = Xty, yty = yty, n = n))
susie_plot(fitted, y="PIP", b=b, main=paste0("ELBO = ", round(susie_get_objective(fitted),2)))
```

Our refine procedure to get out of local optimum is

fit a susie model, \(s\) (suppose it has \(K\) CSs).

for CS in \(s\), set SNPs in CS to have prior weight 0, fit susie model –> we have K susie models: \(t_1, \cdots, t_K\).

for each \(k = 1, \cdots, K\), fit susie with initialization at \(t_k\) (\(\alpha, \mu, \mu^2\)) –> \(s_k\)

if \(\max_k \text{elbo}(s_k) > \text{elbo}(s)\), set \(s = s_{kmax}\) where \(kmax = \arg_k \max \text{elbo}(s_k)\) and go to step 2; if no, break.

We fit susie model with above procedure by setting `refine = TRUE`

.

```
fitted_refine <- with(FinemappingConvergence,
susie_suff_stat(XtX = XtX, Xty = Xty, yty = yty,
n = n, refine=TRUE))
susie_plot(fitted_refine, y="PIP", b=b, main=paste0("ELBO = ", round(susie_get_objective(fitted_refine),2)))
```

With the refine procedure, it identifies 2 CSs with the true signals, and the achieved evidence lower bound (ELBO) is higher.

Here are some details about the computing environment, including the versions of R, and the R packages, used to generate these results.

```
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.7
#
# Matrix products: default
# BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
#
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
#
# attached base packages:
# [1] stats graphics grDevices utils datasets methods base
#
# other attached packages:
# [1] ggplot2_3.3.0 microbenchmark_1.4-7 Matrix_1.2-18
# [4] L0Learn_1.2.0 susieR_0.11.42
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.5 pillar_1.4.3 compiler_3.6.2 plyr_1.8.5
# [5] highr_0.8 tools_3.6.2 digest_0.6.23 evaluate_0.14
# [9] lifecycle_0.1.0 tibble_2.1.3 gtable_0.3.0 lattice_0.20-38
# [13] pkgconfig_2.0.3 rlang_0.4.5 yaml_2.2.0 xfun_0.11
# [17] withr_2.1.2 stringr_1.4.0 dplyr_0.8.3 knitr_1.26
# [21] grid_3.6.2 tidyselect_0.2.5 reshape_0.8.8 glue_1.3.1
# [25] R6_2.4.1 rmarkdown_2.3 mixsqp_0.3-46 irlba_2.3.3
# [29] farver_2.0.1 reshape2_1.4.3 purrr_0.3.3 magrittr_1.5
# [33] scales_1.1.0 htmltools_0.4.0 assertthat_0.2.1 colorspace_1.4-1
# [37] stringi_1.4.3 munsell_0.5.0 crayon_1.3.4
```