briney/dit

dit is a Python package for information theory.

Try dit live:

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 2.7, 3.3, 3.4, 3.5, or 3.6 boltons contextlib2 debtcollector networkx numpy prettytable scipy six

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

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

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

git clone https://github.com/dit/dit.git
cd dit
pip install .

Note

The cython extensions are currently not supported on windows. Please install using the --nocython option.

Testing

$ git clone https://github.com/dit/dit.git
$ cd dit
$ pip install -r requirements_testing.txt
$ py.test

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

Divergences

Variational Distance
Kullback-Leibler Divergence Relative Entropy
Cross Entropy
Jensen-Shannon Divergence
Earth Mover's Distance

Other Measures

Channel Capacity
Complexity Profile
Connected Informations
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_{\downarrow}
I_{proj}
I_{BROJA}
I_{ccs}
I_{\pm}
I_{dep}
I_{RAV}

Secret Key Agreement Bounds

Intrinsic Mutual Information
Reduced Intrinsic Mutual Information
Minimal Intrinsic Mutual Information
Necessary Intrinsic Mutual Information
Secrecy Capacity

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']

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 pleast 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.

briney / dit