Last updated: 2020-10-09

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Knit directory: vgapois/analysis/

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Add text here.

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

Simulate data

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

Compute importance sampling estimate of marginal likelihood

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

Fit variational approximation

fit <- vgapois1(x,y,b0,s0)
mu  <- fit$par["mu"]
s   <- fit$par["s"]
print(compute_elbo_vgapois1(x,y,b0,s0,mu,s),digits = 6)
# N = 2, M = 5 machine precision = 2.22045e-16
# L = -inf 1e-15 
# X0 = 0 1 
# U = inf inf 
# At X0, 0 variables are exactly at the bounds
# At iterate     0  f=       12.879  |proj g|=       2.6103
# Iteration     0
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -1.1137e+01  1.1137e+01
# Distance to the next break point =   4.8095e-01
# Distance to the stationary point =   1.0000e+00
# Variable  2  is fixed.
# 
# GCP found in this segment
# Piece      2 f1, f2 at start point -3.5367e+00  6.8139e+00
# Distance to the stationary point =   5.1905e-01
# Cauchy X =  2.61034 1e-15 
# 
# ---------------- exit CAUCHY----------------------
# 
# 1  variables are free at GCP on iteration 1
# LINE SEARCH 0 times; norm of step = 1
# X = 0.933822 0.642261 
# G = 2.77367 4.7579 
# Iteration     1
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -3.0331e+01  4.5078e+02
# Distance to the next break point =   1.3499e-01
# Distance to the stationary point =   6.7285e-02
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -3.0331e+01  4.5078e+02
# Distance to the stationary point =   6.7285e-02
# Cauchy X =  0.747195 0.322126 
# 
# ---------------- exit CAUCHY----------------------
# 
# Variable 2 enters the set of free variables
# 0 variables leave; 1 variables enter
# 2  variables are free at GCP on iteration 2
# LINE SEARCH 1 times; norm of step = 0.253669
# X = 0.987541 0.394346 
# G = 1.61641 3.27392 
# Iteration     2
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -1.3331e+01  1.7626e+02
# Distance to the next break point =   1.2045e-01
# Distance to the stationary point =   7.5634e-02
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -1.3331e+01  1.7626e+02
# Distance to the stationary point =   7.5634e-02
# Cauchy X =  0.865286 0.146727 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 3
# LINE SEARCH 1 times; norm of step = 0.139074
# X = 0.99301 0.255379 
# G = 0.879783 1.98679 
# Iteration     3
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -4.7214e+00  6.2208e+01
# Distance to the next break point =   1.2854e-01
# Distance to the stationary point =   7.5896e-02
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -4.7214e+00  6.2208e+01
# Distance to the stationary point =   7.5896e-02
# Cauchy X =  0.926238 0.104589 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 4
# LINE SEARCH 1 times; norm of step = 0.0908631
# X = 1.01968 0.168519 
# G = 0.648301 0.761998 
# Iteration     4
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -1.0009e+00  1.5697e+01
# Distance to the next break point =   2.2115e-01
# Distance to the stationary point =   6.3768e-02
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -1.0009e+00  1.5697e+01
# Distance to the stationary point =   6.3768e-02
# Cauchy X =  0.97834 0.119928 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 5
# LINE SEARCH 0 times; norm of step = 0.100318
# X = 0.921671 0.14712 
# G = -0.126003 -0.180958 
# Iteration     5
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 0  breakpoints
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -4.8622e-02  9.4136e-01
# Distance to the stationary point =   5.1651e-02
# Cauchy X =  0.92818 0.156466 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 6
# LINE SEARCH 0 times; norm of step = 0.0151149
# X = 0.935665 0.152832 
# G = -0.0114464 0.0223763 
# Iteration     6
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -6.3172e-04  9.0129e-03
# Distance to the next break point =   6.8301e+00
# Distance to the stationary point =   7.0091e-02
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -6.3172e-04  9.0129e-03
# Distance to the stationary point =   7.0091e-02
# Cauchy X =  0.936468 0.151263 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 7
# LINE SEARCH 0 times; norm of step = 0.0042122
# X = 0.939401 0.150886 
# G = 0.0050822 -0.01076 
# Iteration     7
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 0  breakpoints
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -1.4161e-04  2.6096e-03
# Distance to the stationary point =   5.4263e-02
# Cauchy X =  0.939126 0.15147 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 8
# LINE SEARCH 0 times; norm of step = 0.00132129
# X = 0.938236 0.15151 
# G = -3.54695e-06 0.000115329 
# Iteration     8
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -1.3313e-08  3.3087e-07
# Distance to the next break point =   1.3137e+03
# Distance to the stationary point =   4.0237e-02
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -1.3313e-08  3.3087e-07
# Distance to the stationary point =   4.0237e-02
# Cauchy X =  0.938237 0.151505 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 9
# LINE SEARCH 0 times; norm of step = 6.45625e-06
# X = 0.93824 0.151505 
# G = -1.89279e-07 1.09575e-07 
# Iteration     9
# 
# ---------------- CAUCHY entered-------------------
# 
# There are 1  breakpoints
# 
# Piece      1 f1, f2 at start point -4.7833e-14  3.6496e-13
# Distance to the next break point =   1.3827e+06
# Distance to the stationary point =   1.3107e-01
# 
# GCP found in this segment
# Piece      1 f1, f2 at start point -4.7833e-14  3.6496e-13
# Distance to the stationary point =   1.3107e-01
# Cauchy X =  0.93824 0.151505 
# 
# ---------------- exit CAUCHY----------------------
# 
# 2  variables are free at GCP on iteration 10
# LINE SEARCH 0 times; norm of step = 3.66473e-08
# X = 0.93824 0.151505 
# G = 5.47063e-10 3.91771e-10 
# 
# iterations 10
# function evaluations 14
# segments explored during Cauchy searches 11
# BFGS updates skipped 0
# active bounds at final generalized Cauchy point 0
# norm of the final projected gradient 5.47063e-10
# final function value 10.1593
# 
# X = 0.93824 0.151505 
# F = 10.1593
# final  value 10.159268 
# converged
#       mu 
# -10.1593

Compare exact and approximate posterior distributions

b    <- seq(-1,3,length.out = 1000)
n    <- length(b)
logp <- rep(0,n)
qb   <- dnorm(b,mu,sqrt(s))
for (i in 1:n)
  logp[i] <- compute_logp_pois(x,y,b0,b[i],s0)
maxlp <- max(logp)
plot(b,exp(logp - maxlp),type = "l",lwd = 2,col = "dodgerblue",
     xlab = "b",ylab = "relative posterior")
lines(b,qb/max(qb),col = "magenta",lwd = 2)


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