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Empirical Information Bottleneck

# EMBO - Empirical Bottleneck

A Python implementation of the Information Bottleneck analysis framework [Tishby, Pereira, Bialek 2001], especially geared towards the analysis of concrete, finite-size data sets.

## Requirements

embo requires Python 3, numpy>=1.7 and scipy.

## Installation

To install the latest release, run:

pip install embo


(depending on your system, you may need to use pip3 instead of pip in the command above).

### Testing

(requires setuptools). If embo is already installed on your system, look for the copy of the test_embo.py script installed alongside the rest of the embo files and execute it. For example:

python /usr/lib/python3.X/site-packages/embo/test/test_embo.py


## Usage

### The Information Bottleneck

We refer to [Tishby, Pereira, Bialek 2001] for a general introduction to the Information Bottleneck. Briefly, if X and Y are two random variables, we are interested in finding another random variable M (called the "bottleneck" variable) that solves the following optimisation problem:

min_{p(m|x)}I(M:X) - β I(M:Y)

for any β>0, and where M is constrained to be independent on Y conditional on X:

p(x,m,y) = p(x)p(m|x)p(y|x)

Intuitively, we want to find the stochastic mapping p(M|X) that extracts from X as much information about Y as possible while forgetting all irrelevant information. β is a free parameter that sets the relative importance of forgetting irrelevant information versus remembering useful information. Usually, one is interested in the curve described by I(M:X) and I(M:Y) at the solution of the bottleneck problem for a range of values of β. This curve gives the optimal tradeoff of compression and prediction, telling us what is the minimum amount of information one needs to know about X to be able to predict Y to a certain accuracy, or vice versa, what is the maximum accuracy one can have in predicting Y given a certain amount of information about X.

### Using embo

In embo, we assume that the true joint distribution of X and Y is not available, and that we only have a set of joint empirical observations. We also assume that X and Y both take on a finite number of discrete values. The main point of entry to the package is the EmpiricalBottleneck class. In its constructor, EmpiricalBottleneck takes as arguments an array of observations for X and an (equally long) array of observations for Y, together with other optional parameters (see the docstring for details). In the most basic use case, users can call the get_information_bottleneck method of an EmpiricalBottleneck object, which will return a set of β values and the optimal values of I(M:X) and I(M:Y) corresponding to those β. The optimal tradeoff can then be visualised by plotting I(M:Y) vs I(M:Y).

For instance:

import numpy as np
from matplotlib import pyplot as plt
from embo import EmpiricalBottleneck

# data sequences
x = np.array([0,0,0,1,0,1,0,1,0,1])
y = np.array([0,1,0,1,0,1,0,1,0,1])

# compute the IB bound from the data
I_x,I_y,β = EmpiricalBottleneck(x,y).get_empirical_bottleneck()

# plot the optimal compression-prediction bound
plt.plot(I_x,I_y)


### More examples

A simple example of usage with synthetic data is located at embo/examples/Basic-Example.ipynb. A more meaningful example is located at embo/examples/Markov-Chains.ipynb, where we compute the Information Bottleneck between the past and the future of time series generated from different Markov chains.

### Further details

For more details, please consult the docstrings in EmpiricalBottleneck.

## Authors

embo is maintained by Eugenio Piasini, Alexandre Filipowicz and Jonathan Levine.

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