Last updated: 2020-10-14

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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 log-rates \(\lambda_1, \ldots, \lambda_n\), in which \(\lambda_i = a_i + x_i b\). 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 \(N(b; \mu, s^2)\). See the Overleaf document for a more detailed description of the model and variational approximation.

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: \(y_i \sim \mathrm{Poisson}(e^{\lambda_i})\), in which \(\lambda_i = a_i + b x_i\).

n <- 10
a <- rnorm(n,mean = -2)
b <- 1
x <- rnorm(n)
r <- a + x*b
y <- rpois(n,exp(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,a,b[i])
d    <- max(logw)
logZ <- log(mean(exp(logw - d))) + d

Fit variational approximation

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

fit <- vgapois1(x,y,a,s0)
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:    -6.020537793023
# Variational lower bound: -6.027544173384

Here we see that the ELBO slightly undershoots the marginal likelihood.

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,a,b[i],s0)
par(mar = c(4,4,1,0))
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
7c3e707 Peter Carbonetto 2020-10-14
cf7ea69 Peter Carbonetto 2020-10-09
7fb8a9e Peter Carbonetto 2020-10-09
c02efb3 Peter Carbonetto 2020-10-09
03de25f Peter Carbonetto 2020-10-09

The true posterior is very much “bell shaped”, so as expected the normal approximation is a good fit to the true posterior.


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