Last updated: 2020-10-09

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Here we demonstrate the variational Gaussian approximation for the Poisson-normal in the simplest case when there is one unknown. Under the data model, the counts \(y_1, \ldots, y_n\) are Poisson with rates \(\lambda_1, \ldots, \lambda_n\), in which \(\log \lambda_i = b_0 + x_i b\). The unknown \(b\) is assigned a normal prior with zero mean and standard deviation \(\sigma_0\). Here we use variational methods to approximate the posterior of \(b\) with a normal density \(N(b; \mu, s^2)\).

Load the functions implementing the variational inference algorithms and set the seed.

source("../code/vgapois.R")
set.seed(1)

Simulate data

Simulate counts from the following Poisson model: \(y_i \sim \mathrm{Poisson}(\lambda_i)\), in which \(\log \lambda_i = b_0 + b x_i\).

n  <- 10
b0 <- -1
b  <- 1.5
x  <- rnorm(n)
r  <- exp(b0 + x*b)
y  <- rpois(n,r)

Compute Monte Carlo estimate of marginal likelihood

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   <- 3
ns   <- 1e5
b    <- rnorm(ns,sd = sqrt(s0))
logw <- rep(0,ns)
for (i in 1:ns)
  logw[i] <- compute_loglik_pois(x,y,b0,b[i])
a    <- max(logw)
logZ <- log(mean(exp(logw - a))) + a

Fit variational approximation

Fit the variational Gaussian approximation by optimizing the variational lower bound (the “ELBO”).

fit <- vgapois1(x,y,b0,s0)
cat(sprintf("Monte Carlo estimate:    %0.12f\n",logZ))
cat(sprintf("Variational lower bound: %0.12f\n",-fit$value))
# final  value 10.159268 
# converged
# Monte Carlo estimate:    -10.138730338819
# Variational lower bound: -10.159267582586

Compare exact and approximate posterior distributions

Plot the exact posterior density (dark blue), and compare it against the variational Gaussian approximation (magenta).

ns   <- 1000
b    <- seq(-1,3,length.out = ns)
logp <- rep(0,ns)
for (i in 1:ns)
  logp[i] <- compute_logp_pois(x,y,b0,b[i],s0)
par(mar = c(2,2,0,0))
plot(b,exp(logp - max(logp)),type = "l",lwd = 2,col = "darkblue",
     xlab = "b",ylab = "posterior")
mu <- fit$par["mu"]
s  <- fit$par["s"]
pv <- dnorm(b,mu,sqrt(s))
lines(b,pv/max(pv),col = "magenta",lwd = 2)

Version Author Date
c02efb3 Peter Carbonetto 2020-10-09
03de25f Peter Carbonetto 2020-10-09 

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     
# 
# 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