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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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