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Rmd 321c6eb Matthew Stephens 2020-09-23 workflowr::wflow_publish(“polya_gamma.Rmd”) 
library("BayesLogit")

Introduction

The Pólya–Gamma distribution is used in Bayesian analysis of logistic regression and related models. See Polson et al, henceforth PSW.

However, I could not find an accessible summary of its basic properties, so I decided to summarize them here. For now I am not going to explain how this distribution is used, so you will have to read the primary literature for that.

This is work in progress as I learn about the distribution myself.

Definition

PSW define a random variable \(X\) to have a Pólya–Gamma distribution with parameters \(b>1\) and \(c\ in R\) if it has the same distribution as the following (non-negative) weighted sum of Gamma random variables: \[X = 1/(2\pi^2) \sum_{k=1}^\infty \frac{g_k}{(k-0.5)^2 + c^2/(4\pi)^2}\] where \(g_k \sim \Gamma(b,1)\) are mutually independent.

Note that \(X\) is non-negative. As we shall see, the Pólya–Gamma distribution can have a density that looks somewhat similar to a Gamma distribution, with a mode at zero or a mode away from zero. If you have never come across this distribution before it is perhaps helpful to think of it as most similar to a Gamma distribution (among commonly-used distributions).

Density and exponential tilting

The PG distribution does not have a closed-form density. However the role of the parameters \(b,c\) can be better understood by noting that the density for \(PG(b,c)\) factorizes into a part that depends on \(c\) and a part that depends on \(b\).

Specifically, if \(f(\cdot; b,c)\) denotes the density of \(PG(b,c)\) then \[f(x; b,c) \propto \exp(-c^2x/2) f(x; b,0).\] The phrase “exponential tilting” is sometimes used to describe multiplying a density by an exponential term like this. So we say that \(PG(b,c)\) is obtained from \(PG(b,0)\) by exponential tilting, with \(c^2\) controlling the amount of tilt.

Histograms of samples

The BayesLogit package provides ways to simulate from this distribution. Here we use this to plot some histograms of samples from PG distributions.

rpg_hist = function(b,c,nsamp=10000,xmax=2,log=FALSE,...){
  x = rpg(nsamp,b,c)
  if(log==TRUE){
    x = log(x)
    title = "log(X); "
    breaks = seq(-xmax,xmax,length=100)
    x = x[x<xmax & x>(-xmax)]
  } else {
    title = ""
    breaks = seq(0,xmax,length=100)
    x = x[x<xmax]
  }
  hist(x,breaks = breaks,probability=TRUE,main=paste0(title, "b=",b,", c=",c),...)
}

\(PG(b,0)\)

We start with \(PG(b,0)\) as the base case.

Moderately small \(b\)

In practical applications \(b\) is often an integer. So \(b\geq 1\) is of primary interest and we start there. On the left I give histograms of \(X\) and on the right I plot \(\log(X)\). As \(b\) gets bigger the variance of \(\log(X)\) gets smaller.

par(mfcol=c(4,2),mai=rep(0.3,4))
rpg_hist(1,0,xmax=2,ylim=c(0,4))
rpg_hist(2,0,xmax=2,ylim=c(0,4))
rpg_hist(3,0,xmax=2,ylim=c(0,4))
rpg_hist(4,0,xmax=2,ylim=c(0,4))

rpg_hist(1,0,xmax=5,log=TRUE,ylim=c(0,1))
rpg_hist(2,0,xmax=5,log=TRUE,ylim=c(0,1))
rpg_hist(3,0,xmax=5,log=TRUE,ylim=c(0,1))
rpg_hist(4,0,xmax=5,log=TRUE,ylim=c(0,1))

Large \(b\)

par(mfcol=c(4,2),mai=rep(0.3,4))
rpg_hist(5,0,xmax=35,ylim=c(0,1))
rpg_hist(10,0,xmax=35,ylim=c(0,1))
rpg_hist(50,0,xmax=35,ylim=c(0,1))
rpg_hist(100,0,xmax=35,ylim=c(0,1))

rpg_hist(5,0,xmax=5,ylim=c(0,5),log=TRUE)
rpg_hist(10,0,xmax=5,ylim=c(0,5),log=TRUE)
rpg_hist(50,0,xmax=5,ylim=c(0,5),log=TRUE)
rpg_hist(100,0,xmax=5,ylim=c(0,5),log=TRUE)

Very small \(b\) (\(<1\)).

For completeness we show some plots for very small \(b\) too.

par(mfcol=c(4,2),mai=rep(0.3,4))
rpg_hist(.01,0,xmax=1,ylim=c(0,10))
rpg_hist(.05,0,xmax=1,ylim=c(0,10))
rpg_hist(.1,0,xmax=1,ylim=c(0,10))
rpg_hist(.5,0,xmax=1,ylim=c(0,10))

rpg_hist(.01,0,xmax=15,ylim=c(0,.5),log=TRUE)
rpg_hist(.05,0,xmax=15,ylim=c(0,.5),log=TRUE)
rpg_hist(.1,0,xmax=15,ylim=c(0,.5),log=TRUE)
rpg_hist(.5,0,xmax=15,ylim=c(0,.5),log=TRUE)

\(PG(b,c)\)

If \(c\) is small then the distribution looks similar to \(PG(b,0)\) which has mean \(b/4\).

If \(c\) is large then it becomes concentrated about the mean, which for large \(c\) is approximately \(b/2c\).

Laplace Transform

The Laplace tranform of \(PG(b,0)\) has a nice closed form, and is given by: \[f(t) = [\cosh(\sqrt{t/2})]^{-b}\].

Expectation

Notice that \(E(g_k)=b\) so the expectation of \(X\) scales linearly with \(b\). Clearly it also decreases with \(c\).

In fact the expectation is given by PSW as \[E(X) = (b/2c) \tanh(c/2) = (b/2c) \frac{\exp(c)-1}{\exp(c)+1}\]

For small \(x\), \(\tanh(x) \approx x\) so for small \(c\) we have \(E(X) \approx b/4\). For large \(x\), \(\tanh(x) \approx 1\) so for large \(c\) we have \(E(X) \approx b/(2c)\).

In general the expectation lies betweed 0 and \(b/4\).

We can compute the expectation

epg = function(b,c){
  (b/(2*c)) * tanh(c/2)
}

And then we can plot the expectation for \(b=1\) as a function of \(c\). (For other \(b\) the graph will have the same shape, just multiplied by \(b\)).

c = seq(-10,10,length=100)
plot(c,epg(1,c),type="l",ylim=c(0,0.25), main="Expectation of PG(1,c) as fn of c")

And for a wider range of \(c\):

c = seq(-100,100,length=1000)
plot(c,epg(1,c),type="l",ylim=c(0,0.25), main="Expectation of PG(1,c) as fn of c")


sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6

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] BayesLogit_2.1

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.5       rstudioapi_0.11  whisker_0.4      knitr_1.29      
 [5] magrittr_1.5     workflowr_1.6.2  R6_2.4.1         rlang_0.4.7     
 [9] stringr_1.4.0    tools_3.6.0      xfun_0.16        git2r_0.27.1    
[13] htmltools_0.5.0  ellipsis_0.3.1   yaml_2.2.1       digest_0.6.25   
[17] rprojroot_1.3-2  tibble_3.0.3     lifecycle_0.2.0  crayon_1.3.4    
[21] later_1.1.0.1    vctrs_0.3.4      fs_1.4.2         promises_1.1.1  
[25] glue_1.4.2       evaluate_0.14    rmarkdown_2.3    stringi_1.4.6   
[29] compiler_3.6.0   pillar_1.4.6     backports_1.1.10 httpuv_1.5.4    
[33] pkgconfig_2.0.3

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