Last updated: 2022-09-29

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Rmd ca6fb43 Dongyue Xie 2022-09-29 add inversion method

Introduction

\[y|\mu\sim N(\mu,s^2),\mu\sim g(\cdot)\]

Optimization: \(\theta = E_q\mu\),

\[\min_{\theta,g}h(\theta,g) = \frac{1}{2s^2}(y-\theta)^2+\rho_{g,s}(\theta).\]

We use inversion method for solving the optimization problem.

For another method, see compound method

source("code/normal_mean_model_utils.R")

generate data, and get grid by running ash

set.seed(12345)
n = 200
w0 = 0.9
mu = c(rep(0,round(n*w0)),rep(10,n-round(n*w0)))
w_true = c(w0,1-w0)
grid_true = c(0.01,7)
s = rep(1,n)
y = rnorm(n,mu,s)
library(ashr)
fit.ash = ashr::ash(y,s,mixcompdist = 'normal')
#grid = exp(seq(log(s/100),log(sqrt(max(abs(y^2-s^2)))),by=log(sqrt(2))))
#fit.ash = S(y,s,w_true,grid_true)
#plot(fit.ash$fitted_g$sd,fit.ash$fitted_g$pi)
grid = fit.ash$fitted_g$sd
grid = grid[-1]
K = length(grid)

#plot(y,main='ash fit',col='grey80')
#lines(mu,col='grey60')
#lines(fit.ash$result$PosteriorMean)
#legend('topleft',c('data','true mean','ash posteriorMean'),pch=c(1,NA,NA),lty=c(NA,1,1),col=c('grey80','grey60',1))

Known prior

f_obj_known_g = function(theta,y,s,w,grid,z_range){
  z = S_inv(theta,s,w,grid,z_range)
  return((y-theta)^2/2/s^2 - l_nm(z,s,w,grid)-(z-theta)^2/2/s^2)
}

f_obj_known_g_grad = function(theta,y,s,w,grid,z_range){
  z = S_inv(theta,s,w,grid,z_range)
  return((z-y)/s^2)
}

ebnm_penalized_inversion_known_g = function(x,s,w,grid,theta_init = NULL,z_range=NULL,opt_method = 'L-BFGS-B'){
  n = length(x)
  
  if(length(s)==1){
    s = rep(s,n)
  }
  if(is.null(theta_init)){
    theta_init = rep(0,n)
  }
  if(is.null(z_range)){
    z_range = range(x) + c(-1,1)
  }
  theta = double(n)
  for(i in 1:n){
    theta[i] = optim(theta_init[i],
                     fn=f_obj_known_g,
                     gr = f_obj_known_g_grad,
                   y=x[i],
                   s=s[i],
                   w=w,
                   grid=grid,
                   z_range=z_range,
                   method = opt_method)$par
  }
  return(theta)
}
ploter = function(fit,y,mu,main='known prior'){
  plot(y,main=main,col='grey80')
  lines(mu,col='grey60')
  lines(fit)
  legend('topleft',c('data','true mean','estimated mean'),pch=c(1,NA,NA),lty=c(NA,1,1),col=c('grey80','grey60',1))
}

Init at data \(y\).

fit = ebnm_penalized_inversion_known_g(y,s,w_true,grid_true,theta_init = y,opt_method = 'L-BFGS-B')
ploter(fit,y,mu,main='known prior, init at y')

Init at 0.

fit = ebnm_penalized_inversion_known_g(y,s,w_true,grid_true,theta_init = rep(0,n),opt_method = 'L-BFGS-B')
ploter(fit,y,mu,main='known prior, init at 0')

Estimate prior

#'objective function
#'@param theta (theta,w)
#'@param grid prior sds
f_obj = function(params,y,s,grid,z_range,opt_method='L-BFGS-B'){
  n = length(y)
  w = softmax(params[-(1:n)])
  theta = params[1:n]
  z = S_inv(theta,s,w,grid,z_range)
  return(sum((y-theta)^2/2/s^2 - l_nm(z,s,w,grid)-(z-theta)^2/2/s^2))
}


#'objective function
#'@param theta (theta,w)
#'@param grid prior sds
f_obj_grad = function(params,y,s,grid,z_range,opt_method='L-BFGS-B'){
  n = length(y)
  a = params[-(1:n)]
  w = softmax(a)
  theta = params[1:n]
  z = S_inv(theta,s,w,grid,z_range)
  grad_theta = (z-y)/s^2
  grad_a = -colSums(l_nm_d1_a(z,s,a,grid))
  return(c(grad_theta,c(grad_a)))
}
ebnm_penalized_inversion = function(x,s,grid,theta_init = NULL,w_init=NULL,z_range=NULL,opt_method = 'L-BFGS-B'){
  n = length(x)
  K = length(grid)
  if(is.null(w_init)){
    w_init = rep(1/K,K)
  }
  if(length(s)==1){
    s = rep(s,n)
  }
  if(is.null(theta_init)){
    theta_init = rep(0,n)
  }
  if(is.null(z_range)){
    z_range = range(x) + c(-1,1)
  }
  params = c(theta_init,w_init)
  out = optim(params,
                   fn=f_obj,
                   gr = f_obj_grad,
                 y=x,
                 s=s,
                 grid=grid,
                 z_range=z_range,
                 method = opt_method)
  
  return(list(posteriorMean = out$par[1:n],w = softmax(out$par[-(1:n)]),opt_res = out))
}
ploter = function(fit,y,mu,main='estimate prior'){
  plot(y,main=main,col='grey80')
  lines(mu,col='grey60')
  lines(fit$posteriorMean)
  legend('topleft',c('data','true mean','estimated mean'),pch=c(1,NA,NA),lty=c(NA,1,1),col=c('grey80','grey60',1))
}

Init at data \(y\).

fit = ebnm_penalized_inversion(y,s,grid,theta_init = y,opt_method = 'L-BFGS-B')
plot(grid,fit$w,ylab='w hat')

fit$opt_res$value
[1] 408.2922
ploter(fit,y,mu,main='estimate prior, init at y')

Init at 0.

fit = ebnm_penalized_inversion(y,s,grid,theta_init = rep(0,n),opt_method = 'L-BFGS-B')
plot(grid,fit$w,ylab='w hat')

fit$opt_res$value
[1] 408.2915
ploter(fit,y,mu,main='estimate prior, init at 0')

ash fit

fit.ash = ash(y,s,mixcompdist = 'normal',pointmass=FALSE,prior='uniform',mixsd=grid)
plot(y,main='ash fit',col='grey80')
lines(mu,col='grey60')
lines(fit.ash$result$PosteriorMean)
legend('topleft',c('data','true mean','ash posteriorMean'),pch=c(1,NA,NA),lty=c(NA,1,1),col=c('grey80','grey60',1))


sessionInfo()
R version 4.2.1 (2022-06-23 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 22000)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.utf8 
[2] LC_CTYPE=English_United States.utf8   
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.utf8    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ashr_2.2-54     workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.9       highr_0.9        compiler_4.2.1   pillar_1.8.1    
 [5] bslib_0.4.0      later_1.3.0      git2r_0.30.1     jquerylib_0.1.4 
 [9] tools_4.2.1      getPass_0.2-2    digest_0.6.29    lattice_0.20-45 
[13] jsonlite_1.8.0   evaluate_0.16    tibble_3.1.8     lifecycle_1.0.2 
[17] pkgconfig_2.0.3  rlang_1.0.5      Matrix_1.4-1     cli_3.3.0       
[21] rstudioapi_0.14  yaml_2.3.5       xfun_0.32        fastmap_1.1.0   
[25] invgamma_1.1     httr_1.4.4       stringr_1.4.1    knitr_1.40      
[29] fs_1.5.2         vctrs_0.4.1      sass_0.4.2       grid_4.2.1      
[33] rprojroot_2.0.3  glue_1.6.2       R6_2.5.1         processx_3.7.0  
[37] fansi_1.0.3      rmarkdown_2.16   mixsqp_0.3-43    irlba_2.3.5     
[41] callr_3.7.2      magrittr_2.0.3   whisker_0.4      ps_1.7.1        
[45] promises_1.2.0.1 htmltools_0.5.3  httpuv_1.6.5     utf8_1.2.2      
[49] stringi_1.7.8    truncnorm_1.0-8  SQUAREM_2021.1   cachem_1.0.6