Last updated: 2020-02-20

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Here we compare two different ways of updating the mixture weights in mr-ash: using EM, and using mix-SQP.

Script parameters

These are the data simulation settings.

I suggest trying this example with s = 0 (all predictors are independent) and s = 0.4 (the correlation between all predictors is 0.4).

n  <- 500
p  <- 100
sd <- c(0,   1,    2)
w  <- c(0.9, 0.05, 0.05)
s  <- 0

This specifies the variances for the mixture-of-normals prior on the regression coefficients.

s0 <- 10^seq(-4,0,length.out = 12)

Load functions

These are the packages used in this analysis.

library(ggplot2)
library(cowplot)
library(MASS)
library(mixsqp)

This R code provides a very simple implementation of the mr-ash algorithm.

source("../code/misc.R")
source("../code/mr_ash.R")

Simulate data

The predictors are drawn from the multivariate normal with zero mean and covariance matrix S, in which all diagonal entries are 1, and all off-diagonal entries are s. Setting s = 0.5 reproduces the simulation of the predictors used in Example 3 of Zou & Hastie (2005).

set.seed(2)
S       <- matrix(s,p,p)
diag(S) <- 1
X       <- mvrnorm(n,rep(0,p),S)
k       <- sample(length(w),p,replace = TRUE,prob = w)
beta    <- sd[k] * rnorm(p)
y       <- drop(X %*% beta + rnorm(n))

Fit model

These are the initial estimates of residual variance (s), mixture weights (w0), and posterior mean estimates of the regression coefficients (b).

k  <- length(s0)
se <- 1
w0 <- rep(1/k,k)
b  <- rep(0,p)

Fit the model by running 200 EM updates for the mixture weights.

fit1 <- mr_ash(X,y,se,s0,w0,b,maxiter = 200,verbose = FALSE)

Fit the model a second time using the mix-SQP updates for the mixture weights.

fit2 <- mr_ash_with_mixsqp(X,y,se,s0,w0,b,numiter = 10,maxiter.inner = 50,
                           tol.inner = 1e-8)
# iter                elbo max|w0-w0'| niter
#  100 -7.873409197069e+02 7.60892e-01    15
#  100 -7.597870942033e+02 3.92902e-02    11
#  100 -7.596906035040e+02 1.10491e-04     9
#  100 -7.596906001827e+02 4.52321e-07     6
#  100 -7.596906001829e+02 2.11100e-09     3
#  100 -7.596906001829e+02 4.93716e-12     1
#  100 -7.596906001829e+02 3.64420e-12     1
#  100 -7.596906001829e+02 1.62259e-13     1
#  100 -7.596906001829e+02 1.62134e-13     1
#  100 -7.596906001829e+02 3.88856e-14     1

Review model fit

Compare the posterior mean estimates from the two fits.

ggplot(data.frame(em = fit1$b,mixsqp = fit2$b),
             aes(x = em,y = mixsqp)) +
  geom_point(color = "darkblue") +
  geom_abline(intercept = 0,slope = 1,col = "magenta",lty = "dotted") +
  labs(title = "coefs") +
  theme_cowplot()

Compare the posterior mean estimates against the values used to simulate the data.

ggplot(data.frame(true = beta,mixsqp = fit2$b),
             aes(x = true,y = mixsqp)) +
  geom_point(color = "darkblue") +
  geom_abline(intercept = 0,slope = 1,col = "magenta",lty = "dotted") +
  labs(title = "coefs") +
  theme_cowplot()

Plot the improvement in the solution over time. The EM updates are shown in blue, and the mix-SQP updates are shown in orange.

elbo.best <- max(c(fit1$elbo,fit2$elbo))
pdat      <- rbind(data.frame(update = "em",
                              iter   = 1:length(fit1$elbo),
                              elbo   = fit1$elbo),
                   data.frame(update = "mixsqp",
                              iter   = cumsum(fit2$niter),
                              elbo   = fit2$elbo))
pdat$elbo <- elbo.best - pdat$elbo + 1e-4
ggplot(pdat,aes(x = iter,y = elbo,color = update)) +
  geom_line() +
  geom_point() +
  scale_y_log10() +
  scale_color_manual(values = c("royalblue","darkorange")) +
  labs(y = "distance to \"best\" elbo",title = paste("s =",s)) +
  theme_cowplot()


sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.3
# 
# Matrix products: default
# BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/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] mixsqp_0.3-17 MASS_7.3-51.4 cowplot_1.0.0 ggplot2_3.2.1
# 
# loaded via a namespace (and not attached):
#  [1] Rcpp_1.0.3       compiler_3.6.2   pillar_1.4.3     later_1.0.0     
#  [5] git2r_0.26.1     workflowr_1.6.0  tools_3.6.2      digest_0.6.23   
#  [9] lattice_0.20-38  evaluate_0.14    lifecycle_0.1.0  tibble_2.1.3    
# [13] gtable_0.3.0     pkgconfig_2.0.3  rlang_0.4.2      Matrix_1.2-18   
# [17] yaml_2.2.0       xfun_0.11        withr_2.1.2      stringr_1.4.0   
# [21] dplyr_0.8.3      knitr_1.26       fs_1.3.1         rprojroot_1.3-2 
# [25] grid_3.6.2       tidyselect_0.2.5 glue_1.3.1       R6_2.4.1        
# [29] rmarkdown_2.0    irlba_2.3.3      farver_2.0.1     purrr_0.3.3     
# [33] magrittr_1.5     whisker_0.4      backports_1.1.5  scales_1.1.0    
# [37] promises_1.1.0   htmltools_0.4.0  assertthat_0.2.1 colorspace_1.4-1
# [41] httpuv_1.5.2     labeling_0.3     stringi_1.4.3    lazyeval_0.2.2  
# [45] munsell_0.5.0    crayon_1.3.4