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Python package for information theory.

Project description

dit is a Python package for information theory.

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Try dit live: Run `dit` live!


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.


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

  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 = {}

Basic Information




Optional Dependencies

  • colorama: colored column heads in PID indicating failure modes

  • cython: faster sampling from distributions

  • hypothesis: random sampling of distributions

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

  • numdifftools: numerical evaluation of gradients and hessians during optimization

  • pint: add units to informational values

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


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

Alternatively, you can clone this repository, move into the newly created dit directory, and then install the package:

git clone
cd dit
pip install .


$ git clone
$ cd dit
$ pip install -r requirements_testing.txt
$ py.test

Code and bug tracker


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.


  • 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


  • Variational Distance

  • Kullback-Leibler Divergence Relative Entropy

  • Cross Entropy

  • Jensen-Shannon Divergence

  • Earth Mover’s Distance

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

Common Informations

  • Gacs-Korner Common Information

  • Wyner Common Information

  • Exact Common Information

  • Functional Common Information

  • MSS Common Information

Partial Information Decomposition

  • I_{min}

  • I_{wedge}

  • I_{RR}

  • I_{downarrow}

  • I_{proj}

  • I_{BROJA}

  • I_{ccs}

  • I_{pm}

  • I_{dep}

  • I_{RAV}

  • I_{mmi}

  • I_{prec}

  • I_{RA}

  • I_{SKAR}

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


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

Calculate the Shannon entropy and extropy of the joint distribution.

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

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

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']

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.

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