pymc.PolyaGamma#

class pymc.PolyaGamma(name, *args, rng=None, dims=None, initval=None, observed=None, total_size=None, transform=UNSET, default_transform=UNSET, **kwargs)[source]#

The Polya-Gamma distribution.

The distribution is parametrized by h (shape parameter) and z (exponential tilting parameter). The pdf of this distribution is

\[f(x \mid h, z) = cosh^h(\frac{z}{2})e^{-\frac{1}{2}xz^2}f(x \mid h, 0),\]

where \(f(x \mid h, 0)\) is the pdf of a \(PG(h, 0)\) variable. Notice that the pdf of this distribution is expressed as an alternating-sign sum of inverse-Gaussian densities.

\[X = \Sigma_{k=1}^{\infty}\frac{Ga(h, 1)}{d_k},\]

where \(d_k = 2(k - 0.5)^2\pi^2 + z^2/2\), \(Ga(h, 1)\) is a gamma random variable with shape parameter h and scale parameter 1.

(Source code, png, hires.png, pdf)

../../../_images/pymc-PolyaGamma-1.png

Support

\(x \in (0, \infty)\)

Mean

\(\dfrac{h}{4}\) if \(z=0\), \(\dfrac{tanh(z/2)h}{2z}\) otherwise.

Variance

\(0.041666688h\) if \(z=0\), \(\dfrac{h(sinh(z) - z)(1 - tanh^2(z/2))}{4z^3}\) otherwise.
Parameters:
htensor_like of float, default 1

The shape parameter of the distribution (h > 0).

ztensor_like of float, default 0

The exponential tilting parameter of the distribution.

References

[1]

Polson, Nicholas G., James G. Scott, and Jesse Windle. “Bayesian inference for logistic models using Pólya–Gamma latent variables.” Journal of the American statistical Association 108.504 (2013): 1339-1349.

[2]

Windle, Jesse, Nicholas G. Polson, and James G. Scott. “Sampling Polya-Gamma random variates: alternate and approximate techniques.” arXiv preprint arXiv:1405.0506 (2014).

[3]

Luc Devroye. “On exact simulation algorithms for some distributions related to Jacobi theta functions.” Statistics & Probability Letters, Volume 79, Issue 21, (2009): 2251-2259.

[4]

Windle, J. (2013). Forecasting high-dimensional, time-varying variance-covariance matrices with high-frequency data and sampling Pólya-Gamma random variates for posterior distributions derived from logistic likelihoods.(PhD thesis). Retrieved from http://hdl.handle.net/2152/21842

Examples

rng = np.random.default_rng()
with pm.Model():
    x = pm.PolyaGamma("x", h=1, z=5.5)
with pm.Model():
    x = pm.PolyaGamma("x", h=25, z=-2.3, rng=rng, size=(100, 5))

Methods

PolyaGamma.dist([h, z])

Create a tensor variable corresponding to the cls distribution.