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 \(b\) 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 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 mean and covariance:
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