ndd is a Python package for Bayesian entropy estimation from discrete data. ndd provides the ndd.entropy function, a Bayesian replacement for the scipy.stats.entropy function from the SciPy library, based on an efficient implementation of the Nemenman-Schafee-Bialek (NSB) algorithm. Remarkably, the NSB algorithm allows entropy estimation when the number of samples is much smaller than the number of classes with non-zero probability.

The entropy function takes as input a vector of frequency counts (the observed frequencies for a set of classes or states) and an alphabet size (the number of classes with non-zero probability, including unobserved classes) and returns an entropy estimate (in nats):

>>> import ndd
>>> counts = [4, 12, 4, 5, 3, 1, 5, 1, 2, 2, 2, 2, 11, 3, 4, 12, 12, 1, 2]
>>> ndd.entropy(counts, k=100)
2.8060922529931225

The uncertainty in the entropy estimate can be quantified using the posterior standard deviation (see Eq. 13 in Archer 2013)

>>> ndd.entropy(counts, k=100, return_std=True)
(2.8060922529931225, 0.11945501149743358)

If the alphabet size is unknown or countably infinite, the k argument can be omitted and the entropy function will either use an upper bound estimate for k, or switch to the asymptotic NSB estimator for strongly undersampled distributions (Equations 29, 30 in Nemenman 2011)

>>> import ndd
>>> counts = [4, 12, 4, 5, 3, 1, 5, 1, 2, 2, 2, 2, 11, 3, 4, 12, 12, 1, 2]
>>> ndd.entropy(counts)  # k is omitted
2.8130746489179046

The easiest way to install ndd is via the conda package manager. Packages are provided on the conda-forge Anaconda Cloud channel for Linux, OS X, and Win platforms.

Install the latest stable release using conda with:

conda install --channel conda-forge ndd

Install using pip with:

pip3 install -U ndd

or directly from sources in github for the latest version of the code:

pip3 install git+https://github.com/simomarsili/ndd.git

In order to build ndd with pip, you will need numpy (>= 1.13) and a Fortran compiler installed on your machine. If you are using Debian or a Debian derivative such as Ubuntu, you can install the gfortran compiler using the following command:

sudo apt-get install gfortran

On Windows, you can use the gfortran compiler from the MinGW-w64 project (direct link to the installer).

v1.10.5

Added ndd to the anaconda conda-forge channel.

v1.10

Changed: the signature of the entropy function is::

entropy(nk, k=None, estimator=None, return_std=False)

v1.9

Changed:

if argument k is omitted, the entropy function will guess a reasonable alphabet size and select the best estimator for the sampling regime.

v.1.8.3

Fixed:

integration for huge cardinalities

v1.8

Added:

full Bayesian error estimate (from direct computation of the posterior variance of the entropy)

v1.7

Changed:

estimation is much faster (removed unnecessary checks on input counts)

entropy() function needs cardinality k for the default (NSB) estimator

v1.6.1

Changed: Fixed numerical integration for large alphabet sizes.

v1.6

Changed:

The signature of the entropy function has been changed to allow arbitrary entropy estimators. The new signature is:

entropy(pk, k=None, estimator='NSB', return_std=False)

The available estimators are:

>>> import ndd
>>> ndd.entropy_estimators
['Plugin', 'MillerMadow', 'NSB', 'AsymptoticNSB', 'Grassberger']

Check the function docstring for details.

Added:

• MillerMadow estimator class
• AsymptoticNSB estimator class
• Grassberger estimator class

v1.5

For methods/functions working on data matrices: the default input is a n-by-p 2D array (n samples from p discrete variables, with different samples on different rows). Since release 1.3, the default was a transposed (p-by-n) data matrix. The behavior of functions taking frequency counts as input (e.g. the entropy function) is unchanged.

v1.4

Added the kullback_leibler_divergence function.

v1.1

Added:

• from_data
• mutual_information
• conditional_information
• interaction_information
• coinformation

v1.0

Drop support for Python < 3.4.

v0.9

Added the jensen_shannnon_divergence function.

Some refs:

@article{wolpert1995estimating,
  title={Estimating functions of probability distributions from a finite set of samples},
  author={Wolpert, David H and Wolf, David R},
  journal={Physical Review E},
  volume={52},
  number={6},
  pages={6841},
  year={1995},
  publisher={APS}
}

@inproceedings{nemenman2002entropy,
  title={Entropy and inference, revisited},
  author={Nemenman, Ilya and Shafee, Fariel and Bialek, William},
  booktitle={Advances in neural information processing systems},
  pages={471--478},
  year={2002}
}

@article{paninski2003estimation,
  title={Estimation of entropy and mutual information},
  author={Paninski, Liam},
  journal={Neural computation},
  volume={15},
  number={6},
  pages={1191--1253},
  year={2003},
  publisher={MIT Press}
}

@article{nemenman2004entropy,
  title={Entropy and information in neural spike trains: Progress on the sampling problem},
  author={Nemenman, Ilya and Bialek, William and van Steveninck, Rob de Ruyter},
  journal={Physical Review E},
  volume={69},
  number={5},
  pages={056111},
  year={2004},
  publisher={APS}
}

@article{nemenman2011coincidences,
  title={Coincidences and estimation of entropies of random variables with large cardinalities},
  author={Nemenman, Ilya},
  journal={Entropy},
  volume={13},
  number={12},
  pages={2013--2023},
  year={2011},
  publisher={Molecular Diversity Preservation International}
}

@article{archer2013bayesian,
  title={Bayesian and quasi-Bayesian estimators for mutual information from discrete data},
  author={Archer, Evan and Park, Il Memming and Pillow, Jonathan W},
  journal={Entropy},
  volume={15},
  number={5},
  pages={1738--1755},
  year={2013},
  publisher={Multidisciplinary Digital Publishing Institute}
}

@article{archer2014bayesian,
  title={Bayesian entropy estimation for countable discrete distributions},
  author={Archer, Evan and Park, Il Memming and Pillow, Jonathan W},
  journal={The Journal of Machine Learning Research},
  volume={15},
  number={1},
  pages={2833--2868},
  year={2014},
  publisher={JMLR. org}
}

and interesting links:

• Sebastian Nowozin on Bayesian estimators
• Il Memming Park on discrete entropy estimators

ndd is an OPEN Source Project so please help out by reporting bugs or forking and opening pull requests when possible.

Copyright (c) 2016-2019, Simone Marsili. All rights reserved.

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