Normal mean penalized form optimization
Dongyue Xie
2022-05-25
Last updated: 2023-03-31
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Knit directory: gsmash/
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Introduction
\[y|\mu\sim N(\mu,s^2),\mu\sim g(\cdot)\]
Optimization: \(\theta = E_q\mu\),
\[\min_{\theta,g}h(\theta,g) = \frac{1}{2s^2}(y-\theta)^2+\rho_{g,s}(\theta).\]
\[S_{g,s^2}(z) = z +s^2l'_{NM}(z;g,s^2)\]
The penalty term is only tractable at \(\rho(S(\cdot))\). There are two ways to accommodate this.
Define the penalty evaluated at \(S_{g,s^2}(\cdot)\) as \(c_{g,s^2}(\cdot):=\rho_{g,s^2}(S_{g,s^2}(\cdot))\).
- Compound way
Let \(\theta = S_{g,s^2}(z)\) such that \(\rho_{g,s^2}(\theta) = c_{g,s^2}(z)\), the optimization problem is
\[\min_{z,g} \tilde h(z,g) = \frac{1}{2s^2}(y-S_{g,s^2}(z))^2-l_{NM}(z;g,s^2)-(z-S_{g,s^2}(z))^2/(2s^2)\]
\[\min_{z,g} \tilde h(z,g) = \frac{1}{2s^2}(y-z-s^2l'_{NM}(z;g,s^2))^2-l_{NM}-s^2(l'_{NM})^2/2\]
Then set \(\hat\theta = S(z)\).
- Inversion way
Note that \(c_{g,s^2}(S^{-1}_{g,s^2}(\theta)) = \rho_{g,s^2}(\theta)\), so we can write the optimization problem as
\[\min_{\theta,g} h(\theta,g) = \frac{1}{2s^2}(y-\theta)^2-c_{g,s^2}(S^{-1}_{g,s^2}(\theta))\]
For the inversion method implementation and illustration, see here
library(vebpm)generate data, and get grid by running ash
set.seed(12345)
n = 200
w0 = 0.9
mu = c(rep(0,round(n*w0)),rep(10,n-round(n*w0)))
w_true = c(w0,1-w0)
grid_true = c(0.01,7)
s = rep(1,n)
y = rnorm(n,mu,s)
library(ashr)
grids = ebnm:::get_ashr_grid(y,s)
system.time(fit.ash <- ash(y,s,mixcompdist = 'normal',pointmass=FALSE,prior='uniform',mixsd=grids)) user system elapsed
0.037 0.000 0.037 system.time(fit_inv <- ebnm_penalized_inversion(y,s,g_init = list(sd=grids))) user system elapsed
4.014 0.084 4.098 system.time(fit_compound <- ebnm_penalized_compound(y,s,g_init = list(sd=grids))) user system elapsed
45.542 0.143 45.685 library(ggplot2)
library(gridExtra)
# plot 1
p1 <- ggplot() +
geom_point(aes(x = 1:length(y), y = y), col = 'grey80', pch = 20) +
geom_line(aes(x = 1:length(mu), y = mu), col = 'grey60') +
geom_line(aes(x = 1:length(fit.ash$result$PosteriorMean), y = fit.ash$result$PosteriorMean)) +
labs(title = 'ash', x = '', y = '') +
scale_color_manual(values = c('grey80', 'grey60', 'black')) +
guides(colour = guide_legend(override.aes = list(pch = c(20, NA, NA), lty = c(NA, 1, 1)))) +
theme_bw()
# plot 2
p2 <- ggplot() +
geom_point(aes(x = 1:length(y), y = y), col = 'grey80', pch = 20) +
geom_line(aes(x = 1:length(mu), y = mu), col = 'grey60') +
geom_line(aes(x = 1:length(fit_inv$posterior$mean), y = fit_inv$posterior$mean)) +
labs(title = 'inversion', x = '', y = '') +
scale_color_manual(values = c('grey80', 'grey60', 'black')) +
guides(colour = guide_legend(override.aes = list(pch = c(20, NA, NA), lty = c(NA, 1, 1)))) +
theme_bw()
# plot 3
p3 <- ggplot() +
geom_point(aes(x = 1:length(y), y = y), col = 'grey80', pch = 20) +
geom_line(aes(x = 1:length(mu), y = mu), col = 'grey60') +
geom_line(aes(x = 1:length(fit_compound$posterior$mean), y = fit_compound$posterior$mean)) +
labs(title = 'compound', x = '', y = '') +
scale_color_manual(values = c('grey80', 'grey60', 'black')) +
guides(colour = guide_legend(override.aes = list(pch = c(20, NA, NA), lty = c(NA, 1, 1)))) +
theme_bw()
# arrange the plots
grid.arrange(p1, p2, p3, ncol = 1)
fit.ash$fitted_g$pi [1] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.6639519 0.1973092
[8] 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
[15] 0.1387388 0.0000000 0.0000000 0.0000000fit_inv$fitted_g$pi [1] 1.741773e-07 3.162366e-07 5.916218e-07 2.294661e-06 5.728518e-05
[6] 6.641689e-01 1.970535e-01 6.278710e-07 4.399936e-10 4.788833e-12
[11] 1.370181e-12 3.243880e-12 8.503757e-13 9.971222e-12 1.387154e-01
[16] 8.939451e-07 2.017274e-10 4.877657e-11fit_compound$fitted_g$pi [1] 2.029749e-08 5.524990e-08 1.504608e-07 1.124650e-06 6.832218e-05
[6] 6.641200e-01 1.970686e-01 7.464503e-11 9.854802e-21 1.171594e-27
[11] 6.139681e-31 9.358840e-32 1.014260e-30 1.005443e-24 1.387417e-01
[16] 1.946484e-16 1.105745e-24 2.119400e-28
sessionInfo()R version 4.1.0 (2021-05-18)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: CentOS Linux 7 (Core)
Matrix products: default
BLAS: /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRblas.so
LAPACK: /software/R-4.1.0-no-openblas-el7-x86_64/lib64/R/lib/libRlapack.so
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C LC_TIME=C
[4] LC_COLLATE=C LC_MONETARY=C LC_MESSAGES=C
[7] LC_PAPER=C LC_NAME=C LC_ADDRESS=C
[10] LC_TELEPHONE=C LC_MEASUREMENT=C LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] gridExtra_2.3 ggplot2_3.4.1 ashr_2.2-54 vebpm_0.4.7
[5] workflowr_1.6.2
loaded via a namespace (and not attached):
[1] Rcpp_1.0.10 horseshoe_0.2.0 invgamma_1.1 lattice_0.20-44
[5] rprojroot_2.0.2 digest_0.6.31 utf8_1.2.3 truncnorm_1.0-8
[9] R6_2.5.1 evaluate_0.14 highr_0.9 pillar_1.8.1
[13] rlang_1.0.6 rstudioapi_0.13 ebnm_1.0-11 irlba_2.3.5.1
[17] whisker_0.4 jquerylib_0.1.4 nloptr_1.2.2.2 Matrix_1.5-3
[21] rmarkdown_2.9 labeling_0.4.2 splines_4.1.0 stringr_1.5.0
[25] munsell_0.5.0 mixsqp_0.3-48 compiler_4.1.0 httpuv_1.6.1
[29] xfun_0.24 pkgconfig_2.0.3 SQUAREM_2021.1 htmltools_0.5.4
[33] tidyselect_1.2.0 tibble_3.1.8 matrixStats_0.59.0 fansi_1.0.4
[37] dplyr_1.1.0 withr_2.5.0 later_1.3.0 grid_4.1.0
[41] jsonlite_1.8.4 gtable_0.3.1 lifecycle_1.0.3 git2r_0.28.0
[45] magrittr_2.0.3 scales_1.2.1 cli_3.6.0 stringi_1.6.2
[49] ebpm_0.0.1.3 farver_2.1.1 fs_1.5.0 promises_1.2.0.1
[53] bslib_0.2.5.1 generics_0.1.3 vctrs_0.5.2 trust_0.1-8
[57] tools_4.1.0 glue_1.6.2 parallel_4.1.0 fastmap_1.1.0
[61] yaml_2.3.7 colorspace_2.1-0 deconvolveR_1.2-1 knitr_1.33
[65] sass_0.4.0