Last updated: 2018-10-09

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library(mashr)
Loading required package: ashr

We use EM algorithm to update \(\rho\).

B is the \(n\times R\) true value matrix. \(\mathbf{z}\) is a length n vector.

E step

\[ P(\hat{B},B,\mathbf{z}|\rho, \pi) = \prod_{i=1}^{n} \prod_{p=0}^{P}\left[\pi_{p}N(\hat{b}_{i}; b_{i}, V)N(b_{i}; 0, \Sigma_{p})\right]^{\mathbb{I}(z_{i}=p)} \]

\[ \mathbb{E}_{\mathbf{z},B|\hat{B}} \log P(\hat{B},B,\mathbf{z}|\rho, \pi) = \sum_{i=1}^{n} \sum_{p=0}^{P} \mathbb{E}_{b_{i}|\hat{b}_{i}}\left[ P(z_{i}=p|b_{i})\left( \log \pi_{p} + \log N(\hat{b}_{i}; b_{i}, V) + \log N(b_{i}; 0, \Sigma_{p}) \right) \right] \] \[ \gamma_{z_{i}}(p) = P(z_{i}=p|b_{i}) = \frac{\pi_{p}N(b_{i}; 0, \Sigma_{p})}{\sum_{p'=0}^{P}\pi_{p'}N(b_{i}; 0, \Sigma_{p'})} \]

It is hard to compute the expectation.

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

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] mashr_0.2-15 ashr_2.2-14 

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.19      knitr_1.20        whisker_0.3-2    
 [4] magrittr_1.5      workflowr_1.1.1   MASS_7.3-50      
 [7] pscl_1.5.2        doParallel_1.0.14 SQUAREM_2017.10-1
[10] lattice_0.20-35   foreach_1.4.4     plyr_1.8.4       
[13] stringr_1.3.1     tools_3.5.1       parallel_3.5.1   
[16] grid_3.5.1        R.oo_1.22.0       rmeta_3.0        
[19] git2r_0.23.0      htmltools_0.3.6   iterators_1.0.10 
[22] assertthat_0.2.0  abind_1.4-5       yaml_2.2.0       
[25] rprojroot_1.3-2   digest_0.6.15     Matrix_1.2-14    
[28] codetools_0.2-15  R.utils_2.6.0     evaluate_0.11    
[31] rmarkdown_1.10    stringi_1.2.4     compiler_3.5.1   
[34] backports_1.1.2   R.methodsS3_1.7.1 mvtnorm_1.0-8    
[37] truncnorm_1.0-8

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