Empircal Bayes Normal Means with Bimodal Priors
jhmarcus
2019-05-05
Last updated: 2019-05-06
Checks: 5 1
Knit directory: drift-workflow/analysis/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | 39cad41 | jhmarcus | 2019-05-05 | added links to ebnm bimodal code |
| html | 39cad41 | jhmarcus | 2019-05-05 | added links to ebnm bimodal code |
| Rmd | 44a10f7 | jhmarcus | 2019-05-05 | added bimodal ash exploration |
| html | 44a10f7 | jhmarcus | 2019-05-05 | added bimodal ash exploration |
Here I explore the idea of “hacking” ashr to solve the Empirical Bayes Normal Means problem with a bimodal prior, specifically with the modes of the prior at 0 and 1. The idea is we’d like to “penalize” against estimating intermediate effects i.e. we shrink the effects to 1 if there large enough and 0 if their small enough, accounting for the precision of the estimate and learning the right level to shrink.
Imports
library(ggplot2)
library(dplyr)
library(tidyr)Functions
Here are some helper function for simulation, fitting, and plotting.
sim = function(n0, n1, sigma_e){
n = n0+n1
beta = c(rep(0, n0), rep(1, n1))
s = abs(rnorm(n, 0, sigma_e))
betahat = rnorm(n, beta, s)
return(list(betahat=betahat, s=s, beta=beta, n=n))
}
fit = function(betahat, s, beta, m=20){
b = seq(1.0, 0.0, length=m)
a = seq(0.0, 1.0, length=m)
bimodal_g = ashr:::unimix(rep(0, 2*m), c(rep(0, m),b), c(a, rep(1,m)))
ash_res = ashr::ash(betahat, s, g=bimodal_g, fixg=FALSE, outputlevel=4)
betapm = ash_res$flash_data$postmean
df = data.frame(betahat=betahat, beta=beta, betapm=betapm, s=s, idx=1:length(betahat))
return(df)
}
plot_sim = function(df, title){
gath_df = df %>% gather(variable, value, -idx, -s)
p0 = ggplot(gath_df, aes(x=idx, y=value,
color=factor(variable, levels=c("beta", "betahat", "betapm")))) +
geom_point() +
theme_bw() +
labs(color="") +
xlab("Variable") +
ylab("Value") +
theme(legend.position="bottom")
min_betahat = min(df$betahat)
max_betahat = max(df$betahat)
p1 = ggplot(df, aes(betahat, betapm, color=s)) +
geom_point() + viridis::scale_color_viridis() +
theme_bw() +
theme(legend.position="bottom") +
xlim(c(min_betahat, max_betahat)) +
ylim(c(min_betahat, max_betahat)) +
geom_abline()
p = cowplot::plot_grid(p0, p1, nrow=1)
title = cowplot::ggdraw() + cowplot::draw_label(title)
print(cowplot::plot_grid(title, p, ncol=1, rel_heights=c(0.1, 1)))
}Simulations
I simulated a bunch of normal means scenarios where the true \(\beta\)s are set to 0 or 1. In each simulation I specify the number of zeros n0 the number of ones n1 and standard deviation used to simulate std. errors.
n0 = c(rep(40, 3), rep(25, 3), rep(10, 3), rep(0, 3))
n1 = c(rep(40, 3), rep(55, 3), rep(70, 3), rep(80, 3))
sigma_e = rep(c(.05, .1, .25), 4)
for(i in 1:length(n0)){
sim_res = sim(n0[i], n1[i], sigma_e[i])
betahat = sim_res$betahat
s = sim_res$s
beta = sim_res$beta
df = fit(betahat, s, beta, m=20)
title = paste0("n0=",n0[i], ",n1=", n1[i], ",sigma_e=", sigma_e[i])
plot_sim(df, title)
}
I think the idea roughly works! The most interesting scenarios to compare are when the std. errors of the estmates are high but the number of zeros and ones are different. Maybe we can define a term “bimodality” which I’m thinking is how bimodal the distribution is. When the bimodality is low (i.e. the prior distribution is closer to unimodal) the effects seem to be more correctly estimated. As we can see estimating more bimodal effects is a more difficult problem than unimodal effects.
It would also be interesting to think more about how to weight the prior mixture proportions if that would be helpful.
sessionInfo()R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin13.4.0 (64-bit)
Running under: macOS 10.14.2
Matrix products: default
BLAS/LAPACK: /Users/jhmarcus/miniconda3/lib/R/lib/libRblas.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] tidyr_0.8.2 dplyr_0.8.0.1 ggplot2_3.1.0
loaded via a namespace (and not attached):
[1] Rcpp_1.0.0 compiler_3.5.1 pillar_1.3.1
[4] git2r_0.23.0 plyr_1.8.4 workflowr_1.2.0
[7] viridis_0.5.1 iterators_1.0.10 tools_3.5.1
[10] digest_0.6.18 viridisLite_0.3.0 lattice_0.20-38
[13] evaluate_0.12 tibble_2.0.1 gtable_0.2.0
[16] pkgconfig_2.0.2 rlang_0.3.1 foreach_1.4.4
[19] Matrix_1.2-15 parallel_3.5.1 yaml_2.2.0
[22] xfun_0.4 gridExtra_2.3 withr_2.1.2
[25] stringr_1.4.0 knitr_1.21 fs_1.2.6
[28] cowplot_0.9.4 rprojroot_1.3-2 grid_3.5.1
[31] tidyselect_0.2.5 glue_1.3.0 R6_2.4.0
[34] rmarkdown_1.11 mixsqp_0.1-115 purrr_0.3.0
[37] ashr_2.2-37 magrittr_1.5 whisker_0.3-2
[40] MASS_7.3-51.1 codetools_0.2-16 backports_1.1.3
[43] scales_1.0.0 htmltools_0.3.6 assertthat_0.2.0
[46] colorspace_1.4-0 labeling_0.3 stringi_1.2.4
[49] pscl_1.5.2 doParallel_1.0.14 lazyeval_0.2.1
[52] munsell_0.5.0 truncnorm_1.0-8 SQUAREM_2017.10-1
[55] crayon_1.3.4