log link benchmarking of Poisson means
DongyueXie
2022-11-05
Last updated: 2022-11-09
Checks: 7 0
Knit directory: gsmash/
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Introduction
We compare the methods with log-link on estimating the latent \(\mu\) under the following simulation settings. We generate \(n=1000\) samples from \(y_j\sim \Poi(\exp(\mu_j))\), and \(\mu_j\) are generated under the following different data-generating distributions. Each simulation was repeated for 50 times.
library(vebpm)
library(ggplot2)
library(tidyverse)── Attaching packages ─────────────────────────────────────── tidyverse 1.3.2 ──
✔ tibble 3.1.8 ✔ dplyr 1.0.10
✔ tidyr 1.2.1 ✔ stringr 1.4.1
✔ readr 2.1.3 ✔ forcats 0.5.2
✔ purrr 0.3.5
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()source('code/poisson_mean/simulation_summary.R')\(\mu_j \sim \pi_0\delta_0 + \pi_1N(0,2)\).
out = readRDS('output/poisson_mean_simulation/poisson_mean/log_link50_n_1000_priormean_0_priorvar1_2.rds')get_summary_mean(out)
| | mean| sd|
|:-----------------|-----:|-----:|
|split_mixture | 0.621| 0.190|
|ash_pois_identity | 0.649| 0.184|
|GMGM | 0.651| 0.219|
|penalty_inversion | 0.667| 0.200|
|GMGM_pointmass | 0.686| 0.198|
|penalty_compound | 0.717| 0.204|
|GMG | 0.739| 0.223|
|log1exp | 0.745| 0.202|
|split | 0.750| 0.193|
|GG | 0.804| 0.251|
|ebpm_exp_mixture | 0.812| 0.197|
|nb_pg | 1.425| 0.646|
|nb_lb | 1.425| 0.646|
|ash_pois_log | 1.519| 0.198|
|ebpm_gamma | 2.235| 2.098|Warning: Removed 5 rows containing non-finite values (stat_boxplot).
get_summary_mean_log(out,rm_method = 'ash_pois_log')
| | mean| sd|
|:-----------------|-----:|-----:|
|split_mixture | 0.237| 0.027|
|ash_pois_identity | 0.249| 0.027|
|penalty_inversion | 0.252| 0.030|
|penalty_compound | 0.252| 0.038|
|GMGM | 0.261| 0.041|
|GMGM_pointmass | 0.265| 0.032|
|split | 0.266| 0.031|
|log1exp | 0.272| 0.032|
|GG | 0.319| 0.038|
|GMG | 0.327| 0.059|
|nb_lb | 0.352| 0.047|
|nb_pg | 0.352| 0.047|
|ebpm_exp_mixture | 0.687| 0.043|
|ebpm_gamma | 0.689| 0.290|Warning: Removed 5 rows containing non-finite values (stat_boxplot).
plot(out$sim_data$log_Mean[1,],col='grey80',main='log_mean',ylab='')
plot(out$sim_data$Mean[1,],col='grey80',main='log_mean',ylab='')
\(\mu_j \sim \pi_0\delta_3 + \pi_1N(3,2)\)
out = readRDS('output/poisson_mean_simulation/poisson_mean/log_link50_n_1000_priormean_3_priorvar1_2.rds')get_summary_mean(out)
| | mean| sd|
|:-----------------|------:|------:|
|penalty_compound | 13.386| 2.754|
|split_mixture | 13.389| 2.768|
|ash_pois_log | 13.543| 2.694|
|GMGM_pointmass | 13.605| 2.767|
|GMGM | 14.103| 3.020|
|ash_pois_identity | 14.758| 3.537|
|GMG | 17.478| 6.765|
|split | 17.688| 2.884|
|penalty_inversion | 18.459| 4.435|
|log1exp | 24.120| 4.152|
|GG | 24.304| 2.911|
|ebpm_exp_mixture | 25.635| 2.636|
|ebpm_gamma | 35.205| 7.924|
|nb_lb | 48.847| 19.043|
|nb_pg | 48.862| 19.035|Warning: Removed 34 rows containing non-finite values (stat_boxplot).
get_summary_mean_log(out)
| | mean| sd|
|:-----------------|-----:|-----:|
|split_mixture | 0.029| 0.004|
|penalty_compound | 0.029| 0.004|
|GMGM_pointmass | 0.030| 0.004|
|GMGM | 0.031| 0.004|
|ash_pois_identity | 0.032| 0.005|
|penalty_inversion | 0.038| 0.006|
|GMG | 0.038| 0.006|
|split | 0.054| 0.007|
|ebpm_gamma | 0.059| 0.005|
|ebpm_exp_mixture | 0.064| 0.004|
|log1exp | 0.065| 0.005|
|GG | 0.077| 0.010|
|ash_pois_log | 0.097| 0.044|
|nb_lb | 0.225| 0.037|
|nb_pg | 0.225| 0.037|Warning: Removed 34 rows containing non-finite values (stat_boxplot).
\(\mu_j \sim \pi_0\delta_5 + \pi_1N(5,2)\)
out = readRDS('output/poisson_mean_simulation/poisson_mean/log_link50_n_1000_priormean_5_priorvar1_2.rds')get_summary_mean(out,rm_method = c('nb_lb','nb_pg'))
| | mean| sd|
|:-----------------|-------:|------:|
|GMGM_pointmass | 91.837| 28.550|
|GMGM | 93.187| 29.139|
|split_mixture | 93.566| 29.166|
|penalty_compound | 94.482| 34.093|
|ash_pois_identity | 96.301| 33.296|
|GMG | 105.991| 34.532|
|split | 118.585| 28.737|
|ash_pois_log | 125.847| 38.735|
|penalty_inversion | 188.016| 36.860|
|GG | 196.293| 29.259|
|ebpm_exp_mixture | 197.758| 27.850|
|ebpm_gamma | 210.223| 37.791|
|log1exp | NaN| NA|Warning: Removed 50 rows containing non-finite values (stat_boxplot).
get_summary_mean_log(out,rm_method = c('nb_lb','nb_pg'))
| | mean| sd|
|:-----------------|-----:|-----:|
|ash_pois_identity | 0.004| 0.001|
|penalty_compound | 0.004| 0.001|
|GMGM_pointmass | 0.004| 0.001|
|split_mixture | 0.004| 0.001|
|GMG | 0.004| 0.001|
|split | 0.006| 0.001|
|GMGM | 0.006| 0.009|
|penalty_inversion | 0.008| 0.001|
|ebpm_gamma | 0.009| 0.001|
|ash_pois_log | 0.009| 0.012|
|ebpm_exp_mixture | 0.009| 0.001|
|GG | 0.012| 0.003|
|log1exp | NaN| NA|Warning: Removed 50 rows containing non-finite values (stat_boxplot).
Why NB method performs bad?
Its performance is getting worse as x getting larger.
I set \(r = 2*max(y)\). So sometimes \(r\) is of order \(10^5\), which is too large. We see from the results that setting \(r\) large likely let veb algorithm get stuck at local optimum.
Let see 4 examples when NB methods perform worst of itself.
par(mfrow=(c(2,2)))
for(i in c(1,21,23,24)){
plot(out$sim_data$log_Mean[i,],col='grey80')
lines(out$output[[i]]$fitted_model$nb_lb$posterior$mean_log)
}
The corresponding \(r\) are
for(i in c(1,21,23,24)){
print(2*(max(out$sim_data$X[i,])))
}[1] 33206
[1] 25388
[1] 39004
[1] 33638Let’s reduce \(r\) to 1000 for the 1st simulation and see it improves
for(i in c(1,21,23,24)){
temp = nb_mean_lower_bound(out$sim_data$X[i,],r=1000)
plot(out$sim_data$log_Mean[i,],col='grey80')
lines(temp$posterior$mean_log)
print(mse(out$sim_data$log_Mean[i,],temp$posterior$mean_log))
}
[1] 0.005603312
[1] 0.004091247
[1] 0.003240434
[1] 0.003374317Further reduce \(r\) to 100
for(i in c(1,21,23,24)){
temp = nb_mean_lower_bound(out$sim_data$X[i,],r=100)
plot(out$sim_data$log_Mean[i,],col='grey80')
lines(temp$posterior$mean_log)
print(mse(out$sim_data$log_Mean[i,],temp$posterior$mean_log))
}Warning in nb_mean_lower_bound(out$sim_data$X[i, ], r = 100): An iteration
decreases ELBO. This is likely due to numerical issues.
[1] 0.00644856
[1] 0.004579017[1] 0.00372007Warning in nb_mean_lower_bound(out$sim_data$X[i, ], r = 100): An iteration
decreases ELBO. This is likely due to numerical issues.
[1] 0.003922867It seems that setting \(r=1000\) is slightly better than \(100\), but both are significantly better than \(r\sim10^5\).
What happend to inversion penalization method?
Its performance is getting worse as x getting larger.
It seems that it has some convergence issues here. The estimated prior weights are still at the uniform initialization stage.
out$output[[1]]$fitted_model$penalty_inversion$fitted_g$weight
[1] 0.03105418 0.03105418 0.03105419 0.03105419 0.03105420 0.03105423
[7] 0.03105427 0.03105435 0.03105453 0.03105487 0.03105555 0.03105691
[13] 0.03105960 0.03106485 0.03107505 0.03109569 0.03114548 0.03128040
[19] 0.03154961 0.03183850 0.03196651 0.03191920 0.03178937 0.03164999
[25] 0.03152506 0.03141308 0.03131381 0.03123111 0.03116660 0.03111848
[31] 0.03108351 0.03105844
$mean
[1] 4.996289
$sd
[1] 0.000000e+00 5.930995e-04 8.387693e-04 1.186199e-03 1.677539e-03
[6] 2.372398e-03 3.355077e-03 4.744796e-03 6.710154e-03 9.489591e-03
[11] 1.342031e-02 1.897918e-02 2.684062e-02 3.795837e-02 5.368124e-02
[16] 7.591673e-02 1.073625e-01 1.518335e-01 2.147249e-01 3.036669e-01
[21] 4.294499e-01 6.073339e-01 8.588998e-01 1.214668e+00 1.717800e+00
[26] 2.429335e+00 3.435599e+00 4.858671e+00 6.871198e+00 9.717342e+00
[31] 1.374240e+01 1.943468e+01Check the gradient of posterior mean
plot(vebpm:::f_obj_grad(out$output[[1]]$fitted_model$penalty_inversion$fit$optim_fit$par,out$sim_data$X[1,],out$output[[1]]$fitted_model$penalty_inversion$fitted_g$sd)[1:1000],ylab='gradient')
Check the gradient of prior weight
plot(vebpm:::f_obj_grad(out$output[[1]]$fitted_model$penalty_inversion$fit$optim_fit$par,out$sim_data$X[1,],out$output[[1]]$fitted_model$penalty_inversion$fitted_g$sd)[1001:1032],ylab='gradient')
Check the gradient of prior mean
plot(vebpm:::f_obj_grad(out$output[[1]]$fitted_model$penalty_inversion$fit$optim_fit$par,out$sim_data$X[1,],out$output[[1]]$fitted_model$penalty_inversion$fitted_g$sd)[1033],ylab='gradient')
It’s not converging! Because the LBFGSB in optim uses
relative tol(tol*f_value) for monitoring the convergence and the f_value
is too large so iterations stop before convergence.
Need to increase number of iterations.
temp= pois_mean_penalized_inversion(out$sim_data$X[1,],tol=1e-10)
temp$fitted_g$weight
mse(temp$posterior$mean,out$sim_data$Mean[1,])
sessionInfo()R version 4.2.1 (2022-06-23)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 20.04.5 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.9.0
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.9.0
locale:
[1] LC_CTYPE=C.UTF-8 LC_NUMERIC=C LC_TIME=C.UTF-8
[4] LC_COLLATE=C.UTF-8 LC_MONETARY=C.UTF-8 LC_MESSAGES=C.UTF-8
[7] LC_PAPER=C.UTF-8 LC_NAME=C LC_ADDRESS=C
[10] LC_TELEPHONE=C LC_MEASUREMENT=C.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] forcats_0.5.2 stringr_1.4.1 dplyr_1.0.10 purrr_0.3.5
[5] readr_2.1.3 tidyr_1.2.1 tibble_3.1.8 tidyverse_1.3.2
[9] ggplot2_3.3.6 vebpm_0.1.6 workflowr_1.7.0
loaded via a namespace (and not attached):
[1] matrixStats_0.62.0 fs_1.5.2 lubridate_1.9.0
[4] httr_1.4.4 rprojroot_2.0.3 tools_4.2.1
[7] backports_1.4.1 bslib_0.4.0 utf8_1.2.2
[10] R6_2.5.1 irlba_2.3.5.1 DBI_1.1.3
[13] colorspace_2.0-3 withr_2.5.0 tidyselect_1.2.0
[16] processx_3.7.0 ebpm_0.0.1.3 compiler_4.2.1
[19] git2r_0.30.1 rvest_1.0.3 cli_3.4.1
[22] xml2_1.3.3 labeling_0.4.2 horseshoe_0.2.0
[25] sass_0.4.2 scales_1.2.1 SQUAREM_2021.1
[28] callr_3.7.2 mixsqp_0.3-43 digest_0.6.29
[31] rmarkdown_2.17 deconvolveR_1.2-1 pkgconfig_2.0.3
[34] htmltools_0.5.3 highr_0.9 dbplyr_2.2.1
[37] fastmap_1.1.0 invgamma_1.1 rlang_1.0.6
[40] readxl_1.4.1 rstudioapi_0.14 farver_2.1.1
[43] jquerylib_0.1.4 generics_0.1.3 jsonlite_1.8.2
[46] googlesheets4_1.0.1 magrittr_2.0.3 Matrix_1.5-1
[49] Rcpp_1.0.9 munsell_0.5.0 fansi_1.0.3
[52] lifecycle_1.0.3 stringi_1.7.8 whisker_0.4
[55] yaml_2.3.5 nleqslv_3.3.3 rootSolve_1.8.2.3
[58] plyr_1.8.7 grid_4.2.1 parallel_4.2.1
[61] promises_1.2.0.1 crayon_1.5.2 lattice_0.20-45
[64] haven_2.5.1 splines_4.2.1 hms_1.1.2
[67] knitr_1.40 ps_1.7.1 pillar_1.8.1
[70] reshape2_1.4.4 reprex_2.0.2 glue_1.6.2
[73] evaluate_0.17 trust_0.1-8 getPass_0.2-2
[76] modelr_0.1.9 vctrs_0.4.2 nloptr_2.0.3
[79] tzdb_0.3.0 httpuv_1.6.6 cellranger_1.1.0
[82] gtable_0.3.1 ebnm_1.0-9 assertthat_0.2.1
[85] ashr_2.2-54 cachem_1.0.6 xfun_0.33
[88] broom_1.0.1 later_1.3.0 googledrive_2.0.0
[91] gargle_1.2.1 truncnorm_1.0-8 timechange_0.1.1
[94] ellipsis_0.3.2