Empircal Bayes Normal Means with Bernoulli Priors
jhmarcus
2019-05-23
Last updated: 2019-05-23
Checks: 5 1
Knit directory: drift-workflow/analysis/
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| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | e013545 | jhmarcus | 2019-05-23 | added obs |
| html | e013545 | jhmarcus | 2019-05-23 | added obs |
| Rmd | 221956b | jhmarcus | 2019-05-23 | added exploration of using bernoulli prior for enbm |
| html | 221956b | jhmarcus | 2019-05-23 | added exploration of using bernoulli prior for enbm |
Here I explore solving the Empirical Bayes Normal Means problem with a Bernoulli prior. This could be useful for fitting highly constrained non-negative FLASH.
\[ \beta_j \sim Bern(\pi) \\ y_j | \beta_j \sim N(\beta_j, s^2_j) \] The marginal likelihood for a single obesrvation is given by:
\[ \begin{aligned} p(y_j |\pi) &= p(\beta_j=0)p(y_j | \beta_j = 0) + p(\beta_j=1) p(y_j|\beta_j =1) \\ &= (1-\pi)p(y_j|\beta_j=0) + \pi p(y_j|\beta_j=1) \end{aligned} \]
We can easily maximize the marginal likelihood to estimate \(\hat{\pi}\) using a root finding algorithm, such as Brent’s method. To compute the posterior mean we can simply use Bayes rule and plug in the estimated prior:
\[ \begin{aligned} E(\beta_j|y_j) &= p(\beta_j=1|y_j) \\ &=\frac{p(y_j|\beta_j=1)p(\beta_j=1)}{p(y_j|\beta_j=0)p(\beta_j=0) + p(y_j | \beta_j=1)p(\beta_j=1)} \\ &= \frac{p(y_j|\beta_j=1)\hat{\pi}}{p(y_j|\beta_j=0)(1-\hat{\pi}) + p(y_j | \beta_j=1)\hat{\pi}} \end{aligned} \]
Imports
library(ggplot2)
library(dplyr)
library(tidyr)Functions
Here are some helper function for simulation, fitting, and plotting.
# Simulation
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))
}
# Negative log marginal likelihood
nll = function(pi, data){
betahat = data[[1]]
s = data[[2]]
liks = (1.0-pi) * dnorm(betahat, mean=0, sd=s) + pi * dnorm(betahat, mean=1, sd=s)
loglik = sum(log(liks))
return(-loglik)
}
# Estimate pi by minimizes nll
est_pi = function(betahat, s){
fit = optim(par=c(.5),
fn=nll,
data=list(betahat, s),
method="Brent",
lower=0,
upper=1)
pihat = fit$par
return(pihat)
}
# Estimate the post using bayes rule and plug in of pihat mle
est_post = function(betahat, s, pihat){
c0 = dnorm(betahat, mean=0, sd=s) * (1 - pihat)
c1 = dnorm(betahat, mean=1, sd=s) * pihat
pm = c1 / (c0 + c1)
return(pm)
}
# Viz
plot_sim = function(df, title){
gath_df = df %>% gather(variable, value, -idx, -s)
p0 = ggplot(gath_df, aes(x=idx, y=value,
color=factor(variable, levels=c("beta", "betahat", "betapm")))) +
geom_point() +
theme_bw() +
labs(color="") +
xlab("Variable") +
ylab("Value") +
theme(legend.position="bottom")
min_betahat = min(df$betahat)
max_betahat = max(df$betahat)
p1 = ggplot(df, aes(betahat, betapm, color=s)) +
geom_point() + viridis::scale_color_viridis() +
theme_bw() +
theme(legend.position="bottom") +
xlim(c(min_betahat, max_betahat)) +
ylim(c(min_betahat, max_betahat)) +
geom_abline()
p = cowplot::plot_grid(p0, p1, nrow=1)
title = cowplot::ggdraw() + cowplot::draw_label(title)
print(cowplot::plot_grid(title, p, ncol=1, rel_heights=c(0.1, 1)))
}n0=10
n1=90
sigma_e=.25
sim_res = sim(n0, n1, sigma_e)
betahat = sim_res$betahat
s = sim_res$s
beta = sim_res$beta
pihat = est_pi(betahat, s)
betapm = est_post(betahat, s, pihat)
df = data.frame(betahat=betahat, beta=beta, betapm=betapm, s=s, idx=1:length(betahat))
title = paste0("n0=",n0, ",n1=", n1, ",sigma_e=", sigma_e, ",pihat=", round(pihat, digits=3))
plot_sim(df, title)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
pihat = est_pi(betahat, s)
betapm = est_post(betahat, s, pihat)
df = data.frame(betahat=betahat, beta=beta, betapm=betapm, s=s, idx=1:length(betahat))
title = paste0("n0=",n0[i], ",n1=", n1[i], ",sigma_e=", sigma_e[i], ",pihat=", round(pihat, digits=3))
plot_sim(df, title)
}
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
| Version | Author | Date |
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| 221956b | jhmarcus | 2019-05-23 |
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_1.0.1 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 tools_3.5.1 digest_0.6.18
[10] viridisLite_0.3.0 evaluate_0.12 tibble_2.0.1
[13] gtable_0.2.0 pkgconfig_2.0.2 rlang_0.3.1
[16] yaml_2.2.0 xfun_0.4 gridExtra_2.3
[19] withr_2.1.2 stringr_1.4.0 knitr_1.21
[22] fs_1.2.6 cowplot_0.9.4 rprojroot_1.3-2
[25] grid_3.5.1 tidyselect_0.2.5 glue_1.3.0
[28] R6_2.4.0 rmarkdown_1.11 purrr_0.3.0
[31] magrittr_1.5 whisker_0.3-2 backports_1.1.3
[34] scales_1.0.0 htmltools_0.3.6 assertthat_0.2.1
[37] colorspace_1.4-0 labeling_0.3 stringi_1.2.4
[40] lazyeval_0.2.1 munsell_0.5.0 crayon_1.3.4