MLE

The loglikelihood is (with penalty) \[ l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{k=0}^{K} \pi_{k}N(x_{i}; 0, \Sigma_{k}) + \sum_{k=0}^{K} (\lambda_{k}-1) \log \pi_{k} \]

The penalty on \(\pi\) encourages over-estimation of \(\pi_{0}\), \(\lambda_{k}\geq 1\).

\[ l(\rho, \pi) = \sum_{i=1}^{n} \log \sum_{k=0}^{K} \pi_{k}\frac{1}{2\pi\sigma_{k11}\sigma_{k22}\sqrt{1-\phi_{k}^2}} \exp\left( -\frac{1}{2(1-\phi_{k}^2)}\left[ \frac{x_{i}^2}{\sigma_{k11}^2} + \frac{y_{i}^2}{\sigma_{k22}^2} - \frac{2\phi_{k}x_{i}y_{i}}{\sigma_{k11}\sigma_{k22}}\right] \right) + \sum_{k=0}^{K} (\lambda_{k}-1) \log \pi_{k} \]

Note: This probelm is convex with respect to \(\pi\). In terms of \(\rho\), the covenxity depends on the data.

Algorithm:

Input: X, init_pi, init_rho, Ulist
Compute loglikelihood
delta = 1
while delta > tol
  Given pi, estimate rho by max loglikelihood (optim function)
  Given rho, estimate pi by max loglikelihood (convex problem)
  Compute loglikelihood
  Update delta

Data

\[ \hat{\beta}|\beta \sim N_{2}(\hat{\beta}; \beta, \left(\begin{matrix} 1 & 0.5 \\ 0.5 & 1 \end{matrix}\right)) \]

\[ \beta \sim \frac{1}{4}\delta_{0} + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}\right)) + \frac{1}{4}N_{2}(0, \left(\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}\right)) \]

n = 4000

set.seed(1)
n = 4000; p = 2
Sigma = matrix(c(1,0.5,0.5,1),p,p)
U0 = matrix(0,2,2)
U1 = U0; U1[1,1] = 1
U2 = U0; U2[2,2] = 1
U3 = matrix(1,2,2)
Utrue = list(U0=U0, U1=U1, U2=U2, U3=U3)
data = generate_data(n, p, Sigma, Utrue)

m.data = mash_set_data(data$Bhat, data$Shat)
U.c = cov_canonical(m.data)
grid = mashr:::autoselect_grid(m.data, sqrt(2))
Ulist = mashr:::normalize_Ulist(U.c)
xUlist = mashr:::expand_cov(Ulist,grid,usepointmass =  TRUE)
result <- optimize_pi_rho_times(data$Bhat, xUlist, init_rho = c(-0.7,0,0.7))

Warning in REBayes::KWDual(A, rep(1, k), normalize(w), control = control): estimated mixing distribution has some negative values:
               consider reducing rtol

Warning in mixIP(matrix_lik = structure(c(0.0627889120852815,
0.0114523005348735, : Optimization step yields mixture weights that are
either too small, or negative; weights have been corrected and renormalized
after the optimization.

plot(result[[1]]$loglik)

The estimated \(\rho\) is 0.6124039.

m.data.mle = mash_set_data(data$Bhat, data$Shat, V = matrix(c(1,result[[1]]$rho,result[[1]]$rho,1),2,2))
U.c = cov_canonical(m.data.mle)
m.mle = mash(m.data.mle, U.c, verbose= FALSE)
null.ind = which(apply(data$B,1,sum) == 0)

The log likelihood is -1.23017710^{4}. There are 55 significant samples, 2 false positives. The RRMSE is 0.5964282.

The estimated pi is

barplot(get_estimated_pi(m.mle), las=2, cex.names = 0.7, main='MLE', ylim=c(0,0.8))

The ROC curve:

m.data.correct = mash_set_data(data$Bhat, data$Shat, V=Sigma)
m.correct = mash(m.data.correct, U.c, verbose = FALSE)
m.correct.seq = ROC.table(data$B, m.correct)
m.mle.seq = ROC.table(data$B, m.mle)

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] knitr_1.20       kableExtra_0.9.0 mashr_0.2-11     ashr_2.2-10     

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.18      compiler_3.5.1    pillar_1.3.0     
 [4] plyr_1.8.4        iterators_1.0.10  tools_3.5.1      
 [7] digest_0.6.15     viridisLite_0.3.0 evaluate_0.11    
[10] tibble_1.4.2      lattice_0.20-35   pkgconfig_2.0.1  
[13] rlang_0.2.1       Matrix_1.2-14     foreach_1.4.4    
[16] rstudioapi_0.7    yaml_2.2.0        parallel_3.5.1   
[19] mvtnorm_1.0-8     xml2_1.2.0        httr_1.3.1       
[22] stringr_1.3.1     REBayes_1.3       hms_0.4.2        
[25] rprojroot_1.3-2   grid_3.5.1        R6_2.2.2         
[28] rmarkdown_1.10    rmeta_3.0         readr_1.1.1      
[31] magrittr_1.5      scales_0.5.0      backports_1.1.2  
[34] codetools_0.2-15  htmltools_0.3.6   MASS_7.3-50      
[37] rvest_0.3.2       assertthat_0.2.0  colorspace_1.3-2 
[40] stringi_1.2.4     Rmosek_8.0.69     munsell_0.5.0    
[43] pscl_1.5.2        doParallel_1.0.11 truncnorm_1.0-8  
[46] SQUAREM_2017.10-1 crayon_1.3.4