Poisson mean scale invariant
Dongyue Xie
2019-11-11
Last updated: 2019-11-12
Checks: 7 0
Knit directory: SMF/
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Introduction
See if scale-invariant version could handle large scale case.
library(ebpm)
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(1,50,1)
}
}
lamda = sort(lamda)
hist(lamda,breaks =20)
x = rpois(n,lamda)
s=1
fit = ebpm_gamma_mixture_single_scale(x,s)
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 |
|---|---|---|
| 83f336c | Dongyue Xie | 2019-11-12 |
No shrinkage estimate when \(x=0,1,2\).
How about \(\exp(E\log\lambda)\)?
plot((x/s),col='grey80',pch=16)
lines(exp(fit$posterior$mean_log),type='p',pch=3,col=2)
legend('topleft',c('MLE','exp(Posterior mean log)'),col=c(1,2),pch=c(16,3))
It shrinks \(E\log\lambda\) when \(x\) is large.
Look at estimated values from 70 to 80
(x/s)[70:80] [1] 0 1 0 2 1 3 4 38 38 38 47round(exp(fit$posterior$mean_log[70:80]),3) [1] 0.000 1.005 0.000 2.095 1.005 3.122 4.130 37.691 37.691 37.691
[11] 46.557Look at fitted prior
fit$fitted_g$pi
[1] 0.0000000 0.0000000 0.2311031 0.4836700 0.0000000 0.0000000 0.0000000
[8] 0.0000000 0.0000000 0.0000000 0.2852269 0.0000000 0.0000000
$shape
[1] 0.0007692308 0.0015384615 0.0030769231 0.0061538462 0.0123076923
[6] 0.0246153846 0.0492307692 0.0984615385 0.1969230769 0.3938461538
[11] 0.7876923077 1.5753846154 3.1507692308
$scale
[1] 65 65 65 65 65 65 65 65 65 65 65 65 65
attr(,"class")
[1] "gammamix"
attr(,"row.names")
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13Let’s increase scale to \(s=100\).
set.seed(12345)
s = 100
x = rpois(n,s*lamda)
fit = ebpm_gamma_mixture_single_scale(x,s)
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))
(x/s)[60][1] 0.08round(fit$posterior$mean[60],3)[1] 0.081No shrinkage estimate of \(\lambda\).
How about \(\exp(E\log\lambda)\)?
plot(x/s,exp(fit$posterior$mean_log),xlab = 'MLE',ylab = 'exp(Posterior mean log)',pch=16)
abline(0,1)
| Version | Author | Date |
|---|---|---|
| 83f336c | Dongyue Xie | 2019-11-12 |
plot(x/s,col='grey80',pch=16)
lines(exp(fit$posterior$mean_log),type='p',pch=3,col=2)
legend('topleft',c('MLE','exp(Posterior mean log)'),col=c(1,2),pch=c(16,3))
| Version | Author | Date |
|---|---|---|
| 83f336c | Dongyue Xie | 2019-11-12 |
(x/s)[60][1] 0.08round(exp(fit$posterior$mean_log)[60],3)[1] 0.076Some shrinkage, roughly \(2\%-5\%\) for some values. Maybe this is the reason why
Look at fitted prior
fit$fitted_g$pi
[1] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
[7] 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
[13] 0.00000000 0.79795698 0.02294723 0.00000000 0.00000000 0.17909579
[19] 0.00000000 0.00000000
$shape
[1] 7.450454e-06 1.490091e-05 2.980182e-05 5.960364e-05 1.192073e-04
[6] 2.384145e-04 4.768291e-04 9.536582e-04 1.907316e-03 3.814633e-03
[11] 7.629265e-03 1.525853e-02 3.051706e-02 6.103412e-02 1.220682e-01
[16] 2.441365e-01 4.882730e-01 9.765460e-01 1.953092e+00 3.906184e+00
$scale
[1] 67.11 67.11 67.11 67.11 67.11 67.11 67.11 67.11 67.11 67.11 67.11
[12] 67.11 67.11 67.11 67.11 67.11 67.11 67.11 67.11 67.11
attr(,"class")
[1] "gammamix"
attr(,"row.names")
[1] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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] ebpm_0.0.0.9004
loaded via a namespace (and not attached):
[1] workflowr_1.5.0 Rcpp_1.0.2 gtools_3.8.1 rprojroot_1.3-2
[5] digest_0.6.21 later_1.0.0 R6_2.4.0 backports_1.1.5
[9] git2r_0.26.1 magrittr_1.5 evaluate_0.14 stringi_1.4.3
[13] rlang_0.4.0 fs_1.3.1 promises_1.1.0 whisker_0.4
[17] rmarkdown_1.16 tools_3.6.1 stringr_1.4.0 glue_1.3.1
[21] mixsqp_0.1-97 httpuv_1.5.2 xfun_0.10 yaml_2.2.0
[25] compiler_3.6.1 htmltools_0.4.0 knitr_1.25