The ndd package provides a simple Python interface to an efficient implementation of the Nemenman-Schafee-Bialek (NSB) algorithm, a parameter-free, Bayesian entropy estimator for discrete data.

The entropy function takes as input a vector of frequency counts (the observed frequencies for a set of classes or states) and returns an entropy estimate (in nats):

>>> counts
[7, 3, 5, 8, 9, 1, 3, 3, 1, 0, 2, 5, 2, 11, 4, 23, 5, 0, 8, 0]
>>> import ndd
>>> entropy_estimate = ndd.entropy(counts)
>>> entropy_estimate
2.623634344888532

Optionally, the uncertainty in the entropy estimate can be quantified by computing an approximation for the posterior standard deviation:

>>> entropy_estimate, std = ndd.entropy(counts, return_std=True)
>>> std
0.048675500725595504

ndd provide functions for the estimation of entropic information measures (as linear combinations of single Bayesian entropy estimates):

• jensen_shannon_divergence
• kullback_leibler_divergence
• conditional_entropy
• mutual_information
• interaction_information
• coinformation

See the functions' docstrings for details.

Install using pip:

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 compile ndd, you will need numpy (>= 1.9) 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

Clone the repo, install tests requirements and run the tests with make:

git clone https://github.com/simomarsili/ndd.git
cd ndd
pip install .[test]
make test

• Added kullback_leibler_divergence function.

• For methods/functions working on data matrices:

  the default input is a p-by-n 2D array (n samples from p discrete variables, with different samples on different columns).

Added:

• from_data
• mutual_information
• conditional_information
• interaction_information
• coinformation

• Python3 only (>= 3.4)

• Added 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{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{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}
}

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.

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

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