(Stumbling Blocks) On the Road to Understanding Multivariate Information Theory

A theory is more than just a set of measures. It also attributes meaning to those measures, and ties that meaning to some sort of objective reality. At this point, most understanding of multivariate information theory is flawed. In part, this is due to several counter-intuitive situations which can arise in the study of joint distributions.

In this document, we will discuss many of these examples which generally impact the ability to construct universal understanding of multivariate mutual information theory. It remains to be seen whether this is because multivariate information theory is simply richer and more nuanced than our intuitions would like, or if it is simply inadequate for the jobs it has been tasked with.

Necessity of Common Informations

Our first pitfall is that the mutual information between two random variables can not be embodied. That is, generically there does not exist a variable \(Z\) such that:

\[\begin{split}\I{X : Y | Z} = 0 \\ \textrm{and} \\ \H{Z} - \I{X : Y} = 0\end{split}\]

That is, there is no \(Z\) which captures the entirety of \(\I{X : Y}\) and nothing more.

When embodying the information shared by \(X\) and \(Y\) is desired, one must make a choice. Choosing the variable capturing as much of \(I{X : Y}\) and nothing more results in the Gács-Körner Common Information. Choosing the variable capturing all of \(I{X : Y}\) with as little else as possible results in the Exact Common Information. If one chooses to incorporate only information from \(X\) we arrive as \(Z = X \mss Y\). Other choices are described in Common Informations.

In [1]: d = Distribution(['00', '01', '10'], [1/3]*3)

In [2]: I(d, [[0], [1]])
Out[2]: 0.25162916738782304

In [3]: K(d, [[0], [1]])
Out[3]: 0.0

In [4]: G(d, [[0], [1]])
Out[4]: 0.9182906701593299

In this case, we see a wide gap. The largest random variable capturing nothing outside of \(\I{X : Y}\) is null, indicated by the Gács-Körner common information being zero, while the smallest variable capturing all of \(\I{X : Y}\) is much larger, capturing two-thirds of a bit more than the actual shared information.

Conditional Dependence

Consider the duality between set theory and information theory. One simple inequality in set theory is:

\[| X - Y | \leq | X |\]

and indeed the corresponding information theoretic inequality holds:

\[\H{X | Y} \leq \H{X}\]

Since the intersection of two sets is itself a set, the following inequality also holds:

\[| (X \cap Y) - Z | \leq | X \cap Y |\]

We might then assume that its corresponding information-theoretic inequality would hold:

\[\I{X : Y | Z} \leq \I{X : Y}\]

This, however, has a couple major difficulties. Firstly, the mutual information between two variables does not itself correspond to a random variable, as we saw in Necessity of Common Informations and so the analogy does not hold. Secondly, the inequality does not hold. The most simplest counterexample is the xor distribution:

In [5]: d = Distribution(['000', '011', '101', '110'], [1/4]*4)

In [6]: I(d, [[0], [1]])
Out[6]: 0.0

In [7]: I(d, [[0], [1]], [2])
Out[7]: 1.0

Zero Probabilities

The following implication holds, so long as \(p(w, x, y, z) > 0\):

\[\begin{split}\left. \begin{array}{l} W \perp Z | (X, Y) \\ W \perp Y | (X, Z) \end{array} \right\} \implies W \perp (Y, Z) | X\end{split}\]

This demonstrates that structural properties, such as conditional independence, is sensitive to the distinction between “small” probability and zero probability.

This becomes an issue when, for example, Bayesian methods are used to infer the probability distribution. These methods will generally never set a probability to zero and so will always exhibit this conditional independence even if the underlying reality does not due to null probabilities. In this way, Bayesian methods can systematically mislead a practitioner regarding the structural independencies in a system.

Shannon-like Information Measures Are Insensitive to Structural Differences

Consider two distributions of three variables, each taking on four values. One built by flipping three coins and assigning each to a different pair of variables, the variable’s state is then the concatenation of the two coins it has access to. The second built by again flipping three coins, but this time all variables share one of the coin flips, and then the other two coins and their xor are each assigned to a variable. The first is constructed using solely pairwise (dyadic) interactions, while the second using three-way (triadic) interactions.

In spite of the fact that these two distributions are qualitatively quite distinct, their informational signatures are all identical:

In [8]: from dit.example_dists import dyadic, triadic

In [9]: from dit.profiles import ShannonPartition

In [10]: ShannonPartition(dyadic)
Out[10]: 
+----------+--------+
| measure  |  bits  |
+----------+--------+
| H[0|1,2] |  0.000 |
| H[1|0,2] |  0.000 |
| H[2|0,1] |  0.000 |
| I[0:1|2] |  1.000 |
| I[0:2|1] |  1.000 |
| I[1:2|0] |  1.000 |
| I[0:1:2] |  0.000 |
+----------+--------+

In [11]: ShannonPartition(triadic)
Out[11]: 
+----------+--------+
| measure  |  bits  |
+----------+--------+
| H[0|1,2] |  0.000 |
| H[1|0,2] |  0.000 |
| H[2|0,1] |  0.000 |
| I[0:1|2] |  1.000 |
| I[0:2|1] |  1.000 |
| I[1:2|0] |  1.000 |
| I[0:1:2] |  0.000 |
+----------+--------+

This result implies that any measure built form Shannon-like information measures necessarily can not distinguish between distributions with different scales of interaction.

Local Modifications Can Create Redundancy

It is commonly believed that a non-zero coinformation value is a signature of some sort of triadic interactions. Positive values indicate “redundancy”, for example a giant bit:

In [12]: d = Distribution(['000', '111'], [1/2]*2)

In [13]: I(d)
Out[13]: 1.0

Negative values indicate “synergy”, for example the xor:

In [14]: d = Distribution(['000', '011', '101', '110'], [1/4]*4)

In [15]: I(d)
Out[15]: -1.0

As seen in Shannon-like Information Measures Are Insensitive to Structural Differences, zero coinformation does not indicate a lack of triadic interactions.

If we begin with a distribution lacking triadic interactions by construction, the dyadic distribution from Shannon-like Information Measures Are Insensitive to Structural Differences. If we then allow each variable to be modified independent of the others while maximizing the coinformation, we arrive at the DeWeese-like Measures:

In [16]: from dit.multivariate import deweese_coinformation

In [17]: deweese_coinformation(dyadic)
Out[17]: 0.061278124458192036

This implies that cyclic pairwise interactions can be utilized to construct triadic interactions.

Negative Coinformation Does Not Imply Threeway Interactions

Finally, does a negative coinformation imply triadic interactions? Consider a distribution consisting of two random bits and their logical and. This distribution has a negative coinformation, implying conditional dependence and some sort of triadic interaction. However, if we consider the family of distributions which match and on its pairwise marginals, this family consists of exactly one distribution: the and distribution!

In [18]: d = Distribution(['000', '010', '100', '111'], [1/4]*4)

In [19]: I(d)
Out[19]: -0.18872187554086706

In [20]: maxent_dist(d, [[0, 1], [0, 2], [1, 2]])
Out[20]: 
Class:          Distribution
Alphabet:       ('0', '1') for all rvs
Base:           linear
Outcome Class:  str
Outcome Length: 3
RV Names:       None

x     p(x)
000   1/4
010   1/4
100   1/4
111   1/4

In [21]: from dit.algorithms.distribution_optimizers import MinEntOptimizer

In [22]: meo = MinEntOptimizer(d, [[0, 1], [0, 2], [1, 2]])

In [23]: meo.optimize()

In [24]: meo.construct_dist()
Out[24]: 
Class:          Distribution
Alphabet:       ('0', '1') for all rvs
Base:           linear
Outcome Class:  str
Outcome Length: 3
RV Names:       None

x     p(x)
000   1/4
010   1/4
100   1/4
111   1/4

And so this negative coinformation arises from cyclic, but strictly pairwise interactions. We do note that a negative coinformation is not possible without at least the cyclic pairwise constraints. But this raises an important observation: negative coinformations can be constructed solely with pairwise interactions, and so conditional dependence is not a phenomena which requires triadic interactions.

Closing

At this point one might suspect that information theory is in shambles, and not up for the task of accurately detecting and quantifying dependencies. However, I believe the limitation lies not with information theory but rather with our impression of what it should be.