Pólya-Gamma augmentation trick

An infinite weird RV useful Bayesian Binomial regression (and maybe other things?) (Polson, Scott, and Windle 2013). C&C the optimization-driven approach to a similar problem in Gumbel-max tricks.

See also my former colleague, Louis Tiao, A Primer on Pólya-gamma Random Variables - Part II: Bayesian Logistic Regression.

Gregory Gunderson, in Pólya-Gamma Augmentation, explains the problem we are trying to solve.

…n logistic regression, the dependent variables are assumed to be i.i.d. from a Bernoulli distribution with parameter \(p\), and therefore the likelihood function is \[ \mathcal{L}(p) \propto \prod_{n=1}^{N} p^{y_{n}}(1-p)^{1-y_{n}}=p^{\sum y_{n}}(1-p)^{N-\sum y_{n}} \] The observations interact with the response through a linear relationship with the log-odds, \[ \log \left(\frac{p}{1-p}\right)=\beta_{0}+x_{1} \beta_{1}+x_{2} \beta_{2}+\cdots+x_{D} \beta_{D}=\beta^{\top} \mathbf{x} \] If we solve for \(p\) in (2), we get \[ p=\frac{\exp \left(\boldsymbol{\beta}^{\top} \mathbf{x}_{n}\right)}{1+\exp \left(\boldsymbol{\beta}^{\top} \mathbf{x}_{n}\right)} \] and a likelihood of \[ \mathcal{L}(\boldsymbol{\beta}) \propto \frac{\left[\exp \left(\boldsymbol{\beta}^{\top} \mathbf{x}\right)\right]^{\sum y_{n}}}{\left[1+\exp \left(\boldsymbol{\beta}^{\top} \mathbf{x}\right)\right]^{N}} \] Due to this functional form, Bayesian inference for logistic regression is intractable.

Using Pólya-Gamma RVs we devise an auxiliary variable sample.


Polson, Nicholas G., James G. Scott, and Jesse Windle. 2013. Bayesian Inference for Logistic Models Using Pólya–Gamma Latent Variables.” Journal of the American Statistical Association 108 (504): 1339–49.

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