Package {BayesLogit}

Polya-Gamma Distribution Moments

Description

Compute the first moment (mean), second moment, and variance of the Polya-Gamma distribution PG(b, z).

Usage

pg.m1(b, z)
pg.m2(b, z)
pg.var(b, z)

Arguments

Details

For X \sim \mathrm{PG}(b, z), the moments are derived from the moment generating function

E[e^{tX}] = \left( \frac{\cosh(z/2)}{\cosh(\sqrt{z^2/4 - t/2})} \right)^b.

The first moment is

E[X] = \frac{b \tanh(z/2)}{2z}, \quad z \neq 0,

with E[X] = b/4 when z = 0. Taylor series are used near z = 0 for numerical stability.

pg.m2 returns E[X^2], and pg.var returns \mathrm{Var}(X) = E[X^2] - E[X]^2.

Value

A scalar giving the requested moment.

References

Nicholas G. Polson, James G. Scott, and Jesse Windle. Bayesian inference for logistic models using Polya-Gamma latent variables. https://arxiv.org/abs/1205.0310

See Also

rpg

Examples

## Mean and variance of PG(1, 0)
pg.m1(1, 0)   # 0.25
pg.var(1, 0)  # 1/24

## Compare sample moments to theoretical values
set.seed(1)
x <- rpg(10000, h = 2, z = 1)
mean(x)
pg.m1(2, 1)

var(x)
pg.var(2, 1)

Polya-Gamma Random Variates

Description

Generate random variates from the Polya-Gamma distribution.

Usage

rpg(num=1, h=1, z=0.0)

rpg.gamma(num=1, h=1, z=0.0, trunc=200)

rpg.devroye(num=1, h=1, z=0.0)

rpg.sp(num=1, h=1, z=0.0)

rpg.gamma.R(num=1, h=1, z=0.0, trunc=200)

rpg.devroye.R(num=1, h=1, z=0.0)

rpg.sp.R(num=1, h=1, z=0.0)

Arguments

Details

A random variable X with distribution PG(h,z) is distributed like

X \sim \sum_{k=1}^\infty G(h,1) / ( 2 \pi^2 (k-1/2)^2 + z^2/2).

The density for X may be derived by exponentially tilting the PG(h,0) density:

p(x|h,z) \propto \exp(-x z^2/2) p(x|h,0).

Different methods for generating this random variable are implemented, each of which is useful for certain parameters. The parameters supplied by the user automatically determine which method is used. One may manually call each routine using rpg.METHOD. Functions ending in ".R" are pure R implementations.

You may call rpg when n and z are vectors.

Value

This function returns num Polya-Gamma samples.

References

Nicholas G. Polson, James G. Scott, and Jesse Windle. Bayesian inference for logistic models using Polya-Gamma latent variables. https://arxiv.org/abs/1205.0310

Examples

h = c(1, 2, 3);
z = c(4, 5, 6);

## Devroye-like method -- only use if h contains integers, preferably small integers.
X = rpg.devroye(100, h, z);

h = c(1.2, 2.3, 3.2);
z = c(4, 5, 6);

## Sum of gammas method -- this is slow.
X = rpg.gamma(100, h, z);

h = c(1, 4, 2.3);
z = c(4, 5, 6);

## Hybrid method -- automatically chooses best procedure.
X = rpg(100, h, z);