Last updated: 2019-05-23

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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.

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))
}

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)
}

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)
}

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)
}

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
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
221956b jhmarcus 2019-05-23

Version Author Date
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