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 1001 SNPs. It is simulated to have exactly 2 non-zero effects at 234, 287.

```
library(susieR)
library(curl)
data_file <- tempfile(fileext = ".RData")
data_url <- paste0("https://raw.githubusercontent.com/stephenslab/susieR/",
"master/inst/datafiles/FinemappingConvergence1k.RData")
curl_download(data_url,data_file)
load(data_file)
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))
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
# XtX is not symmetric; forcing XtX to be symmetric by replacing XtX with (XtX + t(XtX))/2
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] curl_4.3 Matrix_1.2-18 L0Learn_1.2.0 susieR_0.11.92
#
# loaded via a namespace (and not attached):
# [1] Rcpp_1.0.7 highr_0.8 plyr_1.8.5 compiler_3.6.2
# [5] pillar_1.6.2 tools_3.6.2 digest_0.6.23 evaluate_0.14
# [9] lifecycle_1.0.0 tibble_3.1.3 gtable_0.3.0 lattice_0.20-38
# [13] pkgconfig_2.0.3 rlang_0.4.11 DBI_1.1.0 parallel_3.6.2
# [17] yaml_2.2.0 xfun_0.11 stringr_1.4.0 dplyr_1.0.7
# [21] knitr_1.26 generics_0.0.2 vctrs_0.3.8 RcppZiggurat_0.1.5
# [25] Rfast_2.0.3 grid_3.6.2 tidyselect_1.1.1 reshape_0.8.8
# [29] glue_1.4.2 R6_2.4.1 fansi_0.4.0 rmarkdown_2.3
# [33] mixsqp_0.3-46 irlba_2.3.3 reshape2_1.4.3 ggplot2_3.3.5
# [37] purrr_0.3.4 magrittr_2.0.1 matrixStats_0.61.0 scales_1.1.0
# [41] htmltools_0.4.0 ellipsis_0.3.2 assertthat_0.2.1 colorspace_1.4-1
# [45] utf8_1.1.4 stringi_1.4.3 munsell_0.5.0 crayon_1.4.1
```