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Introduction

What do the penalties \(\rho_{g,s^2}(\theta)\) and \(c_{g,s^2}(\theta) = \rho_{g,s^2}(S_{g,s^2}(\theta))\) look like?

The first one is the original penalty and we’ll need inversion \(S^{-1}(\cdot)\) to evaluate it; the second one is the compound penalty.

\[\rho_{g,s^2}(\theta) = \rho_{g,s^2}(S_{g,s^2}(S^{-1}_{g,s^2}(\theta))) = c_{g,s^2}(S^{-1}_{g,s^2}(\theta))\]

\[c_{g,s^2}(\theta) = \rho_{g,s^2}(S_{g,s^2}(\theta)) = -l_{NM}(\theta;g,s^2)-\frac{(\theta-S_{g,s}(\theta))^2}{2s^2}\]

(To evaluate \(\rho(\theta)\), we find \(z\) such that \(S(z) = \theta\) then \(\rho(\theta) = c(z)\).)

We also evaluate the second derivative of the penalties:

For the compound penalty \(c(\theta) = -l(\theta) - \frac{s^2(l'(\theta))^2}{2}\)

\[c'(\theta) = -l'(\theta) - s^2l'(\theta)l''(\theta)\]

\[c''(\theta) = -l''(\theta) - s^2(l''(\theta)^2+l'''(\theta)l'(\theta))\]

For the penalty \(\rho(\theta)\):

\[\rho_{g,s^2}'(\theta) = \frac{S^{-1}_{g,s^2}(\theta)-\theta}{s^2}\]

\[\rho_{g,s^2}''(\theta) = \frac{1}{s^2}\left(\frac{1}{1+s^2l''_{nm}(S^{-1}_{g,s^2}(\theta))}-1\right) = \frac{-l''_{nm}(S^{-1}_{g,s^2}(\theta))}{1+s^2l''_{nm}(S^{-1}_{g,s^2}(\theta))}\]

\[\rho''(S(\theta)) = -l''(\theta)/S'(\theta).\]

\[S(\theta) = \theta + s^2l_{NM}'(\theta)\]

\[S'(\theta) = 1 + s^2l_{NM}''(\theta)\]

\[\rho''(S(\theta)) = -l''(\theta)/S'(\theta) = -l''(\theta)/(1+s^2l''(\theta)).\]

source('code/normal_mean_model_utils.R')

# nm = function(z,g,s){
#   fit = ash(z,s,g = g,fixg = T)
#   return(list(l_nm = fit$loglik, pm=fit$result$PosteriorMean))
# }
# 
# penalty = function(theta,g,s){
#   temp = nm(theta,g,s)
#   return(-temp$l_nm - (theta-temp$pm)^2/2/s^2)
# }
# 
# penalty_vec = function(x,g,s){
#   n = length(x)
#   out = rep(0,n)
#   for(i in 1:n){
#     out[i] = penalty(x[i],g,s)
#   }
#   return(out)
# }
# 
# 
# 
# 
# #'posterior mean operator
# S = function(z,g,s){
#   pw = log(g$pi)+dnorm(z,mean=0,sd=sqrt(s^2+g$sd^2),log=TRUE)
#   pw = pw - max(pw)
#   pw = exp(pw)/sum(exp(pw))
#   pm = z/(1+s^2/g$sd^2)
#   return(sum(pw*pm))
# }
 
# #'Solve S(z) = theta for a given theta
# #'@param theta 
# #'@param g a normalmix object with weights pi and sd
# #'@param s known standard error
# inv_S = function(theta,g,s=1){
#   eqn = function(z,theta,g,s){
#     S(z,g,s) - theta
#   }
#   if(theta==0){
#     sol = 0
#   }else if(theta>0){
#     sol = uniroot(eqn,interval=c(0,theta/min(1/(1+s^2/g$sd^2))),theta=theta,g=g,s=s)$root
#   }else{
#     sol = uniroot(eqn,interval=c(theta/min(1/(1+s^2/g$sd^2)),0),theta=theta,g=g,s=s)$root
#   }
#   return(sol)
# }
# 
# inv_S_vec = function(x,g,s=1){
#   n = length(x)
#   out = c()
#   for(i in 1:n){
#     out[i] = inv_S(x[i],g,s)
#   }
#   out
# }


plot_penalty = function(x,s=1,w,grid){

  # plot compound penalty
  compound_penalty = nm_penalty_compound(x,s,w,grid)
  original_penalty = nm_penalty(x,s,w,grid)
  
  plot(x,compound_penalty,type='l',xlab='theta',ylab='penalty',ylim=range(c(compound_penalty,original_penalty)),
       main=paste('w=(',paste(w,collapse = ", "),'); grid=(',paste(grid,collapse = ", "),')',sep=''))
  lines(x,original_penalty,col='blue',lty=2)
  legend('bottomright',c(expression(rho(S(theta))),expression(rho(theta))),lty=c(1,2),col=c(1,'blue'))
}

plot_penalty_hess = function(x,s=1,w,grid){

  # plot compound penalty
  compound_penalty = nm_penalty_compound_hess(x,s,w,grid)
  original_penalty = nm_penalty_hess(x,s,w,grid)
  
  plot(x,compound_penalty,type='l',xlab='theta',ylab='2nd derivative of penalty',ylim=range(c(compound_penalty,original_penalty)),
       main=paste('w=(',paste(w,collapse = ", "),'); grid=(',paste(grid,collapse = ", "),')',sep=''))
  lines(x,original_penalty,col='blue',lty=2)
  legend('topright',c(expression(rho^"''"*(S(theta))),expression(rho^"''"*(theta))),lty=c(1,2),col=c(1,'blue'))
}
n = 101
x = seq(-10,10,length.out = n)
s = rep(1,n)
w = c(0.9,0.1)
par(mfrow=c(2,2))
grid_list=c(1,2,3,5)
for(gg in grid_list){
  plot_penalty(x,s,w,c(0,gg))
}

Version Author Date
6e18a0a Dongyue Xie 2022-09-30 
for(gg in grid_list){
  plot_penalty_hess(x,s,w,c(0,gg))
}

Version Author Date
6e18a0a Dongyue Xie 2022-09-30 
n = 101
x = seq(-10,10,length.out = n)
s = rep(1,n)
w = c(0.5,0.5)
par(mfrow=c(2,2))
grid_list=c(1,2,3,5)
for(gg in grid_list){
  plot_penalty(x,s,w,c(0,gg))
}

Version Author Date
6e18a0a Dongyue Xie 2022-09-30
86c651b Dongyue Xie 2022-06-06 
for(gg in grid_list){
  plot_penalty_hess(x,s,w,c(0,gg))
}

Version Author Date
6e18a0a Dongyue Xie 2022-09-30 
n = 101
x = seq(-10,10,length.out = n)
s = rep(1,n)
w = c(0.1,0.9)
par(mfrow=c(2,2))
grid_list=c(1,2,3,5)
for(gg in grid_list){
  plot_penalty(x,s,w,c(0,gg))
}

Version Author Date
6e18a0a Dongyue Xie 2022-09-30
86c651b Dongyue Xie 2022-06-06 
for(gg in grid_list){
  plot_penalty_hess(x,s,w,c(0,gg))
}

Version Author Date
6e18a0a Dongyue Xie 2022-09-30 
#'softmax
softmax = function(a){
  exp(a-max(a))/sum(exp(a-max(a)))
}
s2 = exp(seq(-8,5,by=log(2)))
w = softmax(log(1/s2))
round(w,2)
round(s2,2)
K = length(w)
g = normalmix(pi = w,mean=rep(0,K),sd = sqrt(s2))
x = seq(-10,10,length.out = 101)
plot_penalty(x,w,s,sqrt(s2))

sessionInfo()
R version 4.2.0 (2022-04-22)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur/Monterey 10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2/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] ashr_2.2-54     workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.8.3     highr_0.9        bslib_0.3.1      compiler_4.2.0  
 [5] pillar_1.7.0     later_1.3.0      git2r_0.30.1     jquerylib_0.1.4 
 [9] tools_4.2.0      getPass_0.2-2    digest_0.6.29    lattice_0.20-45 
[13] jsonlite_1.8.0   evaluate_0.15    tibble_3.1.6     lifecycle_1.0.1 
[17] pkgconfig_2.0.3  rlang_1.0.2      Matrix_1.4-1     cli_3.3.0       
[21] rstudioapi_0.13  yaml_2.3.5       xfun_0.30        fastmap_1.1.0   
[25] invgamma_1.1     httr_1.4.2       stringr_1.4.0    knitr_1.38      
[29] sass_0.4.1       fs_1.5.2         vctrs_0.4.1      grid_4.2.0      
[33] rprojroot_2.0.3  glue_1.6.2       R6_2.5.1         processx_3.5.3  
[37] fansi_1.0.3      rmarkdown_2.13   mixsqp_0.3-47    irlba_2.3.5     
[41] callr_3.7.0      magrittr_2.0.3   whisker_0.4      ps_1.7.0        
[45] promises_1.2.0.1 htmltools_0.5.2  ellipsis_0.3.2   httpuv_1.6.5    
[49] utf8_1.2.2       stringi_1.7.6    truncnorm_1.0-8  SQUAREM_2021.1  
[53] crayon_1.5.1