An example where mr.ash does worse than Lasso
Peter Carbonetto
June 23, 2020
Last updated: 2020-06-24
Checks: 7 0
Knit directory: mr-ash/analysis/
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To our surprise, we discovered that the Lasso (or the Elastic Net) sometimes provides more accurate predictions than mr.ash in examples where there are many predictors having a small effect on the outcome. Here I expand on an example from Matthew’s brief investigation of this phenomenon to better understand the (mis) behaviour of mr.ash.
Load packages
These are the packages used in the analysis.
library(glmnet)
library(varbvs)
library(mr.ash.alpha)
library(ggplot2)
library(cowplot)Simulate data
I simulate the data just as Matthew did, except that I split the data into a training set and a test set.
These are the data simulation settings: the number of samples in training set, “n”; number of simulated variables, “p”; the number of variables affecting the outcome (“p1”); and the proportion of variance in the outcome explained by the variables (“pve”).
n <- 500
p <- 1000
p1 <- 467
pve <- 0.95Simulate a \(2n \times p\) design matrix; the first \(n\) rows is the training set data, and the remaining \(n\) rows are the test data.
set.seed(15)
X <- matrix(rnorm(2*n*p),2*n,p)
X <- scale(X,center = TRUE,scale = TRUE)Simulate the \(p\) regression coefficients; only \(p_1 < p\) of the coefficients are nonzero.
b <- rep(0,p)
j <- sample(p,p1)
b[j] <- rnorm(p1)Simulate the responses so that the target PVE is met.
y <- drop(X %*% b)
se <- sqrt((1 - pve)/pve) * sd(y)
y <- y + rnorm(n,sd = se)Split the data 50-50 into a training set and a test set.
test <- 1:n
Xtest <- X[test,]
ytest <- y[test]
X <- X[-test,]
y <- y[-test]Fit the elastic net and mr.ash models
Fit the Elastic Net model, in which the penalty strength parameter (\(\lambda\)) is chosen via 10-fold cross-validation.
fit.glmnet <- cv.glmnet(X,y,alpha = 0.95,standardize = FALSE)Fit the mr.ash model using the default settings.
fit.mrash <- mr.ash(X,y,standardize = FALSE)Fit the mr.ash model again, but give it some help by providing it with the prior and residual variance used to simulate the data. Also, I initialize the posterior estimates of the regression coefficients are to the Elastic Net estimates.
b <- coef(fit.glmnet)[-1]
w1 <- p1/p
s <- se^2
fit.trueg <- mr.ash(X,y,beta.init = b,update.pi = FALSE,update.sigma2 = FALSE,
sigma2 = s,sa2 = c(0,1/s),pi = c(1 - w1,w1))Now let’s run mr.ash again, providing with some help, but a little less than before: initialize the prior to the settings used to simulate the data, but allow mr.ash to fit the prior. A well-behaving likelihood (ELBO) surface will guide the iterates toward a setting close to the truth.
fit.trueginit <- mr.ash(X,y,beta.init = coef(fit.glmnet)[-1],
update.pi = TRUE,update.sigma2 = FALSE,
sigma2 = s,sa2 = c(0,1/s),pi = c(1 - w1,w1))Evaluate models on test set
Predict the test set outcomes using the fitted models.
y.glmnet <- drop(predict(fit.glmnet,Xtest,s = "lambda.min"))
y.mrash <- predict(fit.mrash,Xtest)
y.trueg <- predict(fit.trueg,Xtest)
y.trueginit <- predict(fit.trueginit,Xtest)Report the accuracy of the test predictions by the root-mean squared error (RMSE).
rmse <- function (x, y) sqrt(mean((x - y)^2))
cat(sprintf("glmnet: %0.3f\n",rmse(ytest,y.glmnet)))
cat(sprintf("mr.ash: %0.3f\n",rmse(ytest,y.mrash)))
cat(sprintf("mr.ash (true prior): %0.3f\n",rmse(ytest,y.trueg)))
cat(sprintf("mr.ash (true prior init): %0.3f\n",rmse(ytest,y.trueginit)))
# glmnet: 15.768
# mr.ash: 18.536
# mr.ash (true prior): 16.432
# mr.ash (true prior init): 19.483A couple surprises
These results are surprising in a couple ways:
The Elastic Net method does very well, despite the fact that we typically think of the method as being best suited for sparse settings in which only a few variables have an effect.
Unsuprisingly, mr.ash does well when the prior is fixed to the true settings. However, initializing the mr.ash prior to the truth, then fitting the prior to the data, does not improve performance at all, and in fact makes things slightly worse (at least in this example).
Let’s investigate this second surprise a little more closely.
Why mr.ash gets \(\pi\) wrong
The (approximate) EM algorithm—which maximizes the ELBO—seems to have identified a local maximum at around \(\pi = 0.25\), whereas the true setting is nearly \(\pi = 0.5\):
fit.trueginit$pi[2]
# [1] 0.2470846We can use varbvs to see how the ELBO changes with \(\pi\):
logodds <- seq(-2,1,length.out = 40)
fit.varbvs <- varbvs(X,NULL,y,sigma = s,sa = 1/s,
logodds = logodds,verbose = FALSE)Indeed, the highest point in the ELBO is around \(\pi = 0.25\), but this is not a case where the objective surface is flat—in fact, it drops off very rapidly as \(\pi\) increases beyond that:
sigmoid10 <- function (x) 1/(1 + 10^(-x))
logw <- fit.varbvs$logw
pdat <- data.frame(w = sigmoid10(logodds),
elbo = logw,sigmoid10(logodds))
ggplot(pdat,aes(x = w,y = elbo)) +
geom_point() +
geom_line() +
scale_x_continuous(trans = "log10",breaks = c(0.001,0.01,0.1,0.5)) +
labs(x = "\u03c0",y = "ELBO") +
theme_cowplot(10)
We attribute this sharp drop-off at larger values to the fact the ELBO, which is a lower bound to the exact marginal log-likelihood, is most accurate for sparser models, and therefore the mr.ash model fitting algorithm will have some bias in favour of sparser models (and, in the case of the spike-and-slab prior, smaller values of \(\pi\)).
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.5
#
# 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] cowplot_1.0.0 ggplot2_3.3.0 mr.ash.alpha_0.1-35
# [4] varbvs_2.6-5 glmnet_4.0-2 Matrix_1.2-18
#
# loaded via a namespace (and not attached):
# [1] shape_1.4.4 tidyselect_0.2.5 xfun_0.11
# [4] purrr_0.3.3 splines_3.6.2 lattice_0.20-38
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# [28] knitr_1.26 httpuv_1.5.2 Rcpp_1.0.3
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# [52] R6_2.4.1 git2r_0.26.1 compiler_3.6.2