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PolyaGammaSamplers.jl

PolyaGammaSamplers.jl provides a method for sampling from the Polya-Gamma distribution ([1]), using the Distributions.jl interface. The Polya-Gamma distribution with parameters b and z has Laplace transform

\[\begin{aligned} \mathcal{L}(t) = \cosh^b(z) \cosh^{-b}(\sqrt{2t + z^2}). \end{aligned}\]

Although somewhat contrived, this distribution is extremely useful for learning Bayesian Logit models ([1]), Bayesian Negative Binomial models ([2]), and their generalizations (see, e.g. [3]).

Currently, the only available sampler is the one proposed in [1] (hereafter, the PSW sampler), but I hope to add the sampler proposed in [4] soon.

References

  1. Polson, N., Scott, J. & Windle, J. (2013) Bayesian inference for logistic models using Pólya–Gamma latent variables, Journal of the American Statistical Association, 108:504, 1339-1349, https://doi.org/10.1080/01621459.2013.829001.
  2. Zhou, M., Li, L., Dunson, D., & Carin, L. (2012). Lognormal and Gamma mixed negative Binomial regression. Proceedings of the International Conference on Machine Learning. International Conference on Machine Learning, 2012, 1343–1350.
  3. Rigon, T., & Durante, D. (2021). Tractable Bayesian density regression via logit stick-breaking priors. Journal of Statistical Planning and Inference, 211, 131–142. https://doi.org/10.1016/j.jspi.2020.05.009.
  4. Windle, J., Polson, N. G., and Scott, J. G. (2014). Sampling Polya-Gamma random variates: Alternate and approximate techniques. arXiv e-prints, page arXiv:1405.0506.