Last updated: 2019-11-10

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Introduction

In poisson mean model, \(x\sim Poisson(s\lambda)\), when \(s\) is large, the posterior mean is very close to \(x/s\). Here is the example.

This is probably why we can not get sparsity in ebpmf. The scale is typically \(300-500\) in ebpmf.

\(s=1\)

library(ebmpmf)
set.seed(12345)
pi0=0.8
lamda=c()
n=100
for(i in 1:100){
  idx = rbinom(1,1,pi0)
  if(idx){
    lamda[i] = rgamma(1,0.1,1)
  }else{
    lamda[i]=rgamma(1,10,2)
  }
}

x = rpois(n,lamda)

#x = seq(0,50)

fit = ebpm_exp_mixture(x,scale=c(0.01,100),shape=1,nullweight = 100)

fit$fitted_g
$pi
[1] 0.8719191 0.1280809

$shape
[1] 1 1

$scale
[1] 1e-02 1e+02

attr(,"class")
[1] "gammamix"
plot(x,fit$posterior$mean,xlab = 'MLE',ylab = 'Posterior mean',pch=16)
abline(0,1)

Version Author Date
bb7d2d1 Dongyue Xie 2019-11-08 
plot(x,col='grey80',pch=16)
lines(fit$posterior$mean,type='p',pch=3,col=2)
legend('topleft',c('MLE','Posterior mean'),col=c(1,2),pch=c(16,3))

Version Author Date
bb7d2d1 Dongyue Xie 2019-11-08

\(s=100\)

s=100
x = rpois(n,s*lamda)

fit = ebpm_exp_mixture(x,s,scale=c(0.0001,10000),shape=1,nullweight = 1000)

fit$fitted_g
$pi
[1] 0.95692011 0.04307989

$shape
[1] 1 1

$scale
[1] 1e-04 1e+04

attr(,"class")
[1] "gammamix"
par(mfrow=c(1,2))

plot(x/s,fit$posterior$mean,xlab = 'MLE',ylab = 'Posterior mean',pch=16)
abline(0,1)

plot(x/s,col='grey80',pch=16)
lines(fit$posterior$mean,type='p',pch=3,col=2)
legend('topleft',c('MLE','Posterior mean'),col=c(1,2),pch=c(16,3))

Version Author Date
bb7d2d1 Dongyue Xie 2019-11-08

If we treat \(x/s\) as observation, then

fit = ebpm_exp_mixture(x/s,scale=c(0.01,100),shape=1)

par(mfrow=c(1,2))

plot(x/s,fit$posterior$mean,xlab = 'MLE',ylab = 'Posterior mean',pch=16)
abline(0,1)

plot(x/s,col='grey80',pch=16)
lines(fit$posterior$mean,type='p',pch=3,col=2)
legend('topright',c('MLE','Posterior mean'),col=c(1,2),pch=c(16,3))

Version Author Date
bb7d2d1 Dongyue Xie 2019-11-08 

sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] ebmpmf_0.1.0

loaded via a namespace (and not attached):
 [1] workflowr_1.5.0 Rcpp_1.0.2      rprojroot_1.3-2 digest_0.6.21  
 [5] later_1.0.0     R6_2.4.0        backports_1.1.5 git2r_0.26.1   
 [9] magrittr_1.5    evaluate_0.14   stringi_1.4.3   rlang_0.4.0    
[13] fs_1.3.1        promises_1.1.0  whisker_0.4     rmarkdown_1.16 
[17] tools_3.6.1     stringr_1.4.0   glue_1.3.1      mixsqp_0.1-97  
[21] httpuv_1.5.2    xfun_0.10       yaml_2.2.0      compiler_3.6.1 
[25] htmltools_0.4.0 knitr_1.25