dit 2.1

Python package for information theory on discrete random variables.

dit is a Python package for information theory.

Introduction

Information theory is a powerful extension to probability and statistics, quantifying dependencies among arbitrary random variables in a way that is consistent and comparable across systems and scales. Information theory was originally developed to quantify how quickly and reliably information could be transmitted across an arbitrary channel. The demands of modern, data-driven science have been coopting and extending these quantities and methods into unknown, multivariate settings where the interpretation and best practices are not known. For example, there are at least four reasonable multivariate generalizations of the mutual information, none of which inherit all the interpretations of the standard bivariate case. Which is best to use is context-dependent. dit implements a vast range of multivariate information measures in an effort to allow information practitioners to study how these various measures behave and interact in a variety of contexts. We hope that having all these measures and techniques implemented in one place will allow the development of robust techniques for the automated quantification of dependencies within a system and concrete interpretation of what those dependencies mean.

Citing

If you use dit in your research, please cite it as:

@article{dit,
  Author = {James, R. G. and Ellison, C. J. and Crutchfield, J. P.},
  Title = {{dit}: a {P}ython package for discrete information theory},
  Journal = {The Journal of Open Source Software},
  Volume = {3},
  Number = {25},
  Pages = {738},
  Year = {2018},
  Doi = {https://doi.org/10.21105/joss.00738}
}

Basic Information

Documentation

http://docs.dit.io

Downloads

https://pypi.org/project/dit/

https://anaconda.org/conda-forge/dit

Dependencies
Python 3.11+ boltons debtcollector lattices loguru networkx numpy PLTable scipy xarray

Optional Dependencies

• colorama: colored column heads in PID indicating failure modes

• cython: faster sampling from distributions

• hypothesis: random sampling of distributions

• jax, jaxlib: JAX-based optimization backend with autodiff support

• matplotlib, python-ternary: plotting of various information-theoretic expansions

• numdifftools: numerical evaluation of gradients and hessians during optimization

• pint: add units to informational values

• pycddlib-standalone, pypoman: polytope vertex enumeration for convex-maximization-based measures

• pytensor: PyTensor-based optimization backend with autodiff support

• scikit-learn: faster nearest-neighbor lookups during entropy/mutual information estimation from samples

• torch: PyTorch-based optimization backend with autodiff and GPU support

Install

The easiest way to install is:

pip install dit

If you want to install dit within a conda environment, you can simply do:

conda install -c conda-forge dit

For development, we recommend uv:

git clone https://github.com/dit/dit.git
cd dit
uv sync --extra dev

This installs dit in editable mode with all development dependencies (tests, docs, linting, type checking, and optional backends).

Testing

# Using uv (recommended)
uv run pytest

# Or with pip
pip install -e ".[test]"
pytest

Code and bug tracker

https://github.com/dit/dit

License

BSD 3-Clause, see LICENSE.txt for details.

Implemented Measures

dit implements the following information measures. Most of these are implemented in multivariate & conditional generality, where such generalizations either exist in the literature or are relatively obvious — for example, though it is not in the literature, the multivariate conditional exact common information is implemented here.

Entropies Shannon Entropy Renyi Entropy Tsallis Entropy Necessary Conditional Entropy Residual Entropy / Independent Information / Variation of Information	Mutual Informations Co-Information Interaction Information Total Correlation / Multi-Information Dual Total Correlation / Binding Information CAEKL Multivariate Mutual Information O-Information Cohesion Transmission Union Information Logarithmic Decomposition Delta^k / Gamma^k DeWeese Mutual Information	Divergences Variational Distance Kullback-Leibler Divergence / Relative Entropy Cross Entropy Jensen-Shannon Divergence Earth Mover’s Distance
Common Informations Gacs-Korner Common Information Wyner Common Information Exact Common Information Functional Common Information MSS Common Information Kamath-Anantharam Dual Common Information Beta Common Information Salamatian-Cohen-Medard Maximum Entropy Function		Other Measures Channel Capacity Complexity Profile Connected Informations Copy Mutual Information Cumulative Residual Entropy Extropy Hypercontractivity Coefficient Information Bottleneck Information Diagrams Information Trimming Lautum Information LMPR Complexity Marginal Utility of Information Maximum Correlation Maximum Entropy Distributions Perplexity Rate-Distortion Theory TSE Complexity Gray-Wyner Network
	Partial Information Decomposition \(I_{min}\) \(I_{\wedge}\) \(I_{RR}\) \(I_{\downarrow}\) \(I_{proj}\) \(I_{BROJA}\) \(I_{ccs}\) \(I_{\pm}\) \(I_{sx}\) \(I_{dep}\) \(I_{RAV}\) \(I_{mmi}\) \(I_{\prec}\) \(I_{RA}\) \(I_{SKAR}\) \(I_{IG}\) \(I_{RDR}\) \(I_{do}\)
Secret Key Agreement Bounds Secrecy Capacity Intrinsic Mutual Information Reduced Intrinsic Mutual Information Minimal Intrinsic Mutual Information Necessary Intrinsic Mutual Information Two-Part Intrinsic Mutual Information

Quickstart

The basic usage of dit corresponds to creating distributions, modifying them if need be, and then computing properties of those distributions. First, we import:

>>> import dit

Suppose we have a really thick coin, one so thick that there is a reasonable chance of it landing on its edge. Here is how we might represent the coin in dit.

>>> d = dit.Distribution(['H', 'T', 'E'], [.4, .4, .2])
>>> print(d)
Class:          Distribution
Alphabet:       ('E', 'H', 'T') for all rvs
Base:           linear
Outcome Class:  str
Outcome Length: 1
RV Names:       None

x   p(x)
E   0.2
H   0.4
T   0.4

Calculate the probability of H and also of the combination H or T.

>>> d['H']
0.4
>>> d.event_probability(['H','T'])
0.8

Calculate the Shannon entropy and extropy of the joint distribution.

>>> dit.shannon.entropy(d)
1.5219280948873621
>>> dit.other.extropy(d)
1.1419011889093373

Create a distribution where Z = xor(X, Y).

>>> import dit.example_dists
>>> d = dit.example_dists.Xor()
>>> d.set_rv_names(['X', 'Y', 'Z'])
>>> print(d)
Class:          Distribution
Alphabet:       ('0', '1') for all rvs
Base:           linear
Outcome Class:  str
Outcome Length: 3
RV Names:       ('X', 'Y', 'Z')

x     p(x)
000   0.25
011   0.25
101   0.25
110   0.25

Calculate the Shannon mutual informations I[X:Z], I[Y:Z], and I[X,Y:Z].

>>> dit.shannon.mutual_information(d, ['X'], ['Z'])
0.0
>>> dit.shannon.mutual_information(d, ['Y'], ['Z'])
0.0
>>> dit.shannon.mutual_information(d, ['X', 'Y'], ['Z'])
1.0

Calculate the marginal distribution P(X,Z). Then print its probabilities as fractions, showing the mask.

>>> d2 = d.marginal(['X', 'Z'])
>>> print(d2.to_string(show_mask=True, exact=True))
Class:          Distribution
Alphabet:       ('0', '1') for all rvs
Base:           linear
Outcome Class:  str
Outcome Length: 2 (mask: 3)
RV Names:       ('X', 'Z')

x     p(x)
0*0   1/4
0*1   1/4
1*0   1/4
1*1   1/4

Convert the distribution probabilities to log (base 3.5) probabilities, and access its probability mass function.

>>> d2.set_base(3.5)
>>> d2.pmf
array([-1.10658951, -1.10658951, -1.10658951, -1.10658951])

Draw 5 random samples from this distribution.

>>> dit.math.prng.seed(1)
>>> d2.rand(5)
['01', '10', '00', '01', '00']

Source and Channel Coding

Beyond measures, dit builds explicit codes and ships a catalog of channels.

The dit.coding module constructs lossless source codes (Shannon, Fano, Shannon-Fano-Elias, Huffman, length-limited Huffman, Golomb/Rice, Tunstall, and the universal integer codes) and reports their code-theoretic properties (rate, redundancy, efficiency, the Kraft sum, and whether the code is prefix-free / uniquely decodable / optimal).

>>> from dit.coding import huffman
>>> d = dit.Distribution(['a', 'b', 'c', 'd', 'e'], [0.4, 0.2, 0.2, 0.1, 0.1])
>>> code = huffman(d)
>>> code.average_length()
2.2
>>> float(code.source_entropy())
2.1219280948873624
>>> code.is_optimal(), code.is_prefix_free()
(True, True)

It also builds binary (GF(2)) channel codes — linear block codes (repetition, parity-check, Hamming, Reed-Muller, Golay) as well as LDPC, polar, and convolutional codes — and evaluates them against a noisy channel supplied as a conditional Distribution p(Y|X).

>>> from dit.coding import hamming
>>> code = hamming(3)
>>> code.length, code.dimension, code.minimum_distance()
(7, 4, 3)
>>> bsc = dit.example_channels.binary_symmetric_channel(0.05)
>>> float(code.probability_of_error(bsc, method='exact'))
0.04438054218749993

The dit.example_channels module is a catalog of canonical discrete memoryless channels (binary symmetric, binary erasure, Z-channel, q-ary symmetric/erasure, noisy typewriter, …). Each constructor returns a conditional Distribution p(Y|X) ready for dit.algorithms.channel_capacity or the coding layer above.

>>> from dit.algorithms import channel_capacity
>>> bec = dit.example_channels.binary_erasure_channel(0.25)
>>> float(channel_capacity(bec)[0])
0.75

Contributions & Help

If you’d like to feature added to dit, please file an issue. Or, better yet, open a pull request. Ideally, all code should be tested and documented, but please don’t let this be a barrier to contributing. We’ll work with you to ensure that all pull requests are in a mergable state.

If you’d like to get in contact about anything, you can reach us through our slack channel.