polyagamma 1.3.0rc2

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

polya-gamma

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

Features

• polyagamma is written in C and optimized for performance.
• Very light and easy to install (pre-built wheels).
• It is flexible and allows the user to sample using one of 4 available methods.
• Implements functions to compute the CDF and density of the distribution as well as their logarithms.
• Input parameters can be scalars, arrays or both; allowing for easy generation of multi-dimensional samples without specifying the size.
• 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.

Dependencies

• Numpy >= 1.19.0

Installation

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 polyagamma

To install the latest pre-release version, use:

$ pip install --pre -U polyagamma

Alternatively, once can install from source by cloning the repo. This requires an installation of poetry and the following shell commands:

$ git clone https://github.com/zoj613/polya-gamma.git
$ cd polya-gamma/
$ poetry install
# add package to python's path
$ export PYTHONPATH=$PWD:$PYTHONPATH 

Examples

Python

import numpy as np
from polyagamma import random_polyagamma

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

# Get a 5 by 10 array of PG(1, 2) variates.
o = random_polyagamma(z=2, size=(5, 10))

# Pass sequences as input. Numpy's broadcasting rules apply here.
h = [[1.5, 2, 0.75, 4, 5],
     [9.5, 8, 7, 6, 0.9]]
o = random_polyagamma(h, -2.5)

# Pass an output array
out = np.empty(5)
random_polyagamma(out=out)
print(out)

# one can choose a sampling method from {devroye, alternate, gamma, saddle}.
# If not given, the default behaviour is a hybrid sampler that picks a method
# based on the parameter 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 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]], log=True)
# array([[   -8.03172733,  -489.17101125]
#        [   -3.82023942, -1987.09156971]])
>>> polyagamma_cdf(4, z=[-100, 0, 2], log=True)
# array([ 3.72007598e-44, -3.40628215e-09, -1.25463528e-12])

Cython

The package also provides 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, BitGenerator
import numpy as np

# assuming there exists an instance of the Generator class called `rng`.
cdef BitGenerator 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 nogil:  # you probably want to acquire a thread lock here as well for thread safety.
    random_polyagamma_fill(bitgen, h, z, DEVROYE, <int>out.shape[0], &out[0])
print(out.base)
...

C

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

Benchmarks

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 v1.1.1.

Generally:

• The gamma method is slowest and should be avoided in cases where speed is paramount.
• For h > 25, the saddle method is the fastest for any value of z.
• For 0 <= z <= 2 and integer h < 15, the devroye method should be preferred.
• For z > 2 and integer/non-integer h <= 25, 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 use the pgdrawv function which takes arrays as input. Below are runtime plots of 20000 samples for each value of h and z. Values of h range from 0.1 to 60, 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. The exception is when very small (e.g h < 1). Although not shown here, for values of h larger than 50, we use the normal approximation method. It is also worth noting that the pypolygamma package is on average faster than ours at generating exactly one sample 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. For example, on an iPython session:

In [4]: %timeit random_polyagamma()
83.2 µs ± 1.02 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

In [5]: %timeit random_polyagamma(random_state=rng)
2.68 µs ± 29.7 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

In [6]: %timeit random_polyagamma(random_state=rng, disable_checks=True)
2.58 µs ± 22.3 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

To generate the above plots locally, run python scripts/benchmark.py --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.

Contributing

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. Setup the dev environment with poetry install. All dependencies will be installed.
3. Start writing your changes, including unittests.
4. Once finished, run make install to build the project with the new changes.
5. Once build is successful, run tests to make sure they all pass with make test.
6. Once finished, you can submit a PR for review.

References

• 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 http://hdl.handle.net/2152/21842 .