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Introduction

\[y_i\sim Poisson(exp(\mu_j)),\mu_j|b_j\sim N(b_j,\sigma^2),b_j\sim g(\cdot).\]

We show in a simple POisson mean example, there are two local optimums - one where \(g_b\) is a point mass, one where \(g_b\) is the one we are interested in.

library(vebpm)
n = 1000
set.seed(12345)
mu = c(rep(4,100),rep(0,n-100))
x = rpois(n,exp(mu))

plot(x,col='grey80')
lines(exp(mu))

Version Author Date
bdc3619 DongyueXie 2023-02-06

In the first example we initialize \(\sigma^2=0.5\), ans we can see that the final \(sigma^2\) is around 0.2

fit0 = pois_mean_split(x,sigma2=0.5)
Warning in pois_mean_split(x, sigma2 = 0.5): An iteration decreases ELBO. This
is likely due to numerical issues.
plot(x,col='grey80')
lines(fit0$posterior$mean_exp_b)

Version Author Date
bdc3619 DongyueXie 2023-02-06 
plot(fit0$fitted_g$sigma2_trace)

Version Author Date
bdc3619 DongyueXie 2023-02-06 
fit0$fitted_g$g_b
$pi
[1] 0.8836418 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
[8] 0.1163582 0.0000000

$mean
[1] -0.06950088 -0.06950088 -0.06950088 -0.06950088 -0.06950088 -0.06950088
[7] -0.06950088 -0.06950088 -0.06950088

$sd
[1] 0.0000000 0.4050770 0.6765755 0.9916300 1.3870662 1.8993115 2.5729789
[8] 3.4658753 4.6542967

attr(,"class")
[1] "normalmix"
attr(,"row.names")
[1] 1 2 3 4 5 6 7 8 9
fit0$elbo
[1] -3170.676

WE try to initialize \(\sigma^2=0.1\), and we can see that the final \(sigma^2\) is still around 0.14

fit0 = pois_mean_split(x,sigma2=0.1)
plot(x,col='grey80')
lines(fit0$posterior$mean_exp_b)

Version Author Date
bdc3619 DongyueXie 2023-02-06 
plot(fit0$fitted_g$sigma2_trace)

Version Author Date
bdc3619 DongyueXie 2023-02-06 
fit0$fitted_g$g_b
$pi
 [1] 0.8885623 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
 [8] 0.0000000 0.1114377 0.0000000

$mean
 [1] -0.02641456 -0.02641456 -0.02641456 -0.02641456 -0.02641456 -0.02641456
 [7] -0.02641456 -0.02641456 -0.02641456 -0.02641456

$sd
 [1] 0.0000000 0.3227005 0.5389870 0.7899720 1.1049923 1.5130673 2.0497376
 [8] 2.7610545 3.7077984 4.9707654

attr(,"class")
[1] "normalmix"
attr(,"row.names")
 [1]  1  2  3  4  5  6  7  8  9 10
fit0$elbo
[1] -3168.611

So this is a local optimum and is the one that gives the results we’d love to see.

ON the other hand If we initialize \(\sigma^2\) to be bigger at \(\sigma^2 = 2\), then \(\sigma^2\) converges to 3 and \(g_b\) goes to a point mass.

fit0 = pois_mean_split(x,sigma2=2)
plot(x,col='grey80')
lines(fit0$posterior$mean_exp_b)

Version Author Date
bdc3619 DongyueXie 2023-02-06 
fit0$fitted_g$sigma2_trace
[1] 2.476947 2.685522 2.843666 2.949401 3.013449 3.050891 3.072491 3.084879
[9] 3.091961
fit0$fitted_g$g_b
$pi
[1] 1 0 0 0 0

$mean
[1] -0.02755852 -0.02755852 -0.02755852 -0.02755852 -0.02755852

$sd
[1] 0.000000 1.560810 2.606926 3.820868 5.344530

attr(,"class")
[1] "normalmix"
attr(,"row.names")
[1] 1 2 3 4 5
fit0$elbo
[1] -3279.522

Try a larger init value of \(\sigma^2\) at 5, then same thing happens.

fit0 = pois_mean_split(x,sigma2=5)
plot(x,col='grey80')
lines(fit0$posterior$mean_exp_b)

Version Author Date
bdc3619 DongyueXie 2023-02-06 
fit0$fitted_g$sigma2_trace
[1] 4.012418 3.401599 3.249432 3.183250 3.147588 3.127588 3.116270 3.109837
fit0$fitted_g$g_b
$pi
[1] 1 0 0 0 0

$mean
[1] -0.03821734 -0.03821734 -0.03821734 -0.03821734 -0.03821734

$sd
[1] 0.000000 1.568731 2.620156 3.840259 5.371654

attr(,"class")
[1] "normalmix"
attr(,"row.names")
[1] 1 2 3 4 5
fit0$elbo
[1] -3279.521

Maybe we should not start with a too large \(\sigma^2\) because \(g_b\) being a point mass is a local optimum and in most cases we are not very interested in. I find that the starting \(\sigma^2\) should not be greater than \(var(\bar\mu_i)\), and perhaps smaller.

Observations

Apparently if \(g_b\) goes to a pointmass, then \(\sigma^2\) is large. This converges to the problem of putting a Gaussian prior on the parameter \(\mu\).

On the other hand, if \(g_b\) goes to a non-pointmass, then \(\sigma^2\) can converge to a smaller value, but not 0. In this simulation we know that true \(\sigma^2\) is 0, and in this case the estimated \(\sigma^2\) converges to 0.2.


sessionInfo()
R version 4.2.2 Patched (2022-11-10 r83330)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 22.04.1 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

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

other attached packages:
[1] vebpm_0.4.0     workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.9         horseshoe_0.2.0    invgamma_1.1       lattice_0.20-45   
 [5] getPass_0.2-2      ps_1.7.2           rprojroot_2.0.3    digest_0.6.31     
 [9] utf8_1.2.2         truncnorm_1.0-8    R6_2.5.1           evaluate_0.19     
[13] highr_0.9          httr_1.4.4         ggplot2_3.4.0      pillar_1.8.1      
[17] rlang_1.0.6        rstudioapi_0.14    ebnm_1.0-11        irlba_2.3.5.1     
[21] whisker_0.4.1      callr_3.7.3        jquerylib_0.1.4    nloptr_2.0.3      
[25] Matrix_1.5-3       rmarkdown_2.19     splines_4.2.2      stringr_1.5.0     
[29] munsell_0.5.0      mixsqp_0.3-48      compiler_4.2.2     httpuv_1.6.7      
[33] xfun_0.35          pkgconfig_2.0.3    SQUAREM_2021.1     htmltools_0.5.4   
[37] tidyselect_1.2.0   tibble_3.1.8       matrixStats_0.63.0 fansi_1.0.3       
[41] dplyr_1.0.10       later_1.3.0        grid_4.2.2         jsonlite_1.8.4    
[45] gtable_0.3.1       lifecycle_1.0.3    git2r_0.30.1       magrittr_2.0.3    
[49] scales_1.2.1       ebpm_0.0.1.3       cli_3.4.1          stringi_1.7.8     
[53] cachem_1.0.6       fs_1.5.2           promises_1.2.0.1   bslib_0.4.2       
[57] generics_0.1.3     vctrs_0.5.1        trust_0.1-8        tools_4.2.2       
[61] glue_1.6.2         processx_3.8.0     parallel_4.2.2     fastmap_1.1.0     
[65] yaml_2.3.6         colorspace_2.0-3   ashr_2.2-54        deconvolveR_1.2-1 
[69] knitr_1.41         sass_0.4.4