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X follows polya-gamma distribution with parameters \(b>0\) and \(c\in R\) if \[X\overset{D}{=}\frac{1}{2\pi^2}\sum_{k=1}^\infty\frac{g_k}{(k-1/2)^2+c^2/(4\pi^2)},\] where \(g_k\sim Gamma(b,1)\).
Binomial likelihoods parametrized by log odds can be represented as mixtures of Gaussians with respect to a P´olya-Gamma distribution.
\[\frac{\left(e^{\psi}\right)^{a}}{\left(1+e^{\psi}\right)^{b}}=2^{-b} e^{\kappa \psi} \int_{0}^{\infty} e^{-\omega \psi^{2} / 2} p(\omega) d \omega,\] where \(\kappa = a-b/2\), and \(\omega\sim PG(b,0)\).
The density of a Polya-Gamma random variable can be expressed as an alternating-sign sum of inverse-Gaussian densities \[f(x \mid b, c)=\left\{\cosh ^{b}(c / 2)\right\} \frac{2^{b-1}}{\Gamma(b)} \sum_{n=0}^{\infty}(-1)^{n} \frac{\Gamma(n+b)}{\Gamma(n+1)} \frac{(2 n+b)}{\sqrt{2 \pi x^{3}}} e^{-\frac{(2 n+b)^{2}}{8 x}-\frac{c^{2}}{2} x}\]
All finite moments of a Polya-Gamma random variable are available in closed form. In particular, the expectation may be calculated directly. This allows the Polya-Gamma scheme to be used in EM algorithms. If \(\omega\sim PG(b,c)\), then \(E(\omega) = \frac{b}{2c}tanh(c/2) = \frac{b}{2c}(\frac{e^c-1}{1+e^c})\).
If \(w_1\sim PG(b_1,c)\) and \(w_2\sim PG(b_2,c)\) then \(w_1+w_2\sim PG(b_1+b_2,c)\)
Let \(y_i\sim Binomial(n_i,\frac{1}{1+e^{-\phi_i}})\), where \(\phi_i\) are log odds of success. In logistic regression, \(\phi_i = x_i^T\beta\).
THe likelihood contribution of observation \(i\) is
\[L_i(\phi_i) = \frac{(\exp\phi_i)^{y_i}}{(1+\exp(\phi_i))^{n_i}}.\]
In logistic regression, the likelihood is
\[\begin{aligned} L_{i}(\boldsymbol{\beta}) &=\frac{\left\{\exp \left(x_{i}^{T} \boldsymbol{\beta}\right)\right\}^{y_{i}}}{(1+\exp \left(x_{i}^{T} \boldsymbol{\beta}\right))^{n_i}} \\ & \propto \exp \left(\kappa_{i} x_{i}^{T} \boldsymbol{\beta}\right) \int_{0}^{\infty} \exp \left\{-\omega_{i}\left(x_{i}^{T} \boldsymbol{\beta}\right)^{2} / 2\right\} p\left(\omega_{i} \mid n_{i}, 0\right) \end{aligned},\]
where \(\kappa_i - y_i-n_i/2\).
The conditional posterior of \(\beta\) is \[p(\beta|w,y)\propto p(\beta)\exp\{-\frac{1}{2}(z-X\beta)^T\Omega(z-X\beta)\} = p(\beta)\exp\{-\frac{1}{2}(\beta-X^{-1}z)^TX^T\Omega X(\beta-X^{-1}z)\},\] where \(z = (\kappa_1/w_1,...,\kappa_n/w_n)\) and \(\Omega = diag(w_1,...,w_n)\).
If the prior of \(\beta\) is Gaussian, then the canditional posterior of \(\beta\) is also Gaussian. So the Gibbs sampler iteratively samples from \((\omega_i|\beta)\sim PG(n_i,x_i^T\beta), (\beta|y,\Omega)\sim N(m,V)\).
library()Polson, N. G., Scott, J. G., & Windle, J. (2013). Bayesian inference for logistic models using Pólya–Gamma latent variables. Journal of the American statistical Association, 108(504), 1339-1349.
sessionInfo()R version 3.5.1 (2018-07-02)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Scientific Linux 7.4 (Nitrogen)
Matrix products: default
BLAS/LAPACK: /software/openblas-0.2.19-el7-x86_64/lib/libopenblas_haswellp-r0.2.19.so
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] workflowr_1.6.2 Rcpp_1.0.4.6 digest_0.6.18 later_0.7.5
[5] rprojroot_1.3-2 R6_2.3.0 backports_1.1.2 git2r_0.26.1
[9] magrittr_1.5 evaluate_0.12 stringi_1.2.4 fs_1.3.1
[13] promises_1.0.1 whisker_0.3-2 rmarkdown_1.10 tools_3.5.1
[17] stringr_1.3.1 glue_1.3.0 httpuv_1.4.5 yaml_2.2.0
[21] compiler_3.5.1 htmltools_0.3.6 knitr_1.20