Last updated: 2020-10-23
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Knit directory: vgapois/analysis/
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Here we demonstrate the variational Gaussian approximation for the multivariate Poisson-normal with two unknowns. Under the data model, the counts \(y_{ij}\) are Poisson with log-rates \(\eta_{ij}\), in which \(\eta_i = a_{ij} + x_{ij} b_j\). The unknown vector \(b\) is assigned a multivariate normal prior with zero mean and covariance \(S_0\). Here we use variational methods to approximate the posterior of \(b\) with a multivariate normal density \(N(b; \mu, S)\). See the Overleaf document for a more detailed description of the model and variational approximation.
Load the mvtnorm package, the functions implementing the variational inference algorithms, and set the seed.
library(mvtnorm)
source("../code/vgapois.R")
set.seed(1)Simulate counts from the Poisson model.
n <- 16
b <- c(1.3,1.5)
A <- matrix(rnorm(2*n,mean = -2),n,2)
X <- matrix(rnorm(2*n),n,2)
Y <- matrix(0,n,2)
R <- A + scalecols(X,b)
Y <- matrix(rpois(2*n,exp(R)),n,2)Here we compute an importance sampling estimate of the marginal log-likelihood. We will compare this against the lower bound to the marginal likelihood obtained by the variational approximation.
S0 <- rbind(c(2,1.9),
c(1.9,2))
ns <- 1e5
B <- rmvnorm(ns,sigma = S0)
logw <- rep(0,ns)
for (i in 1:ns)
logw[i] <- compute_loglik_pois(X,Y,A,B[i,])
d <- max(logw)
logZ <- log(mean(exp(logw - d))) + dCompute importance sampling estimates of the mean and variance.
w <- exp(logw - d)
w <- w/sum(w)
mu.mc <- drop(w %*% B)
S.mc <- crossprod(sqrt(w)*B) - tcrossprod(mu.mc)Fit the variational Gaussian approximation by optimizing the variational lower bound (the “ELBO”).
fit <- vgapois(X,Y,A,S0)
mu <- fit$mu
S <- fit$S
cat(fit$message,"\n")
cat(sprintf("Monte Carlo estimate: %0.12f\n",logZ))
cat(sprintf("Variational lower bound: %0.12f\n",-fit$value))
# CONVERGENCE: REL_REDUCTION_OF_F <= FACTR*EPSMCH
# Monte Carlo estimate: -18.648407180493
# Variational lower bound: -18.661488145893Here we see that the ELBO slightly undershoots the marginal likelihood.
Compare the importance sampling and variational estimates of the posterior mean and covariance. The covariances are very similar, and the means are almost identical.
cat("Monte Carlo estimates:\n")
print(mu.mc)
print(S.mc)
cat("Variational estimates:\n")
print(mu)
print(S)
# Monte Carlo estimates:
# [1] 1.688477 1.405992
# [,1] [,2]
# [1,] 0.12211591 0.02349623
# [2,] 0.02349623 0.04420254
# Variational estimates:
# [1] 1.685749 1.403398
# [,1] [,2]
# [1,] 0.11957952 0.02188348
# [2,] 0.02188348 0.04156955
sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.6
#
# 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
#
# other attached packages:
# [1] mvtnorm_1.0-11
#
# loaded via a namespace (and not attached):
# [1] workflowr_1.6.2.9000 Rcpp_1.0.5 rprojroot_1.3-2
# [4] digest_0.6.23 later_1.0.0 R6_2.4.1
# [7] backports_1.1.5 git2r_0.26.1 magrittr_1.5
# [10] evaluate_0.14 stringi_1.4.3 rlang_0.4.5
# [13] fs_1.3.1 promises_1.1.0 whisker_0.4
# [16] rmarkdown_2.3 tools_3.6.2 stringr_1.4.0
# [19] glue_1.3.1 httpuv_1.5.2 xfun_0.11
# [22] yaml_2.2.0 compiler_3.6.2 htmltools_0.4.0
# [25] knitr_1.26