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Model

Consider the model

\[\begin{equation} \begin{split} &y_i\sim NB(r,p_i), \\ &\log\frac{p_i}{1-p_i} = \mu_i\sim g(\cdot), \end{split} \end{equation}\] where \(p(y;r,p)\propto p^y(1-p)^r\).

We use PG augmentation proposed in Polson et al.(2013) to perform posterior inference.

source('code/poisson_mean/neg_binom_mean_polya_gamma.R')
set.seed(12345)
n = 1000 
w = 0.2
mu = c(rep(0.1,n*(1-w)),rep(10,n*w))
x = rnbinom(n, size = 10, mu=mu)

When r is known.

Assume r is given, estimate p.

point_laplace, mean = estimate

out = nb_mean_polya_gamma(x=x,r=10,
                          maxiter = 2000,
                          tol=1e-8,
                          ebnm_params=list(mode='estimate',prior_family='point_laplace'))
plot(out$obj,type='l',ylab='ELBO',xlab='iteration')

out$ebnm_res$fitted_g
$pi
[1] 0.7049466 0.2950534

$mean
[1] -3.925342 -3.925342

$scale
[1] 0.000000 2.630529

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2
plot(x,main='',col='grey80',pch=20)
lines(mu,col='grey50')
lines(out$mean_est,col=4)
legend('topleft',c('data','true mean','estimated mean'),pch=c(20,NA,NA),lty=c(NA,1,1),col=c('grey80','grey50',4))

point_laplace, mean = 0

out = nb_mean_polya_gamma(x=x,r=10,
                          maxiter = 2000,
                          tol=1e-8,
                          ebnm_params=list(mode=0,prior_family='point_laplace'))
plot(out$obj,type='l',ylab='ELBO',xlab='iteration')

out$ebnm_res$fitted_g
$pi
[1] 8.149807e-07 9.999992e-01

$mean
[1] 0 0

$scale
[1] 0.000000 2.984682

attr(,"class")
[1] "laplacemix"
attr(,"row.names")
[1] 1 2
plot(x,main='',col='grey80',pch=20)
lines(mu,col='grey50')
lines(out$mean_est,col=4)
legend('topleft',c('data','true mean','estimated mean'),pch=c(20,NA,NA),lty=c(NA,1,1),col=c('grey80','grey50',4))

When r is unknown, estimating r is much harder

We start with letting r_init = true r = 10,

out = nb_mean_polya_gamma(x=x,r=10,est_r=T,update_r_every = 1,
                          maxiter = 300,
                          ebnm_params=list(mode='estimate',prior_family='point_laplace'))
plot(out$obj,type='l',ylab='ELBO',xlab='iteration')

out$r
[1] 0.117094
out$r_trace
 [1] 10.0000000  5.3994464  4.2571362  3.7459465  3.4417621  3.2254966
 [7]  3.0513310  2.8979003  2.7537618  2.6118909  2.4670139  2.3137253
[13]  2.1435736  1.9312473  1.6695600  1.4635529  1.2882753  1.1361468
[19]  1.0028800  0.8852373  0.7803753  0.6855529  0.5978500  0.5137165
[25]  0.4278534  0.3297926  0.2073982  0.1621014  0.1482636  0.1398205
[31]  0.1341019  0.1300825  0.1271699  0.1250137  0.1231944  0.1219864
[37]  0.1209781  0.1202742  0.1197466  0.1191310  0.1188525  0.1186333
[43]  0.1184591  0.1183066  0.1134649  0.1170940
plot(x,main='',col='grey80',pch=20)
lines(mu,col='grey50')
lines(out$mean_est,col=4)
legend('topleft',c('data','true mean','estimated mean'),pch=c(20,NA,NA),lty=c(NA,1,1),col=c('grey80','grey50',4))

It gets stuck at a local optimum - mean converges to constant, r converges to 0.

Maybe this is due to the inaccurate posterior mean and variance at the beginning, we can try to update r after a couple iterations and see what will happen.

out = nb_mean_polya_gamma(x=x,r=10,est_r=T,update_r_every = 10,
                          tol=1e-8,
                          maxiter = 2000,
                          ebnm_params=list(mode='estimate',prior_family='point_laplace'))
plot(out$obj,type='l',ylab='ELBO',xlab='iteration')

out$r
[1] 0.2891371
out$r_trace
 [1] 10.0000000  9.2062539  8.8316224  8.5246025  8.2358090  7.9567243
 [7]  7.6856679  7.4221153  7.1657307  6.9162184  6.6732948  6.4366781
[13]  6.2060935  5.9812643  5.7619190  5.5477817  5.3385686  5.1339978
[19]  4.9337735  4.7375875  4.5451150  4.3560113  4.1698981  3.9863607
[25]  3.8049339  3.6250867  3.4461748  3.2674442  3.0879181  2.9063051
[31]  2.7207795  2.5285211  2.3245429  2.0972697  1.7649082  1.4894809
[37]  1.2621661  1.0746246  0.9193255  0.7895226  0.6791600  0.5826206
[43]  0.4940814  0.4054052  0.2891371
plot(x,main='',col='grey80',pch=20)
lines(mu,col='grey50')
lines(out$mean_est,col=4)
legend('topleft',c('data','true mean','estimated mean'),pch=c(20,NA,NA),lty=c(NA,1,1),col=c('grey80','grey50',4))

out = nb_mean_polya_gamma(x=x,r=10,est_r=T,update_r_every = 20,
                          maxiter = 2000,
                          tol=1e-8,
                          ebnm_params=list(mode='estimate',prior_family='point_laplace'))
plot(out$obj,type='l',ylab='ELBO',xlab='iteration')

out$r
[1] 2.075251
out$r_trace
 [1] 10.000000  9.568768  9.238541  8.922058  8.615424  8.318104  8.029711
 [8]  7.749870  7.478213  7.214412  6.958113  6.708991  6.466721  6.230989
[15]  6.001481  5.777892  5.559911  5.347226  5.139526  4.936488  4.737777
[22]  4.543045  4.351924  4.164013  3.978873  3.796019  3.614893  3.434830
[29]  3.255042  3.074518  2.891917  2.705337  2.511818  2.306060  2.075251
plot(x,main='',col='grey80',pch=20)
lines(mu,col='grey50')
lines(out$mean_est,col=4)
legend('topleft',c('data','true mean','estimated mean'),pch=c(20,NA,NA),lty=c(NA,1,1),col=c('grey80','grey50',4))

increase n

set.seed(12345)
n = 10000
w = 0.2
mu = c(rep(0.1,n*(1-w)),rep(10,n*w))
x = rnbinom(n, size = 10, mu=mu)
out = nb_mean_polya_gamma(x=x,r=10,est_r=T,update_r_every = 20,
                          maxiter = 2000,
                          tol=1e-8,
                          ebnm_params=list(mode='estimate',prior_family='point_laplace'))
plot(out$obj,type='l',ylab='ELBO',xlab='iteration')

out$r
[1] 1.744737
out$r_trace
 [1] 10.000000  9.595019  9.291391  9.000791  8.718828  8.445000  8.178980
 [8]  7.920465  7.669183  7.424824  7.187114  6.955787  6.730584  6.511253
[15]  6.297550  6.089234  5.886059  5.687790  5.494193  5.305028  5.120051
[22]  4.939017  4.761670  4.587746  4.416969  4.249042  4.083652  3.920458
[29]  3.759085  3.599111  3.440046  3.281320  3.122231  2.961903  2.799168
[36]  2.632390  2.459088  2.274988  2.070797  1.744737
plot(x,main='',col='grey80',pch=20)
lines(mu,col='grey50')
lines(out$mean_est,col=4)
legend('topleft',c('data','true mean','estimated mean'),pch=c(20,NA,NA),lty=c(NA,1,1),col=c('grey80','grey50',4))

m = out$m
v = out$v
delta = -sum(m)/2-sum(log(cosh(sqrt(m^2+v)/2)))-n*log(2)
r_vec = 1:50
obj_val = c()
for(r in r_vec){
  obj_val = c(obj_val,Fr(log(r),x,m,v,delta))
}
plot(r_vec,obj_val,type='l')

obj_val = c()
for(r in r_vec){
  obj_val = c(obj_val,Fr_d1(log(r),x,m,v,delta))
}
plot(r_vec,obj_val,type='l')


sessionInfo()
R version 4.2.1 (2022-06-23 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 22000)

Matrix products: default

locale:
[1] LC_COLLATE=English_United States.utf8 
[2] LC_CTYPE=English_United States.utf8   
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.utf8    

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

other attached packages:
[1] ebnm_1.0-11     workflowr_1.7.0

loaded via a namespace (and not attached):
 [1] tidyselect_1.1.2  xfun_0.32         bslib_0.4.0       ashr_2.2-54      
 [5] purrr_0.3.4       splines_4.2.1     lattice_0.20-45   colorspace_2.0-3 
 [9] vctrs_0.4.1       generics_0.1.3    htmltools_0.5.3   yaml_2.3.5       
[13] utf8_1.2.2        rlang_1.0.5       mixsqp_0.3-43     jquerylib_0.1.4  
[17] later_1.3.0       pillar_1.8.1      glue_1.6.2        trust_0.1-8      
[21] lifecycle_1.0.2   stringr_1.4.1     munsell_0.5.0     gtable_0.3.1     
[25] evaluate_0.16     knitr_1.40        callr_3.7.2       fastmap_1.1.0    
[29] httpuv_1.6.5      ps_1.7.1          invgamma_1.1      irlba_2.3.5      
[33] fansi_1.0.3       highr_0.9         Rcpp_1.0.9        scales_1.2.1     
[37] promises_1.2.0.1  cachem_1.0.6      horseshoe_0.2.0   jsonlite_1.8.0   
[41] truncnorm_1.0-8   fs_1.5.2          deconvolveR_1.2-1 ggplot2_3.3.6    
[45] digest_0.6.29     stringi_1.7.8     processx_3.7.0    dplyr_1.0.10     
[49] getPass_0.2-2     rprojroot_2.0.3   grid_4.2.1        cli_3.3.0        
[53] tools_4.2.1       magrittr_2.0.3    sass_0.4.2        tibble_3.1.8     
[57] whisker_0.4       pkgconfig_2.0.3   Matrix_1.4-1      SQUAREM_2021.1   
[61] rmarkdown_2.16    httr_1.4.4        rstudioapi_0.14   R6_2.5.1         
[65] git2r_0.30.1      compiler_4.2.1