Last updated: 2020-06-04

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Rmd 6513b8e Matthew Stephens 2020-06-04 workflowr::wflow_publish(“blasso_bimodal_example.Rmd”)

Introduction

I’m going to look at the example from Park and Casella Figure 4.

This posterior is from the equation (7) in that paper, in the cases \(p=1\) variable (so b is a scalar).

post = function(s2, b, yty=26, xtx=1, xty=5, n=10, lambda=3){
  (1/s2) * ((s2)^(-(n-1)/2))  * exp(-0.5*(1/s2)*(yty+b^2*xtx-2*xty*b) - lambda*abs(b) )
}

This was an attempt to reproduce their Figure 4. It has qualitatively similar features, but the mode near the null is not as big as theirs.

b = seq(-1,6,length=100)
ls2 = seq(-3,2,length=100)
z = matrix(0,nrow=100, ncol=100)
for(i in 1:length(b)){
  for(j in 1:length(ls2)){
    z[i,j] = post(exp(ls2[j]),b[i],lambda=3)
  }
}
contour(b,ls2,z,nlevels = 100,main="lambda=3",xlab="b",ylab="log(sigma^2)")

However, a slight change in lambda produces a more similar plot. It could just be that appearance is sensitive to details of how the contour plot is produced.

for(i in 1:length(b)){
  for(j in 1:length(ls2)){
    z[i,j] = post(exp(ls2[j]),b[i],lambda=3.5)
  }
}
contour(b,ls2,z,nlevels = 20,main="lambda=3.5",xlab="b",ylab="log(sigma^2)")

In sigma space

I was interested to see how the picture changes if we plot in sigma space instead of log(sigma). It is still bimodal.

contour(b,exp(ls2),z,nlevels = 20, main="lambda=3.5",xlab="b",ylab="sigma^2")

Scaled prior

Here we change so that the prior on \(b\) is scaled by \(\sigma\).

post2 = function(s2, b, yty=26, xtx=1, xty=5, n=10, lambda=3){
  (1/s2) * ((s2)^(-(n-1)/2)) * exp(-0.5*(1/s2)*(yty+b^2*xtx-2*xty*b) - (lambda/sqrt(s2))*abs(b) )
}

z2 = matrix(0,nrow=100, ncol=100)
for(i in 1:length(b)){
  for(j in 1:length(ls2)){
    z2[i,j] = post2(exp(ls2[j]),b[i],lambda=3.5)
  }
}
contour(b,ls2,z2)

And try with some other lambda values:

for(i in 1:length(b)){
  for(j in 1:length(ls2)){
    z2[i,j] = post2(exp(ls2[j]),b[i],lambda=1)
  }
}
contour(b,ls2,z2,main="lambda=1",xlab="b",ylab="log(sigma^2)")

contour(b,exp(ls2),z2,main="lambda=1", ,xlab="b",ylab="sigma^2")

for(i in 1:length(b)){
  for(j in 1:length(ls2)){
    z2[i,j] = post2(exp(ls2[j]),b[i],lambda=1.5)
  }
}
contour(b,ls2,z2,main="lambda=1.5",xlab="b",ylab="log(sigma^2)")

contour(b,exp(ls2),z2,main="lambda=1.5", ,xlab="b",ylab="sigma^2")

for(i in 1:length(b)){
  for(j in 1:length(ls2)){
    z2[i,j] = post2(exp(ls2[j]),b[i],lambda=2.5)
  }
}
contour(b,ls2,z2,main="lambda=2.5",xlab="b",ylab="log(sigma^2)")

contour(b,exp(ls2),z2,main="lambda=2.5", ,xlab="b",ylab="sigma^2")


sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6

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):
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[25] knitr_1.28