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Introduction

Here we study if splitting method work for binomial and Bernoulli data.

The basic model is

\[x_i\sim Binomial(n_i,\sigma(\mu_i))\] \[\mu_i|b_i\sim N(b_i,\sigma^2)\] \[b_i\sim g(\cdot).\]

I generated \(\mu_i\sim \pi_0\delta_0 + \pi_1 N(0,1)\) where \(\pi_0 = 0.8\).

source("code/binomial_mean/binomial_mean_splitting.R")
Loading required package: Rcpp
Loading required package: RcppZiggurat

Binomial n = 100

set.seed(12345)
n = 500
nb = rep(100,n)
w = 0.8
mu = c(rep(0,n*w),rnorm(round(n*(1-w))))
p = sigmoid(mu)
x = rbinom(n,nb,p)
library(ashr)
fit_ash = ashr::ash(rep(0,n),1,lik=lik_binom(x,nb,link='identity'))
fit_split = binomial_mean_splitting(x,nb,printevery = 100,n_gh=10,
                                    b_pm_init = NULL,sigma2_init = NULL,
                                    ebnm_params = list(prior_family='normal_scale_mixture'))
Warning in binomial_mean_splitting(x, nb, printevery = 100, n_gh = 10, b_pm_init
= NULL, : An iteration decreases ELBO. This is likely due to numerical issues.
fit_GG = binomial_mean_GG(x,nb,printevery = 100,n_gh=10)
Warning in binomial_mean_GG(x, nb, printevery = 100, n_gh = 10): An iteration
decreases ELBO. This is likely due to numerical issues.
plot(x/nb,col='grey80',main='ash fit')
lines(p,col='grey80')
lines(fit_ash$result$PosteriorMean,col=2)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17 
plot(x/nb,col='grey80',main='splitting fit latent m')
lines(p,col='grey80')
lines(fit_split$posterior$mean,col=4)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17 
plot(x/nb,col='grey80',main='splitting fit latent b')
lines(p,col='grey80')
lines(sigmoid(fit_split$posterior$mean_b),col=4)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17 
plot(x/nb,col='grey80',main='vga binom fit')
lines(p,col='grey80')
lines(fit_GG$posterior$mean,col=3)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17

Binomial n = 10

set.seed(12345)
n = 500
nb = rep(10,n)
w = 0.8
mu = c(rep(0,n*w),rnorm(round(n*(1-w))))
p = sigmoid(mu)
x = rbinom(n,nb,p)
library(ashr)
fit_ash = ashr::ash(rep(0,n),1,lik=lik_binom(x,nb,link='identity'))
fit_split = binomial_mean_splitting(x,nb,printevery = 100,n_gh=10,
                                    b_pm_init = NULL,sigma2_init = NULL,
                                    ebnm_params = list(prior_family='normal_scale_mixture'))
fit_GG = binomial_mean_GG(x,nb,printevery = 100,n_gh=10)
[1] "At iter 100 elbo= -881.405"
plot(x/nb,col='grey80',main='ash fit')
lines(p,col='grey80')
lines(fit_ash$result$PosteriorMean,col=2)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17 
plot(x/nb,col='grey80',main='splitting fit latent m')
lines(p,col='grey80')
lines(fit_split$posterior$mean,col=4)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17 
plot(x/nb,col='grey80',main='splitting fit latent b')
lines(p,col='grey80')
lines(sigmoid(fit_split$posterior$mean_b),col=4)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17 
plot(x/nb,col='grey80',main='vga binom fit')
lines(p,col='grey80')
lines(fit_GG$posterior$mean,col=3)

Version Author Date
e08c844 Dongyue Xie 2022-12-16
b4d61ba DongyueXie 2022-11-17

Bernoulli

set.seed(12345)
n = 500
nb = rep(1,n)
w = 0.8
mu = c(rep(0,n*w),rnorm(round(n*(1-w))))
p = sigmoid(mu)
x = rbinom(n,nb,p)
library(ashr)
fit_ash = ashr::ash(rep(0,n),1,lik=lik_binom(x,nb,link='identity'))
fit_split = binomial_mean_splitting(x,nb,printevery = 100,n_gh=10,
                                    b_pm_init = NULL,sigma2_init = NULL,
                                    ebnm_params = list(prior_family='normal_scale_mixture'))
[1] "At iter 100 elbo= -846.413 sigma2= 2.447"
[1] "At iter 200 elbo= -846.208 sigma2= 1.847"
[1] "At iter 300 elbo= -846.133 sigma2= 1.555"
[1] "At iter 400 elbo= -846.095 sigma2= 1.376"
[1] "At iter 500 elbo= -846.073 sigma2= 1.253"
[1] "At iter 600 elbo= -846.059 sigma2= 1.161"
[1] "At iter 700 elbo= -846.049 sigma2= 1.09"
[1] "At iter 800 elbo= -846.041 sigma2= 1.032"
[1] "At iter 900 elbo= -846.036 sigma2= 0.985"
[1] "At iter 1000 elbo= -846.031 sigma2= 0.944"
fit_GG = binomial_mean_GG(x,nb,printevery = 100,n_gh=10)
[1] "At iter 100 elbo= -943.612"
plot(x/nb,col='grey80',main='ash fit')
lines(p,col='grey80')
lines(fit_ash$result$PosteriorMean,col=2)

plot(x/nb,col='grey80',main='splitting fit latent m')
lines(p,col='grey80')
lines(fit_split$posterior$mean,col=4)

plot(x/nb,col='grey80',main='splitting fit latent b')
lines(p,col='grey80')
lines(sigmoid(fit_split$posterior$mean_b),col=4)

plot(x/nb,col='grey80',main='vga binom fit')
lines(p,col='grey80')
lines(fit_GG$posterior$mean,col=3)


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] ashr_2.2-54        Rfast_2.0.6        RcppZiggurat_0.1.6 ebnm_1.0-11       
[5] fastGHQuad_1.0.1   Rcpp_1.0.9         vebpm_0.4.0        workflowr_1.7.0   

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