Last updated: 2020-02-20

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html 1b60536 Peter Carbonetto 2020-02-20 Re-built mr_ash_demo after a few minor edits.
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An illustration of the “mr-ash” co-ordinate ascent algorithm applied to a small, simulated data set.

Script parameters

These are the data simulation settings.

n  <- 500
p  <- 1000
sd <- c(0,    1,    2)
w  <- c(0.98, 0.01, 0.01)

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

s0 <- c(0.01,0.5,1)^2

Load functions

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

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

Simulate data

set.seed(1)
X    <- matrix(rnorm(n*p),n,p)
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).

b  <- rep(0,p)
s  <- 1
w0 <- c(0.5,0.25,0.25)

Fit the model by running 20 EM updates.

fit <- mr_ash(X,y,s,s0,w0,b,20)
# iter                elbo max|b-b'|
#    1 -1.756557184890e+03 6.927e-01
#    2 -1.099123931192e+03 4.780e-01
#    3 -8.578720448759e+02 2.738e-01
#    4 -8.091518626699e+02 5.838e-02
#    5 -8.058755580814e+02 9.888e-03
#    6 -8.057118084929e+02 3.166e-03
#    7 -8.056811273699e+02 8.567e-04
#    8 -8.056629085272e+02 3.231e-04
#    9 -8.056505966230e+02 1.970e-04
#   10 -8.056421813525e+02 1.526e-04
#   11 -8.056363928186e+02 1.260e-04
#   12 -8.056323873106e+02 1.056e-04
#   13 -8.056296004148e+02 8.869e-05
#   14 -8.056276519072e+02 7.458e-05
#   15 -8.056262837295e+02 6.277e-05
#   16 -8.056253194612e+02 5.288e-05
#   17 -8.056246376769e+02 4.458e-05
#   18 -8.056241542917e+02 3.761e-05
#   19 -8.056238107616e+02 3.176e-05
#   20 -8.056235661300e+02 2.684e-05

Review model fit

Compare the mr-ash estimates against the values used to simulate the data.

plot(beta,fit$b,pch = 20,col = "black")
abline(a = 0,b = 1,col = "skyblue",lty = "dotted",xlab = "true",
       ylab = "estimated")

Version Author Date
582e465 Peter Carbonetto 2020-02-20

Plot the improvement in the model fit over time, as measured by the ELBO.

plot(max(fit$elbo) - fit$elbo + 1e-5,type = "l",col = "dodgerblue",
     lwd = 2,log = "y",xlab = "iteration",ylab = "distance to \"best\" ELBO")

Version Author Date
582e465 Peter Carbonetto 2020-02-20 

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     
# 
# loaded via a namespace (and not attached):
#  [1] workflowr_1.6.0 Rcpp_1.0.3      rprojroot_1.3-2 digest_0.6.23  
#  [5] later_1.0.0     R6_2.4.1        backports_1.1.5 git2r_0.26.1   
#  [9] magrittr_1.5    evaluate_0.14   stringi_1.4.3   rlang_0.4.2    
# [13] fs_1.3.1        promises_1.1.0  whisker_0.4     rmarkdown_2.0  
# [17] tools_3.6.2     stringr_1.4.0   glue_1.3.1      httpuv_1.5.2   
# [21] xfun_0.11       yaml_2.2.0      compiler_3.6.2  htmltools_0.4.0
# [25] knitr_1.26