Last updated: 2019-10-15
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Knit directory: ebpmf_demo/
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| Rmd | 95f7212 | zihao12 | 2019-10-15 | rank1 issue |
In my experiments with rank-1 ebpmf-point-gamma, I observe the following things (reproduced below):
* After the first iteration, ELBO only has small changes (not monotonic!!) and \(\sum_i <l_i>\), \(\sum_j <f_j>\) also only has small changes.
* Different initialization results in similar ELBO (though estimates for L, F are very different individually) adn similar estimate of \(\Lambda\)
first, after writing down the update euqation for \(s : =\sum_i <l_i>\), it should be changing each iteration (it is a very complicated form). If we simplify and suppose the point mass at 0 are all 0, we have \[s_{t+1} = \frac{n a_L + \sum_{ij} X_{ij}}{b_L + \frac{pa_F + \sum_{ij} X_{ij}}{b_F + s_t}}\] As in our data we have \(\sum_{ij} X_{ij >>n a_L, pa_F}\) and \(s >> b_L, b_F\), we can see \(s_{t+1} \approx s_t\). But analytically we cannot draw the equivalence. Thus the changes in ELBO probably are not just due to numerical issues, and we need to checkj if the ELBO is wrong.\
we can draw the connection to the EM for the MLE of PMF (lee’s multiplicative update). In the rank one case, we can arrive at optimal in one step. Why? Since, given \(f_j\)s, the optimal \(l_i\) is \(l_i = \frac{\sum_j X_{ij}}{\sum_j f_{j}}\). This is equivalent to the MLE for the poisson mean problem, where \(x_i := \sum_j X_{ij}\) and \(s_i := \sum_j f_j\). Then we can easily see that \(\sum_i l_i\), \(\sum_j f_j\) stays the same throught iterations, so \(l_i, f_j\) will not be changing after the first iteration. But in ebpmf-point-gamma, we have a nonlinear shrinkage operator on the the MLE result, so we don’t expect to see the same constant bahavior here.
rm(list = ls())
library(NNLM)
#library(gtools)
#library(ebpmf)
simulate_data <- function(n, p, K, params, seed = 123){
set.seed(seed)
L = matrix(rgamma(n = n*K, shape = params$al, rate = params$bl), ncol = K)
F = matrix(rgamma(n = p*K, shape = params$af, rate = params$bf), ncol = K)
Lam = L %*% t(F)
X = matrix(rpois(n*p, Lam), nrow = n)
Y = matrix(rpois(n*p, Lam), nrow = n)
return(list(params = params,Lam = Lam,X = X, Y = Y))
}
n = 100
p = 200
K = 5
params = list(al = 10, bl = 10, af = 10, bf = 10, a = 1)
sim = simulate_data(n, p, K, params)
init1 = list(mean = runif(n, 0, 1))
init100 = list(mean = 100*runif(n, 0, 1))
out_ebpmf1 = ebpmf::ebpmf_rank1_point_gamma(sim$X, maxiter = 10, init = init1, verbose = T)[1] "iter ELBO KL_L KL_F sum_El sum_Ef"
[1] " 1 -511495.957052 568708.0442338070 213.2413395753 49.3418228878 1957.9536062943"
[1] " 2 -511495.957052 568708.0442340682 213.2413395758 49.3418228693 1957.9536071596"
[1] " 3 -511495.957052 568708.0442340955 213.2413395651 49.3418228959 1957.9536060949"
[1] " 4 -511495.957676 568708.0454792172 213.2413395722 49.3418229227 1957.9536051548"
[1] " 5 -511495.957051 568708.0442301770 213.2413395485 49.3418263234 1957.9534701104"
[1] " 6 -511495.957052 568708.0442337667 213.2413395706 49.3418263465 1957.9534691747"
[1] " 7 -511495.957052 568708.0442341160 213.2413395621 49.3418263724 1957.9534681764"
[1] " 8 -511495.957053 568708.0442341956 213.2413395700 49.3418263994 1957.9534670471"
[1] " 9 -511495.957696 568708.0455181362 213.2413395641 49.3418264274 1957.9534660565"
[1] "10 -511495.957052 568708.0442340459 213.2413395680 49.3418318155 1957.9532522913"out_ebpmf100 = ebpmf::ebpmf_rank1_point_gamma(sim$X, maxiter = 10, init = init100, verbose = T)[1] "iter ELBO KL_L KL_F sum_El sum_Ef"
[1] " 1 -511495.957252 568708.0450631959 213.2413423988 4705.7240442140 20.5301032735"
[1] " 2 -511495.957252 568708.0450622196 213.2412989758 4705.7241134741 20.5301017279"
[1] " 3 -511495.956826 568708.0442109936 213.2413422507 4705.7244674216 20.5301014132"
[1] " 4 -511495.957252 568708.0450626983 213.2413418952 4705.7244714504 20.5301014106"
[1] " 5 -511495.957252 568708.0450628576 213.2413424034 4705.7245406111 20.5301011088"
[1] " 6 -511495.957253 568708.0450643523 213.2413424149 4705.7246100241 20.5301008050"
[1] " 7 -511495.957252 568708.0450631579 213.2413309093 4705.7246793734 20.5301008415"
[1] " 8 -511495.957252 568708.0450627547 213.2413424975 4705.7246710149 20.5301005235"
[1] " 9 -511495.957253 568708.0450639572 213.2413423800 4705.7247441811 20.5301002223"
[1] "10 -511495.957252 568708.0450622501 213.2413288995 4705.7248132644 20.5300996850"Let’s see compare the posterior estimate of \(\Lambda\) from different initializations.
lam1 = out_ebpmf1$ql$mean %*% t(out_ebpmf1$qf$mean)
lam100 = out_ebpmf100$ql$mean %*% t(out_ebpmf100$qf$mean)
max(abs((lam1 - lam100)/(lam1 + lam100)))[1] 2.331426e-06
sessionInfo()R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS 10.14
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] NNLM_0.4.2
loaded via a namespace (and not attached):
[1] workflowr_1.4.0 Rcpp_1.0.2 gtools_3.8.1 digest_0.6.21
[5] rprojroot_1.3-2 ebpmf_0.1.0 backports_1.1.5 git2r_0.25.2
[9] magrittr_1.5 evaluate_0.14 ebpm_0.0.0.9001 stringi_1.4.3
[13] fs_1.3.1 whisker_0.3-2 rmarkdown_1.13 tools_3.5.1
[17] stringr_1.4.0 glue_1.3.1 mixsqp_0.1-121 xfun_0.8
[21] yaml_2.2.0 compiler_3.5.1 htmltools_0.3.6 knitr_1.25