**Last updated:** 2020-09-23

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`library("BayesLogit")`

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.

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).

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.

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),...)
}
```

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

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))
```

```
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)
```

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)
```

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 \(b/2c\).

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

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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