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Efficiently generate samples from the Polya-Gamma distribution using a NumPy/SciPy compatible interface.

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Efficiently generate samples from the Polya-Gamma distribution using a NumPy/SciPy compatible interface.


If you are reading this, you probably have already used the pypolyagamma package before. It is a great package that I have also used in the past, however I encountered several issues:

  • Generating an array of samples is awkward because it requires using a list comprehension if parameter values are scalars or have pre-allocated arrays of a known size to pass for both the parameters and the output array. Moreover, broadcasting of input is not supported and thus requiring the user to write another layer to support it.
  • It requires extra effort to be used in multiprocessing because pickling of the sampler is not supported.
  • There is no parameter validation supported meaning it is easy to get the wrong samples if you do not check the inputs manually.
  • The sampling API is very different from the ones used by popular packages like numpy/scipy, making it harder to just "plug-n-play" in existing code bases.
  • It does not allow passing in an instance of a np.random.RandomState or np.random.Generator for seeding, requiring extra effort when changing the seed if used in a larger code base.
  • The C++ code wrapped by the package is GPLv3 licensed, making it difficult to use the source code in a project that prefers licenses like MIT/Apache/BSD.

The above issues are the reason why this package exists. And the aim of polyagamma is to "fix" them.


  • Input parameters can be scalars, arrays or both; allowing for easy generation of multi-dimensional samples without specifying the size.
  • Input validation is done internally with clear error messages upon failure.
  • It is flexible and allows the user to sample using one of 4 available algorithms.
  • Implements functions to compute the CDF and density of the distribution as well as their logarithms.
  • Random number generation is thread safe.
  • The functional API resembles that of common numpy/scipy functions, therefore making it easy to plugin to existing libraries.
  • polyagamma is optimized for performance and tests show that it is faster than other implementations.
  • Pre-built wheels are provided for easy installation on Linux, MacOS and Windows.



import array
import numpy as np
from polyagamma import random_polyagamma

# generate a PG(1, 0) sample
o = random_polyagamma()

# Get a 5 by 1 array of PG(1, 2) variates.
o = random_polyagamma(z=2, size=5)

# We can pass sequences as input. Numpy's broadcasting rules apply here.
# Get a 10 by 2 array where column 1 is PG(2, -10) and column 2 is PG(1, 10)
o = random_polyagamma([2, 1], [-10, 10], size=(10, 2))
z = [[1.5, 2, -0.75, 4, 5],
     [9.5, -8, 7, 6, -0.9]]
o = random_polyagamma(1, z)

# We can pass an output array using the `out` parameter. It does not have to be
# a numpy array. it can be any object that implements the array or buffer protocols.
# As long as its type is 64bit float, contiguous in memory and aligned (e.g. Python's array object).
numpy_out = np.empty(5)
array_out = array.array('d', [0] * 5)

# one can choose a sampling method from {devroye, alternate, gamma, saddle}.
# If not given, the default behaviour is a hybrid sampler that picks the most
# efficient method based on the input values.
o = random_polyagamma(method="saddle")

# one can also use an existing instance of `numpy.random.Generator` as a parameter.
# This is useful to reproduce samples generated via a given seed.
rng = np.random.default_rng(12345)
o = random_polyagamma(random_state=rng)

# If one is using a `numpy.random.RandomState` instance instead of the `Generator`
# class, the object's underlying bitgenerator can be passed as the value of random_state
bit_gen = np.random.RandomState(12345)._bit_generator
o = random_polyagamma(random_state=bit_gen)

# When passing a large input array for the shape parameter `h`, parameter value
# validation checks can be disabled if the values are guaranteed to be positive
# to avoid some overhead, which may boost performance.
large_h = np.ones(1000000)
o = random_polyagamma(large_h, disable_checks=True)

Functions to compute the density and CDF are available. Broadcasting of input is supported.

from polyagamma import polyagamma_pdf, polyagamma_cdf

>>> polyagamma_pdf(0.1)
# 3.613955566329298
>>> polyagamma_cdf([1, 2], h=2, z=1)
# array([0.95637847, 0.99963397])
>>> polyagamma_pdf([2, 0.1], h=[[1, 2], [3, 4]], return_log=True)
# array([[   -8.03172733,  -489.17101125]
#        [   -3.82023942, -1987.09156971]])
>>> polyagamma_cdf(4, z=[-100, 0, 2], return_log=True)
# array([ 3.72007598e-44, -3.40628215e-09, -1.25463528e-12])


The package also provides low-level functions that can be imported in cython modules. They are:

  • random_polyagamma
  • random_polyagamma_fill
  • random_polyagamma_fill2

Refer to the pgm_random.h header file for more info about the function signatures. Below is an example of how these functions can be used.

from cpython.pycapsule cimport PyCapsule_GetPointer
from polyagamma cimport random_polyagamma_fill, DEVROYE
from numpy.random cimport bitgen_t
import numpy as np

# assuming there exists an instance of the Generator class called `rng`.
bitgenerator = rng._bit_generator
# get pointer to the underlying bitgenerator struct
cdef bitgen_t* bitgen = <bitgen_t*>PyCapsule_GetPointer(bitgenerator.capsule, "BitGenerator")
# set distribution parameters
cdef double h = 1, z = 0
# get a memory view of the array to store samples in
cdef double[:] out = np.empty(300)
with bitgenerator.lock, nogil:
    random_polyagamma_fill(bitgen, h, z, DEVROYE, <size_t>out.shape[0], &out[0])


As of pymc>=4.0.0b1, this distribution can be accessed as a PyMC distribution object. See the pymc documentation for more details.


For an example of how to use polyagamma in a C program, see here.


  • Numpy >= 1.19.0


To get the latest version of the package, one can install it by downloading the wheel/source distribution from the releases page, or using pip with the following shell command:

$ pip install --pre -U polyagamma

or using conda with the following command:

$ conda install -c conda-forge polyagamma

Alternatively, once can install from source with the following shell commands:

$ git clone
$ cd polyagamma/
$ pip install .


Below are runtime plots of 20000 samples generated for various values of h and z, using each method. We restrict h to integer values to accomodate the devroye method, which cannot be used for non-integer h. The version of the package used to generate them is v1.3.1.


  • The gamma method is slowest and should be avoided in cases where speed is paramount.
  • For h >= 8, the saddle method is the fastest for any value of z.
  • For 0 <= z <= 1 and integer h <= 4, the devroye method should be preferred.
  • For z > 1 and 1 < h < 8, the alternate method is the most efficient.
  • For h > 50 (or any value large enough), the normal approximation to the distribution is fastest (not reported in the above plot but it is around 10 times faster than the saddle method and also equally accurate).

Therefore, we devise a "hybrid/default" sampler that picks a sampler based on the above guidelines.

We also benchmark the hybrid sampler runtime with the sampler found in the pypolyagamma package (version 1.2.3). The version of NumPy we use is 1.19.0. We compare our sampler to the pgdrawv functions provided by the package. Below are runtime plots of 20000 samples for each value of h and z. Values of h range from 0.1 to 50, while z is set to 0, 2.5, 5, and 10.

It can be seen that when generating many samples at once for any given combination of parameters, polyagamma outperforms the pypolyagamma package by a large margin. The exception is when the scale parameter is very small (e.g h < 1). It is also worth noting that the pypolygamma package is on average faster than ours at generating exactly 1 sample value from the distribution. This is mainly due to the overhead introduced by creating the bitgenerator + acquiring/releasing the thread lock + doing parameter validation checks at every call to the function. This overhead can somewhat be mitigated by passing in a random generator instance at every call to the polyagamma function. To eliminate this overhead, it is best to use the Cython functions directly. Below is a timing example to demonstrate the benefit of passing a generator explicitly:

In [3]: rng = np.random.SFC64(1)

In [4]: %timeit random_polyagamma()
90 µs ± 1.65 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

In [5]: %timeit random_polyagamma(random_state=rng)
1.69 µs ± 6.96 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)

To generate the above plots locally, run

$ pip install -r scripts/requirements.txt
$ python scripts/ --size=<some size> --z=<z value>

Note that the runtimes may differ than the ones reported here, depending on the machine this script is ran on.

Distribution Plots

Below is a visualization of the Cumulative distribution and density functions for various values of the parameters.

We can compare these plots to the Kernel density estimate and empirical CDF plots generated from 20000 random samples using each of the available methods.


All contributions, bug reports, bug fixes, documentation improvements, enhancements, and ideas are welcome.

To submit a PR, follow the steps below:

  1. Fork the repo.
  2. Install and setup the dev environment with pip install -r requirements-dev.txt or make dev.
  3. Start writing your changes, including unittests.
  4. Run tests to make sure they all pass with make test.
  5. Once finished, you can submit a PR for review.


  • Luc Devroye. "On exact simulation algorithms for some distributions related to Jacobi theta functions." Statistics & Probability Letters, Volume 79, Issue 21, (2009): 2251-2259.
  • 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.
  • J. Windle, N. G. Polson, and J. G. Scott. "Improved Polya-gamma sampling". Technical Report, University of Texas at Austin, 2013b.
  • Windle, Jesse, Nicholas G. Polson, and James G. Scott. "Sampling Polya-Gamma random variates: alternate and approximate techniques." arXiv preprint arXiv:1405.0506 (2014)
  • 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 .

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