Mutual information


In probability theory and information theory, the mutual information of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the "amount of information" obtained about one random variable through observing the other random variable. The concept of mutual information is intimately linked to that of entropy of a random variable, a fundamental notion in information theory that quantifies the expected "amount of information" held in a random variable.
Not limited to real-valued random variables and linear dependence like the correlation coefficient, MI is more general and determines how different the joint distribution of the pair is to the product of the marginal distributions of and. MI is the expected value of the pointwise mutual information.
Mutual Information is also known as information gain.

Definition

Let be a pair of random variables with values over the space. If their joint distribution is and the marginal distributions are and, the mutual information is defined as
where is the Kullback–Leibler divergence.
Notice, as per property of the Kullback–Leibler divergence, that is equal to zero precisely when the joint distribution coincides with the product of the marginals, i.e. when and are independent. In general is non-negative, it is a measure of the price for encoding as a pair of independent random variables, when in reality they are not.

In terms of PMFs for discrete distributions

The mutual information of two jointly discrete random variables and is calculated as a double sum:
where is the joint probability mass function of and, and and are the marginal probability mass functions of and respectively.

In terms of PDFs for continuous distributions

In the case of jointly continuous random variables, the double sum is replaced by a double integral:
where is now the joint probability density function of and, and and are the marginal probability density functions of and respectively.
If the log base 2 is used, the units of mutual information are bits.

Motivation

Intuitively, mutual information measures the information that and share: It measures how much knowing one of these variables reduces uncertainty about the other. For example, if and are independent, then knowing does not give any information about and vice versa, so their mutual information is zero. At the other extreme, if is a deterministic function of and is a deterministic function of then all information conveyed by is shared with : knowing determines the value of and vice versa. As a result, in this case the mutual information is the same as the uncertainty contained in alone, namely the entropy of . Moreover, this mutual information is the same as the entropy of and as the entropy of.
Mutual information is a measure of the inherent dependence expressed in the joint distribution of and relative to the marginal distribution of and under the assumption of independence. Mutual information therefore measures dependence in the following sense: if and only if and are independent random variables. This is easy to see in one direction: if and are independent, then, and therefore:
Moreover, mutual information is nonnegative and symmetric.

Relation to other quantities

Nonnegativity

Using Jensen's inequality on the definition of mutual information we can show that is non-negative, i.e.

Symmetry

Relation to conditional and joint entropy

Mutual information can be equivalently expressed as:
where and are the marginal entropies, and are the conditional entropies, and is the joint entropy of and.
Notice the analogy to the union, difference, and intersection of two sets: in this respect, all the formulas given above are apparent from the Venn diagram reported at the beginning of the article.
In terms of a communication channel in which the output is a noisy version of the input, these relations are summarised in the figure:
Because is non-negative, consequently,. Here we give the detailed deduction of for the case of jointly discrete random variables:
The proofs of the other identities above are similar. The proof of the general case is similar, with integrals replacing sums.
Intuitively, if entropy is regarded as a measure of uncertainty about a random variable, then is a measure of what does not say about. This is "the amount of uncertainty remaining about after is known", and thus the right side of the second of these equalities can be read as "the amount of uncertainty in, minus the amount of uncertainty in which remains after is known", which is equivalent to "the amount of uncertainty in which is removed by knowing ". This corroborates the intuitive meaning of mutual information as the amount of information that knowing either variable provides about the other.
Note that in the discrete case and therefore. Thus, and one can formulate the basic principle that a variable contains at least as much information about itself as any other variable can provide.

Relation to Kullback–Leibler divergence

For jointly discrete or jointly continuous pairs,
mutual information is the Kullback–Leibler divergence of the product of the marginal distributions,, from the joint distribution, that is,
Furthermore, let be the conditional mass or density function. Then, we have the identity
The proof for jointly discrete random variables is as follows:
Similarly this identity can be established for jointly continuous random variables.
Note that here the Kullback–Leibler divergence involves integration over the values of the random variable only, and the expression still denotes a random variable because is random. Thus mutual information can also be understood as the expectation of the Kullback–Leibler divergence of the univariate distribution of from the conditional distribution of given : the more different the distributions and are on average, the greater the information gain.

Bayesian estimation of mutual information

It is well-understood how to do Bayesian estimation of the mutual information
of a joint distribution based on samples of that distribution. The
first work to do this, which also showed how to do Bayesian estimation of many
other information-theoretic properties besides mutual information, was. Subsequent researchers have rederived
and extended
this analysis. See
for a recent paper based on a prior specifically tailored to estimation of mutual
information per se. Besides, recently an estimation method accounting for continuous and multivariate outputs, , was proposed in

Independence assumptions

The Kullback-Leibler divergence formulation of the mutual information is predicated on that one is interested in comparing to the fully factorized outer product. In many problems, such as non-negative matrix factorization, one is interested in less extreme factorizations; specifically, one wishes to compare to a low-rank matrix approximation in some unknown variable ; that is, to what degree one might have
Alternately, one might be interested in knowing how much more information carries over its factorization. In such a case, the excess information that the full distribution carries over the matrix factorization is given by the Kullback-Leibler divergence
The conventional definition of the mutual information is recovered in the extreme case that the process has only one value for.

Variations

Several variations on mutual information have been proposed to suit various needs. Among these are normalized variants and generalizations to more than two variables.

Metric

Many applications require a metric, that is, a distance measure between pairs of points. The quantity
satisfies the properties of a metric. This distance metric is also known as the variation of information.
If are discrete random variables then all the entropy terms are non-negative, so and one can define a normalized distance
The metric is a universal metric, in that if any other distance measure places and close-by, then the will also judge them close.
Plugging in the definitions shows that
In a set-theoretic interpretation of information, this is effectively the Jaccard distance between and.
Finally,
is also a metric.

Conditional mutual information

Sometimes it is useful to express the mutual information of two random variables conditioned on a third.
For jointly discrete random variables this takes the form
which can be simplified as
For jointly continuous random variables this takes the form
which can be simplified as
Conditioning on a third random variable may either increase or decrease the mutual information, but it is always true that
for discrete, jointly distributed random variables. This result has been used as a basic building block for proving other inequalities in information theory.

Multivariate mutual information

Several generalizations of mutual information to more than two random variables have been proposed, such as total correlation and interaction information. The expression and study of multivariate higher-degree mutual-information was achieved in two seemingly independent works: McGill who called these functions “interaction information”, and Hu Kuo Ting who also first proved the possible negativity of mutual-information for degrees higher than 2 and justified algebraically the intuitive correspondence to Venn diagrams
and for
where we define

Multivariate statistical independence

The multivariate mutual-information functions generalize the pairwise independence case that states that if and only if, to arbitrary numerous variable. n variables are mutually independent if and only if the mutual information functions vanish with . In this sense, the can be used as a refined statistical independence criterion.

Applications

For 3 variables, Brenner et al. applied multivariate mutual information to neural coding and called its negativity "synergy" and Watkinson et al. applied it to genetic expression. For arbitrary k variables, Tapia et al. applied multivariate mutual information to gene expression ). It can be zero, positive, or negative. The positivity corresponds to relations generalizing the pairwise correlations, nullity corresponds to a refined notion of independence, and negativity detects high dimensional "emergent" relations and clusterized datapoints ).
One high-dimensional generalization scheme which maximizes the mutual information between the joint distribution and other target variables is found to be useful in feature selection.
Mutual information is also used in the area of signal processing as a measure of similarity between two signals. For example, FMI metric is an image fusion performance measure that makes use of mutual information in order to measure the amount of information that the fused image contains about the source images. The Matlab code for this metric can be found at.

Directed information

,, measures the amount of information that flows from the process to, where denotes the vector and denotes. The term directed information was coined by James Massey and is defined as
Note that if, the directed information becomes the mutual information. Directed information has many applications in problems where causality plays an important role, such as capacity of channel with feedback.

Normalized variants

Normalized variants of the mutual information are provided by the coefficients of constraint, uncertainty coefficient or proficiency:
The two coefficients have a value ranging in , but are not necessarily equal. In some cases a symmetric measure may be desired, such as the following redundancy measure:
which attains a minimum of zero when the variables are independent and a maximum value of
when one variable becomes completely redundant with the knowledge of the other. See also Redundancy .
Another symmetrical measure is the symmetric uncertainty, given by
which represents the harmonic mean of the two uncertainty coefficients.
If we consider mutual information as a special case of the total correlation or dual total correlation, the normalized version are respectively,
This normalized version also known as Information Quality Ratio which quantifies the amount of information of a variable based on another variable against total uncertainty:
There's a normalization which derives from first thinking of mutual information as an analogue to covariance. Then the normalized mutual information is calculated akin to the Pearson correlation coefficient,

Weighted variants

In the traditional formulation of the mutual information,
each event or object specified by is weighted by the corresponding probability. This assumes that all objects or events are equivalent apart from their probability of occurrence. However, in some applications it may be the case that certain objects or events are more significant than others, or that certain patterns of association are more semantically important than others.
For example, the deterministic mapping may be viewed as stronger than the deterministic mapping, although these relationships would yield the same mutual information. This is because the mutual information is not sensitive at all to any inherent ordering in the variable values, and is therefore not sensitive at all to the form of the relational mapping between the associated variables. If it is desired that the former relation—showing agreement on all variable values—be judged stronger than the later relation, then it is possible to use the following weighted mutual information.
which places a weight on the probability of each variable value co-occurrence,. This allows that certain probabilities may carry more or less significance than others, thereby allowing the quantification of relevant holistic or Prägnanz factors. In the above example, using larger relative weights for,, and would have the effect of assessing greater informativeness for the relation than for the relation, which may be desirable in some cases of pattern recognition, and the like. This weighted mutual information is a form of weighted KL-Divergence, which is known to take negative values for some inputs, and there are examples where the weighted mutual information also takes negative values.

Adjusted mutual information

A probability distribution can be viewed as a partition of a set. One may then ask: if a set were partitioned randomly, what would the distribution of probabilities be? What would the expectation value of the mutual information be? The adjusted mutual information or AMI subtracts the expectation value of the MI, so that the AMI is zero when two different distributions are random, and one when two distributions are identical. The AMI is defined in analogy to the adjusted Rand index of two different partitions of a set.

Absolute mutual information

Using the ideas of Kolmogorov complexity, one can consider the mutual information of two sequences independent of any probability distribution:
To establish that this quantity is symmetric up to a logarithmic factor one requires the chain rule for Kolmogorov complexity. Approximations of this quantity via compression can be used to define a distance measure to perform a hierarchical clustering of sequences without having any domain knowledge of the sequences.

Linear correlation

Unlike correlation coefficients, such as the product moment correlation coefficient, mutual information contains information about all dependence—linear and nonlinear—and not just linear dependence as the correlation coefficient measures. However, in the narrow case that the joint distribution for and is a bivariate normal distribution, there is an exact relationship between and the correlation coefficient .
The equation above can be derived as follows for a bivariate Gaussian:
Therefore,

For discrete data

When and are limited to be in a discrete number of states, observation data is summarized in a contingency table, with row variable and column variable . Mutual information is one of the measures of association or correlation between the row and column variables. Other measures of association include Pearson's chi-squared test statistics, G-test statistics, etc. In fact, mutual information is equal to G-test statistics divided by, where is the sample size.

Applications

In many applications, one wants to maximize mutual information, which is often equivalent to minimizing conditional entropy. Examples include: