Mutual Information

"Information is a difference that makes a difference."

• Gregory Bateson

Overview

Mutual information is the canonical quantitative measure of statistical dependence. If entropy asks "how uncertain is a random variable?" and KL divergence asks "how different are two distributions?", then mutual information asks the question that matters for learning: "how much does observing one random variable reduce uncertainty about another?"

That single question appears everywhere in AI. Feature selection ranks variables by how much they tell us about labels. Decision trees split on information gain. Contrastive learning objectives such as CPC and InfoNCE are motivated as lower bounds on mutual information. CLIP-style image-text alignment is built around preserving information shared across modalities. Active learning asks which next observation would be most informative about unknown parameters or labels. The information bottleneck formalizes representation learning as a trade-off between compression and task relevance. When we say a representation is "useful," we often mean it preserves the right mutual information and discards the wrong mutual information.

This section develops mutual information from first principles, but keeps the scope disciplined. The full treatments of entropy and KL divergence live in 01-Entropy and 02-KL-Divergence. Here we use them as building blocks and focus on what is special about mutual information itself: dependence, uncertainty reduction, conditional information flow, data processing, estimation, and its role in modern machine learning.

Prerequisites

• Entropy and conditional entropy - definitions, chain rule, and basic properties - 01-Entropy
• KL divergence - non-negativity, asymmetry, and interpretation as relative entropy - 02-KL-Divergence
• Conditional probability and Bayes' rule - from Probability Theory
• Markov chains and conditional independence - for data processing and sufficiency - 07-Markov-Chains
• Convexity and optimization basics - useful for variational bounds and information bottleneck objectives - 08-Optimization

Companion Notebooks

Learning Objectives

After completing this section, you will:

1. Define mutual information in discrete and continuous settings and explain why it measures dependence rather than mere correlation
2. Derive the equivalent forms $I(X;Y) = H(X) - H(X \mid Y) = H(Y) - H(Y \mid X) = D_{\mathrm{KL}}(p_{XY} \Vert p_X p_Y)$
3. Interpret pointwise mutual information and explain why PMI is central in co-occurrence models and embedding methods
4. State and prove the most important properties of mutual information: symmetry, non-negativity, upper bounds, and the independence criterion
5. Use chain rules for mutual information and conditional mutual information to reason about information flow in multi-variable systems
6. State and apply the data processing inequality to learned representations and Markov chains
7. Compute mutual information for canonical examples such as binary channels and Gaussian variables
8. Explain channel capacity as the maximal achievable mutual information over an input distribution
9. Understand why estimating MI from high-dimensional samples is difficult and compare classical and variational estimators
10. Derive and interpret InfoNCE, MINE, and the information bottleneck objective in modern ML terms
11. Connect mutual information to feature selection, contrastive learning, multimodal alignment, and active learning

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Table of Contents

• Mutual Information
  • Overview
  • Prerequisites
  • Companion Notebooks
  • Learning Objectives
  • Table of Contents
  • 1. Intuition
    • 1.1 Shared Information, Not Just Correlation
    • 1.2 Mutual Information as Uncertainty Reduction
    • 1.3 Why Mutual Information Matters for AI
    • 1.4 Historical Timeline
  • 2. Formal Definitions
    • 2.1 Discrete Mutual Information
    • 2.2 Mutual Information as KL Divergence
    • 2.3 Pointwise Mutual Information (PMI)
    • 2.4 Continuous Mutual Information
    • 2.5 Conditional Mutual Information
  • 3. Core Theory I: Fundamental Properties
    • 3.1 Symmetry and Nonnegativity
    • 3.2 Bounds and Equality Cases
    • 3.3 Independence If and Only If MI Is Zero
    • 3.4 Chain Rules for Mutual Information
    • 3.5 Deterministic Functions and Invariance Facts
  • 4. Core Theory II: Information Flow, Conditioning, and Structure
    • 4.1 Conditional Mutual Information and Explaining Away
    • 4.2 Data Processing Inequality
    • 4.3 Markov Chains, Sufficiency, and Representation
    • 4.4 Interaction Information and Synergy
    • 4.5 Total Correlation and Multi-Information
  • 5. Core Theory III: Communication, Prediction, and Limits
    • 5.1 MI as Expected Log-Likelihood Ratio
    • 5.2 Channel Mutual Information
    • 5.3 Channel Capacity
    • 5.4 Binary Symmetric and Gaussian Channel Examples
    • 5.5 Fano's Inequality and Prediction Limits
  • 6. Advanced Topics: Estimation and Optimization
    • 6.1 Why Estimating MI Is Hard in High Dimensions
    • 6.2 Classical Estimators
    • 6.3 Variational Bounds: Donsker-Varadhan, NWJ, and InfoNCE
    • 6.4 MINE and Neural Estimators
    • 6.5 Information Bottleneck
  • 7. Applications in Machine Learning
    • 7.1 Information Gain and Feature Selection
    • 7.2 Contrastive Predictive Coding and Self-Supervised Learning
    • 7.3 CLIP-Style Multimodal Alignment
    • 7.4 Information Bottleneck in Representation Learning
    • 7.5 Active Learning and Bayesian Experimental Design
  • 8. Common Mistakes
  • 9. Exercises
  • 10. Why This Matters for AI (2026 Perspective)
  • 11. Conceptual Bridge
  • References

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1. Intuition

1.1 Shared Information, Not Just Correlation

The easiest way to misunderstand mutual information is to think of it as "just another dependence statistic." It is not. Correlation measures linear association. Mutual information measures any statistical dependence that changes the conditional distribution of one variable after observing the other.

Consider three examples:

1. If $Y = 2X + 1$ with no noise, then $X$ and $Y$ are perfectly dependent and both correlation and mutual information are large.
2. If $X \in \{-1,+1\}$ uniformly and $Y = X^2$, then $Y$ is constant. Correlation is zero and mutual information is also zero because observing $Y$ teaches us nothing.
3. If $X \sim \mathcal{N}(0,1)$ and $Y = X^2$, then correlation can be zero because the relationship is symmetric around zero, yet mutual information is positive because knowing $Y$ narrows down what values of $X$ are plausible.

This third example is the important one. Zero covariance does not imply independence except in special families such as jointly Gaussian distributions. Mutual information detects dependence whenever the joint distribution $p_{XY}$ fails to factorize as $p_X p_Y$, regardless of whether the dependence is linear, nonlinear, monotone, multimodal, or structurally conditional.

DEPENDENCE VS CORRELATION
===============================================================

Case A: linear dependence
  X  ---------->  Y = 2X + 1
  Corr != 0
  MI   >  0

Case B: nonlinear symmetric dependence
  X  ---------->  Y = X^2
  Corr may = 0
  MI   >  0

Case C: independence
  X      Y
   \    /
    no link
  Corr = 0
  MI   = 0

Mutual information asks:
  "Does observing Y change the distribution of X?"
not merely:
  "Is there a linear trend between X and Y?"
===============================================================

This is why mutual information matters so much in representation learning. Useful latent variables need not be linearly predictive of labels or downstream variables; they only need to preserve the right uncertainty-reducing structure.

1.2 Mutual Information as Uncertainty Reduction

Entropy gives a baseline uncertainty. Conditional entropy gives the uncertainty that remains after an observation. Mutual information is the gap:

$$I(X;Y) = H(X) - H(X \mid Y).$$

That formula is worth reading slowly. Before observing $Y$, uncertainty about $X$ is $H(X)$. After observing $Y$, the average remaining uncertainty is $H(X \mid Y)$. Their difference is the average reduction in uncertainty due to the observation of $Y$.

Because the same reasoning works in the other direction,

$$I(X;Y) = H(Y) - H(Y \mid X),$$

mutual information is symmetric. The quantity measures shared uncertainty reduction.

For a concrete discrete example, imagine a noisy binary label $Y$ generated from a feature $X$:

• if $X$ is uninformative, then $H(Y \mid X) \approx H(Y)$, so $I(X;Y) \approx 0$
• if $X$ almost determines $Y$, then $H(Y \mid X)$ is small, so $I(X;Y)$ is close to $H(Y)$

This is the logic behind information gain in decision trees and many feature-selection criteria.

UNCERTAINTY REDUCTION VIEW
===============================================================

Before seeing Y:
  uncertainty about X = H(X)

After seeing Y:
  remaining uncertainty = H(X|Y)

Shared information:
  I(X;Y) = H(X) - H(X|Y)

If Y tells us nothing:
  H(X|Y) = H(X)   -> I(X;Y) = 0

If Y determines X exactly:
  H(X|Y) = 0      -> I(X;Y) = H(X)
===============================================================

Recall: the full theory of entropy and conditional entropy lives in 01-Entropy. In this section we use those quantities as primitives and focus on the new object created by their difference.

1.3 Why Mutual Information Matters for AI

Mutual information appears whenever a learning system must preserve some aspects of a signal while discarding others.

The core pattern is always the same:

1. There is a source of uncertainty.
2. There is a candidate observation, representation, or transformation.
3. The system wants to know how much that object decreases uncertainty about something important.

Mutual information answers that question in a model-agnostic way. It does not care whether the variables are bits, words, images, embeddings, or actions in a reinforcement-learning environment.

For modern AI, this is especially powerful because the same formalism links classical and modern methods:

• information gain in ID3/C4.5 decision trees
• the information bottleneck objective in supervised representation learning
• InfoNCE-style lower bounds used in CPC, SimCLR, and CLIP-style contrastive learning
• BALD-style active learning, where one asks how much an observation informs the posterior over parameters or labels

Mutual information is therefore not a niche information-theory quantity. It is one of the main bridges between probability, learning theory, and practical objective design.

1.4 Historical Timeline

MUTUAL INFORMATION -- KEY MILESTONES
===============================================================

  1948  Shannon
        Defines information quantities for communication systems.
        Mutual information emerges as the rate of information flow
        through a noisy channel.

  1954  McGill
        Formalizes multivariate information ideas beyond the pairwise
        setting and helps shape later interaction-information work.

  1959  Fano
        Connects mutual information to prediction limits and decoding
        error through what is now called Fano's inequality.

  1970s Channel coding theory
        Mutual information becomes central to channel capacity and
        reliable communication limits.

  1980s-1990s Information geometry
        MI is connected more deeply to dependence, sufficiency,
        and statistical structure.

  1999  Tishby, Pereira, Bialek
        Information Bottleneck frames representation learning as
        compression-versus-relevance optimization.

  2018  CPC (van den Oord et al.)
        Contrastive Predictive Coding popularizes InfoNCE as a
        practical MI-inspired learning objective.

  2018  Belghazi et al.
        MINE proposes neural mutual-information estimation.

  2021  CLIP (Radford et al.)
        Contrastive image-text alignment makes MI-style objectives
        central to multimodal representation learning.

  2024-2026
        Mutual-information ideas remain active in self-supervised
        learning, active learning, multimodal alignment, and theory
        of representation compression.
===============================================================

Mutual information began life in communication theory, but today it is equally a language for reasoning about what learned representations retain, discard, and reveal.

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2. Formal Definitions

2.1 Discrete Mutual Information

Let $X$ and $Y$ be discrete random variables with joint PMF $p_{XY}(x,y)$ and marginals $p_X(x)$, $p_Y(y)$. The discrete mutual information is

$$I(X;Y) = \sum_{x,y} p_{XY}(x,y)\log\frac{p_{XY}(x,y)}{p_X(x)p_Y(y)}.$$

Equivalent entropy forms are

$$I(X;Y) = H(X) + H(Y) - H(X,Y)$$

and

$$I(X;Y) = H(X) - H(X\mid Y) = H(Y) - H(Y\mid X).$$

These formulas are equivalent whenever the entropies are finite.

Worked example. Suppose $X,Y \in \{0,1\}$ with

$$\begin{array}{c|cc} & Y=0 & Y=1 \\ \hline X=0 & 0.4 & 0.1 \\ X=1 & 0.1 & 0.4 \end{array}$$

Then $p_X(0)=p_X(1)=0.5$ and $p_Y(0)=p_Y(1)=0.5$. So

$$I(X;Y) = 0.4\log\frac{0.4}{0.25} + 0.1\log\frac{0.1}{0.25} + 0.1\log\frac{0.1}{0.25} + 0.4\log\frac{0.4}{0.25}.$$

The positive contributions come from outcomes that occur more often jointly than independence would predict. Negative pointwise contributions can also occur for individually "anti-associated" outcomes, but the overall sum is always nonnegative.

2.2 Mutual Information as KL Divergence

The identity

$$I(X;Y) = D_{\mathrm{KL}}(p_{XY} \Vert p_X p_Y)$$

is one of the most important equations in the chapter.

It says mutual information is the KL divergence between:

• the actual joint distribution $p_{XY}$, and
• the independence model $p_X p_Y$ built from the marginals.

This is conceptually powerful because it turns a dependence question into a distribution-comparison question:

• if $X$ and $Y$ are independent, then $p_{XY}=p_X p_Y$, so the divergence is zero
• if they are dependent, the joint law differs from the product law, and the divergence is positive

Backward link: the general properties of $D_{\mathrm{KL}}$ belong to 02-KL-Divergence. In this section we use that theory to inherit non-negativity, but the object of interest is specifically the divergence to the product-of-marginals baseline.

This perspective also clarifies why mutual information is invariant under invertible reparameterizations in both arguments: KL divergence between corresponding transformed measures is unchanged.

2.3 Pointwise Mutual Information (PMI)

The pointwise mutual information between a specific pair $(x,y)$ is

$$\operatorname{pmi}(x,y) = \log\frac{p_{XY}(x,y)}{p_X(x)p_Y(y)}.$$

Mutual information is the expectation of PMI:

$$I(X;Y) = \mathbb{E}_{(X,Y)\sim p_{XY}}[\operatorname{pmi}(X,Y)].$$

PMI is more local than MI:

• $\operatorname{pmi}(x,y) > 0$ means the pair co-occurs more often than independence predicts
• $\operatorname{pmi}(x,y) = 0$ means the pair occurs exactly as often as independence predicts
• $\operatorname{pmi}(x,y) < 0$ means the pair co-occurs less often than independence predicts

In NLP, PMI has historically been used to score word associations:

• "New" with "York" has large positive PMI
• "the" with many other words may have lower PMI than raw co-occurrence counts suggest, because "the" is common with almost everything

This matters because co-occurrence counts alone are dominated by frequent tokens. PMI rescales by marginal frequency and therefore asks whether the pair is surprisingly associated, not merely common.

Caution. PMI can become arbitrarily large for rare-but-exclusive pairs, which makes it unstable in low-sample regimes. Positive PMI (PPMI) and smoothed PMI variants are often used in practice.

2.4 Continuous Mutual Information

For continuous random variables with joint density $p_{XY}(x,y)$ and marginal densities $p_X(x)$, $p_Y(y)$, the mutual information is

$$I(X;Y) = \int \int p_{XY}(x,y)\log\frac{p_{XY}(x,y)}{p_X(x)p_Y(y)}\,dx\,dy,$$

provided the integral is well defined.

This is not simply a difference of differential entropies in the naive sense unless the needed quantities exist. But when they do,

$$I(X;Y) = h(X) - h(X\mid Y) = h(Y) - h(Y\mid X),$$

where $h$ denotes differential entropy.

The subtle but crucial fact is:

• differential entropy can be negative and is coordinate-sensitive
• mutual information remains nonnegative and invariant under smooth invertible transformations

That is one reason mutual information is often a safer continuous information measure than raw differential entropy.

2.5 Conditional Mutual Information

The conditional mutual information is

$$I(X;Y\mid Z) = H(X\mid Z) - H(X\mid Y,Z).$$

It can also be written as

$$I(X;Y\mid Z) = \sum_z p_Z(z)\, D_{\mathrm{KL}}(p_{XY\mid Z=z} \Vert p_{X\mid Z=z} p_{Y\mid Z=z}),$$

or, in expectation form,

$$I(X;Y\mid Z) = \mathbb{E}\left[\log\frac{p_{XY\mid Z}(X,Y\mid Z)}{p_{X\mid Z}(X\mid Z)p_{Y\mid Z}(Y\mid Z)}\right].$$

The interpretation is straightforward:

• first condition on $Z$
• then ask how much extra uncertainty about $X$ is reduced by seeing $Y$

So $I(X;Y\mid Z)$ measures information shared by $X$ and $Y$ beyond what $Z$ already explains.

This quantity becomes central in graphical models, causal reasoning, and active-learning design because it separates direct information from information already mediated by other variables.

---

3. Core Theory I: Fundamental Properties

3.1 Symmetry and Nonnegativity

Mutual information is symmetric:

$$I(X;Y)=I(Y;X).$$

This is immediate from the entropy formula $H(X)+H(Y)-H(X,Y)$, which is symmetric in $X$ and $Y$.

It is also nonnegative:

$$I(X;Y)\ge 0.$$

The cleanest proof uses the KL representation:

$$I(X;Y)=D_{\mathrm{KL}}(p_{XY} \Vert p_X p_Y)\ge 0.$$

Equality holds if and only if $p_{XY}=p_X p_Y$ almost everywhere, i.e. $X$ and $Y$ are independent.

This already tells us something profound: mutual information is a dependence measure with a mathematically exact zero point. Many statistics do not enjoy that property.

3.2 Bounds and Equality Cases

Because conditioning cannot increase entropy,

$$H(X\mid Y)\le H(X).$$

Therefore

$$I(X;Y)=H(X)-H(X\mid Y)\le H(X).$$

By symmetry,

$$I(X;Y)\le H(Y),$$

so overall

$$0 \le I(X;Y) \le \min\{H(X),H(Y)\}.$$

When do we attain the upper bound?

• $I(X;Y)=H(X)$ if and only if $H(X\mid Y)=0$, meaning $X$ is a deterministic function of $Y$
• $I(X;Y)=H(Y)$ if and only if $H(Y\mid X)=0$, meaning $Y$ is a deterministic function of $X$

If both hold, then $X$ and $Y$ determine one another almost surely: they are invertibly linked up to null sets.

BOUNDS ON MUTUAL INFORMATION
===============================================================

0 <= I(X;Y) <= min{H(X), H(Y)}

Interpretation:
  MI can never exceed the entire uncertainty of either variable.

Extreme cases:
  Independence:
    I(X;Y) = 0

  X determined by Y:
    I(X;Y) = H(X)

  Y determined by X:
    I(X;Y) = H(Y)

  Perfect reversible link:
    I(X;Y) = H(X) = H(Y)
===============================================================

3.3 Independence If and Only If MI Is Zero

This property is so important that it deserves its own subsection:

$$I(X;Y)=0 \quad \Longleftrightarrow \quad X \perp\!\!\!\perp Y.$$

The forward implication follows from KL non-negativity and equality conditions. The reverse implication follows by direct substitution: if $p_{XY}=p_X p_Y$, every log-ratio term becomes zero.

This criterion is much stronger than:

• zero correlation
• zero covariance
• zero linear regression coefficient

Those can all vanish in the presence of nonlinear dependence. Mutual information vanishes only when there is no statistical dependence at all.

Gaussian exception. If $(X,Y)$ are jointly Gaussian, then zero covariance does imply independence. That is why, in Gaussian settings, MI can be expressed entirely through covariance structure. Outside that setting, covariance is only a partial picture.

3.4 Chain Rules for Mutual Information

The mutual-information chain rule states

$$I(X_1,\dots,X_n;Y) = \sum_{i=1}^n I(X_i;Y\mid X_1,\dots,X_{i-1}).$$

For two variables this is

$$I(X,Z;Y)=I(X;Y)+I(Z;Y\mid X).$$

Derivation. Start from entropy:

$$I(X,Z;Y) = H(Y)-H(Y\mid X,Z).$$

Add and subtract $H(Y\mid X)$:

$$I(X,Z;Y) = \big(H(Y)-H(Y\mid X)\big) + \big(H(Y\mid X)-H(Y\mid X,Z)\big) = I(X;Y)+I(Z;Y\mid X).$$

This is the information-theoretic analogue of decomposing predictive contribution into what is already explained and what is newly explained after conditioning.

In ML, chain rules appear whenever information is accumulated sequentially:

• sequential feature selection
• autoregressive modeling
• layered representation analysis
• sensor fusion and multimodal inference

3.5 Deterministic Functions and Invariance Facts

If $T=f(X)$ is a deterministic function of $X$, then

$$I(T;Y)\le I(X;Y).$$

This is already a special case of the data processing inequality, but it is worth interpreting directly:

• deterministic post-processing can discard information
• it cannot create new information about $Y$ that was not already present in $X$

If $f$ is bijective, then no information is lost:

$$I(f(X);Y)=I(X;Y).$$

This invariance under invertible reparameterization is one reason mutual information is conceptually robust. It depends on the dependence structure between variables, not the arbitrary coordinates in which the variables are represented.

However, coarse-graining or compression can strictly reduce MI. If we bin a high-resolution image into a low-resolution thumbnail, the thumbnail may preserve much of the task-relevant information but not all of it. That tension between useful preservation and irreversible loss lies at the heart of learned representation design.

---

4. Core Theory II: Information Flow, Conditioning, and Structure

4.1 Conditional Mutual Information and Explaining Away

Conditional mutual information is where dependence structure becomes interesting. Unconditioned variables can appear independent but become dependent after conditioning, or vice versa.

Two canonical patterns matter:

1. Common cause: if $Z$ influences both $X$ and $Y$, then $X$ and $Y$ may be dependent marginally, but conditioning on $Z$ can remove the dependence.
2. Common effect (collider): if $X$ and $Y$ both influence $Z$, then $X$ and $Y$ may be marginally independent but become dependent after conditioning on $Z$.

This second phenomenon is often called explaining away. In diagnosis, two independent diseases can become negatively associated once a common symptom is observed: if one disease is present, it partly explains the symptom, reducing the need to attribute it to the other.

Conditional MI quantifies exactly how much residual dependence remains after the conditioning variable is known.

EXPLAINING AWAY
===============================================================

X  ----\
        >---- Z observed
Y  ----/

Before conditioning on Z:
  X and Y may be independent

After conditioning on Z:
  X and Y can become dependent

Reason:
  evidence for X partially "explains away" the need for Y
===============================================================

4.2 Data Processing Inequality

If $X \to T \to Y$ forms a Markov chain, then

$$I(X;Y)\le I(X;T).$$

More commonly one writes, for $X \to T \to Y$,

$$I(X;Y)\le I(X;T) \quad\text{and}\quad I(T;Y)\le I(X;Y)$$

only when the chain is ordered appropriately. The important statement is: processing cannot increase information about the original source beyond what the intermediate representation already contains.

One proof uses the chain rule:

$$I(X;T,Y)=I(X;T)+I(X;Y\mid T).$$

Because $X \to T \to Y$ means $X$ and $Y$ are conditionally independent given $T$, we have $I(X;Y\mid T)=0$. Hence

$$I(X;T,Y)=I(X;T).$$

Also,

$$I(X;T,Y)=I(X;Y)+I(X;T\mid Y)\ge I(X;Y),$$

so $I(X;Y)\le I(X;T)$.

This theorem is foundational for representation learning:

• any learned representation $T=f(X)$ is a processed form of the input
• therefore $T$ cannot contain more information about a downstream target than the input could possibly supply
• the design problem is not to create information, but to preserve the right information while discarding nuisance variation

An equivalent way to say this is that representation learning is always constrained by an information budget inherited from the raw signal. If an image does not contain enough bits about the identity of a bird species, no architectural trick can manufacture that missing evidence. Better models can extract weak cues more efficiently, but they cannot violate the information theoretic limit set by the observation process itself.

This is why DPI is such a useful sanity check in modern ML discussions. It prevents a common confusion between:

• better extraction of existing signal, and
• creation of new task-relevant signal

Large foundation models often feel as if they "discover" new information, but what they actually do is transform raw data into representations that preserve subtle structure humans or weaker models fail to exploit. Data processing inequality tells us that this gain is about organization and accessibility, not about violating conservation of information.

DATA PROCESSING INEQUALITY IN ML
===============================================================

raw input X  --->  learned representation T  --->  predictor output

Possible:
  T removes nuisance variation
  T makes downstream prediction easier
  T exposes structure already present in X

Impossible:
  T contains target information that was never present in X

So the design goal is:
  preserve the right information, not invent new information
===============================================================

4.3 Markov Chains, Sufficiency, and Representation

A statistic $T(X)$ is sufficient for a parameter $\Theta$ if conditioning on $T$ renders $X$ irrelevant to $\Theta$. In information terms, sufficiency can be characterized as

$$I(\Theta;X\mid T)=0.$$

That is a beautifully compact statement: once you know $T$, the raw data $X$ tells you nothing further about $\Theta$.

In representation learning, this suggests an ideal:

• compress the raw input $X$ into a representation $T$
• but preserve the information in $X$ that matters for the target $Y$

The resulting design tension is the conceptual starting point for the information bottleneck:

$$\text{small } I(X;T) \quad\text{but large } I(T;Y).$$

This perspective is more general than any particular neural architecture. It is a way of asking what a representation should remember and what it should forget.

There is also an important statistical lesson here. A sufficient statistic is not merely a "compressed summary." It is a summary that is lossless for the inference problem at hand. That problem-specific phrase matters. A statistic may be sufficient for estimating one parameter and insufficient for another. In machine learning language, the same embedding can be excellent for sentiment classification, mediocre for topic prediction, and useless for speaker identification, because those tasks preserve different information.

This is one reason mutual-information language scales well from classical statistics to modern representation learning. It gives us a way to state the quality of a summary relative to a target, rather than relative to some vague notion of "informativeness." Informativeness is never absolute. It is always about information about what?

4.4 Interaction Information and Synergy

Pairwise mutual information does not capture all multi-variable structure. Suppose two bits $X_1$ and $X_2$ are independent fair coins, and $Y = X_1 \oplus X_2$ is their XOR.

Then:

• $I(X_1;Y)=0$
• $I(X_2;Y)=0$
• but $I((X_1,X_2);Y)=1$ bit

So neither variable individually tells us anything about $Y$, but together they determine it completely. This is synergy.

Interaction information and related multivariate quantities attempt to formalize such effects. The sign conventions can be subtle, which is why many applied papers prefer partial-information decompositions or total correlation instead of relying on interaction information alone.

The key lesson is not the exact choice of multivariate information measure. It is that pairwise MI can miss structurally important dependence that only appears jointly.

One common definition of three-way interaction information is

$$I(X;Y;Z) = I(X;Y) - I(X;Y\mid Z),$$

with equivalent permutations obtained by symmetry. Under this sign convention, interaction information can be positive or negative:

• positive values are often associated with redundancy
• negative values are often associated with synergy

The XOR example illustrates the danger of relying only on pairwise statistics. Each input bit is individually useless for predicting the output, yet the pair is perfectly informative. This is a recurring theme in ML systems with compositional structure. Individual neurons, tokens, or sensors may appear weakly informative in isolation while being decisively informative as a group.

That is also why feature-selection methods based only on marginal MI can fail. They may discard variables that are only useful in combination. Good practice therefore depends on the problem structure:

• if the signal is mostly additive, pairwise screening may work well
• if the signal is highly combinatorial, we need conditional or multivariate information analysis

4.5 Total Correlation and Multi-Information

For random variables $X_1,\dots,X_n$, the total correlation (also called multi-information in one common convention) is

$$\operatorname{TC}(X_1,\dots,X_n) = D_{\mathrm{KL}}\!\left(p_{X_1,\dots,X_n}\,\middle\Vert \,\prod_{i=1}^n p_{X_i}\right).$$

Equivalently,

$$\operatorname{TC}(X_1,\dots,X_n) = \sum_{i=1}^n H(X_i) - H(X_1,\dots,X_n).$$

This is the multivariate analogue of mutual information:

• it is zero iff the variables are jointly independent
• it grows as joint dependence strengthens

In disentanglement and latent-variable learning, total correlation is often penalized to encourage factorized latent representations. A model with low total correlation across latent coordinates is closer to one whose latent dimensions behave independently.

Another way to read total correlation is as the KL gap between the true joint latent law and the fantasy world in which latent coordinates are independent. That makes it especially natural in variational latent-variable models, where the factorization assumption is often built directly into the encoder or prior.

It is worth separating three related questions:

1. Are pairwise correlations small?
2. Are pairwise mutual informations small?
3. Is the full joint law close to a product distribution?

Only the third question is answered by total correlation. A latent space can have weak pairwise dependence while still exhibiting strong higher-order dependence. Total correlation is designed to see that broader failure of factorization.

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5. Core Theory III: Communication, Prediction, and Limits

5.1 MI as Expected Log-Likelihood Ratio

Mutual information can be written as

$$I(X;Y)=\mathbb{E}\left[\log\frac{p_{Y\mid X}(Y\mid X)}{p_Y(Y)}\right].$$

This form is especially intuitive:

• $p_Y(Y)$ is the baseline probability of the observation
• $p_{Y\mid X}(Y\mid X)$ is the probability after knowing $X$
• the log-ratio is how much the observation becomes more or less likely once $X$ is known

Averaging gives the expected log-evidence gained from seeing $X$.

This is why MI naturally appears in communication, statistics, Bayesian experimental design, and likelihood-ratio thinking. It quantifies average evidence transfer.

5.2 Channel Mutual Information

In a communication channel, $X$ is the channel input and $Y$ is the output after noise. The channel law is $p(y\mid x)$. Once an input distribution $p(x)$ is chosen, the joint law becomes

$$p(x,y)=p(x)p(y\mid x).$$

The mutual information

$$I(X;Y)$$

measures how much information the output carries about the input under that chosen input distribution.

This is not yet channel capacity, because capacity optimizes over the input law. But it is the basic performance quantity for a given signaling strategy.

Another useful identity is

$$I(X;Y) = \sum_x p(x)\,D_{\mathrm{KL}}\!\big(p_{Y\mid X=x}\,\Vert\,p_Y\big).$$

This says mutual information is the average KL divergence between:

• the output distribution after a specific input symbol $x$, and
• the unconditional output distribution before we know which symbol was sent

So a channel has high MI when different inputs induce clearly distinguishable output distributions. If all the conditionals $p(y\mid x)$ look nearly the same, then the output barely reveals which input was chosen and the channel carries little information.

That interpretation transfers directly to ML. A representation is useful for a labeling problem when different classes induce distinguishable representation distributions. The more separable those induced distributions are, the more class information the representation preserves.

5.3 Channel Capacity

The channel capacity is

$$C = \max_{p(x)} I(X;Y).$$

Capacity is the largest rate at which information can be transmitted reliably through the channel in the limit of long block length. The theorem-level content belongs to full information-theory courses, but the conceptual meaning is essential:

• the channel law determines how inputs get corrupted
• the input distribution determines how cleverly we use the channel
• capacity is the best achievable mutual information over that design choice

In machine learning terms, capacity is a maximization problem over an input distribution rather than over model weights, but it shares the same spirit as choosing a representation or policy to maximize useful information flow.

Two cautions are helpful here.

First, channel capacity is not model capacity in the usual deep-learning sense. In neural-network practice, "capacity" often means expressiveness of a function class. In information theory, channel capacity means the maximal reliable communication rate of a noisy mechanism. The word is the same, but the objects are different.

Second, the maximizing input distribution is part of the problem. A channel can be mediocre under one signaling strategy and much better under another. For finite discrete channels, algorithms such as Blahut-Arimoto compute the optimal input law numerically. For Gaussian channels under power constraints, the capacity-achieving distribution is itself Gaussian. These results are reminders that information flow depends jointly on the mechanism and the way we excite that mechanism.

5.4 Binary Symmetric and Gaussian Channel Examples

Binary symmetric channel (BSC). Let $X \in \{0,1\}$ and suppose the channel flips the bit with probability $\varepsilon$. If the input is Bernoulli-$\tfrac12$, then

$$I(X;Y)=1-h_2(\varepsilon),$$

where $h_2$ is the binary entropy function.

This formula is perfect intuition:

• if $\varepsilon=0$, the channel is noiseless and MI is 1 bit
• if $\varepsilon=\tfrac12$, output is pure noise and MI is 0

Gaussian channel. For $Y=X+N$ with Gaussian noise $N\sim \mathcal{N}(0,\sigma^2)$ and power constraint $\mathbb{E}[X^2]\le P$, the capacity is

$$C = \frac12 \log\left(1+\frac{P}{\sigma^2}\right)$$

in nats per channel use.

This is one of the great formulas in information theory: information grows logarithmically with signal-to-noise ratio, not linearly.

That logarithmic law is deeply relevant for AI systems that operate under noisy measurement, bandwidth, or token-budget constraints. Doubling the available power or bandwidth does not double reliable information transmission. Gains are real but sublinear. This is one reason compression, representation design, and clever coding matter so much in practice: we rarely win by brute-force scaling alone.

Also notice the difference between the discrete and continuous examples:

• in the BSC, noise is a flip probability on a finite alphabet
• in the Gaussian channel, noise is additive and the key control variable is signal-to-noise ratio

The same mutual-information formalism handles both cases, which is part of its power. The details of the channel law change, but the central question stays the same: how distinguishable are the outputs induced by different inputs?

5.5 Fano's Inequality and Prediction Limits

Fano's inequality connects mutual information to classification error. In one common form, if $Y$ is a label drawn from $M$ classes and $\hat{Y}$ is an estimate based on observed data $X$, then

$$H(Y\mid X)\le h_2(P_e)+P_e\log(M-1),$$

where $P_e = \Pr[\hat{Y}\ne Y]$.

Rearranging with $I(X;Y)=H(Y)-H(Y\mid X)$ shows:

• small MI implies substantial irreducible prediction error
• high prediction accuracy requires enough information about the label to flow through the observation

This is conceptually important for ML because it separates optimization failure from information-theoretic impossibility. Sometimes a model fails because training is bad. Sometimes it fails because the input simply does not carry enough label information.

That distinction is healthy for scientific thinking. When a benchmark is hard, we should ask at least three different questions:

1. Is the optimization procedure effective?
2. Is the model class expressive enough?
3. Does the input actually contain enough information for the target task?

Fano-style reasoning speaks primarily to the third question. It does not tell us how to optimize better, but it does tell us when better optimization alone cannot close the gap.

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6. Advanced Topics: Estimation and Optimization

6.1 Why Estimating MI Is Hard in High Dimensions

Mutual information is easy to define but hard to estimate from samples, especially in high dimensions.

The main reasons are:

1. Density estimation is hard. MI depends on joint and marginal densities or PMFs. In high dimensions these are expensive to estimate well.
2. Bias from discretization. Histogram estimates depend strongly on binning and can be wildly biased.
3. Curse of dimensionality. Local neighborhoods become sparse, hurting KDE and k-NN methods.
4. Rare-event instability. Log-ratio quantities amplify small-probability errors.

This is why papers that talk about maximizing MI in learned representations usually rely on tractable lower bounds, surrogate objectives, or neural estimators rather than direct plug-in estimation.

6.2 Classical Estimators

Three classical families are common:

• Histogram / plug-in estimators: simple for low-dimensional discrete variables, but binning-sensitive
• Kernel density estimators (KDE): smooth, but degrade quickly with dimension
• k-nearest-neighbor estimators such as Kraskov-style methods: often the strongest classical option for moderate dimensions

A useful rule of thumb:

• for small discrete tables, exact tabulation is best
• for modest continuous dimensions, k-NN estimators are often preferable
• for deep high-dimensional embeddings, classical estimators are usually too brittle to drive learning directly

Each estimator family comes with a characteristic failure mode:

The practical lesson is that MI estimation is never detached from modeling assumptions. Even "nonparametric" estimators hide assumptions through metrics, bandwidths, neighborhood sizes, truncation rules, or sampling schemes. In research writing, it is therefore not enough to report a single MI number. One should also report how it was estimated and why that estimator is suitable for the geometry of the data.

6.3 Variational Bounds: Donsker-Varadhan, NWJ, and InfoNCE

Because direct estimation is hard, many ML objectives optimize lower bounds on MI.

Donsker-Varadhan (DV) bound. Since MI is a KL divergence,

$$I(X;Y)=D_{\mathrm{KL}}(p_{XY} \Vert p_X p_Y),$$

and KL admits variational representations. One such form is

$$D_{\mathrm{KL}}(P \Vert Q) = \sup_T \left(\mathbb{E}_P[T]-\log \mathbb{E}_Q[e^T]\right).$$

Applying this with $P=p_{XY}$ and $Q=p_X p_Y$ yields a lower-bound objective over critic functions $T$.

NWJ bound. The Nguyen-Wainwright-Jordan family gives related lower bounds with different optimization behavior.

InfoNCE. For a positive pair $(x,y)$ and negatives $y_1^-,\dots,y_{K-1}^-$, the InfoNCE objective has the form

$$\mathcal{L}_{\mathrm{InfoNCE}} = -\mathbb{E}\left[ \log \frac{\exp s(x,y)} {\exp s(x,y)+\sum_{j=1}^{K-1}\exp s(x,y_j^-)} \right],$$

and implies the lower bound

$$I(X;Y)\ge \log K - \mathcal{L}_{\mathrm{InfoNCE}}$$

under the relevant sampling assumptions.

This is crucial:

• InfoNCE is not the mutual information itself
• it is a lower bound whose tightness depends on critic quality, negative sampling, and the number of negatives

That distinction matters especially in large-scale representation learning. A model can improve the InfoNCE objective for reasons that do not correspond to a clean increase in the true underlying MI:

• better exploitation of the finite negative-sampling scheme
• easier optimization of the critic class
• batch effects that change the hardness of the contrastive task

In other words, MI lower bounds are learning objectives, not merely passive estimators. They shape the geometry of representations because they reward some distinctions and ignore others. This is often exactly what we want, but it should be described honestly.

There is also a saturation issue. InfoNCE lower bounds become tighter as the number of negatives grows, but in practice batch size, memory limits, and sampling structure cap how far that strategy can go. This is one reason modern contrastive systems often rely on queue-based negatives, distributed batches, temperature tuning, or architectural inductive bias in addition to the raw bound itself.

6.4 MINE and Neural Estimators

Belghazi et al. (2018) introduced MINE: Mutual Information Neural Estimation. The core idea is to parameterize the critic $T_\phi(x,y)$ with a neural network and optimize the DV lower bound by gradient descent.

This made MI estimation compatible with deep learning pipelines, but brought new complications:

• optimization bias from finite-batch estimates
• variance explosion from exponentials in the DV term
• instability in high-dimensional settings

MINE matters conceptually because it showed MI could be estimated and optimized inside modern neural systems. But practitioners should not treat neural MI estimates as ground truth. They are estimator-dependent and often best interpreted comparatively rather than absolutely.

This is a recurring research best practice in 2026: use neural MI estimators as diagnostic instruments or training surrogates, but avoid overclaiming absolute semantic meaning for the numeric value unless the estimator has been carefully calibrated. A curve that moves in the expected direction can be valuable even if the axis itself is not an oracle.

6.5 Information Bottleneck

The information bottleneck objective introduces a representation $T$ of input $X$ and target $Y$, then optimizes

$$\min_{p(t\mid x)} I(X;T)-\beta I(T;Y).$$

This expresses a compression-relevance trade-off:

• $I(X;T)$ penalizes how much raw input detail the representation keeps
• $I(T;Y)$ rewards how much target-relevant information survives

At a conceptual level, this is one of the cleanest mathematical statements of what a useful representation should do.

At a practical level, there are difficulties:

• both MI terms can be hard to estimate
• the optimization is nontrivial
• apparent "information planes" in deep networks depend strongly on estimator choices

Still, the information bottleneck remains one of the most influential ideas linking information theory to representation learning.

Several modern objectives can be understood as relaxed descendants of this idea. Variational bottleneck methods replace exact MI terms with tractable bounds. Latent-variable methods often control a KL term that acts as a proxy for how much information flows through the latent code. Some disentanglement objectives can also be read through a bottleneck lens, though the match is not exact and should not be overstated.

The strongest way to carry the idea forward is not to memorize one formula, but to internalize the design question it encodes:

What is the smallest representation that still preserves the information the task actually needs?

That question survives even when the exact bottleneck objective is too hard to optimize literally.

INFORMATION BOTTLENECK
===============================================================

Input X  ---->  Representation T  ---->  Target Y

Want:
  small I(X;T)   -> compress nuisance detail
  large I(T;Y)   -> preserve task-relevant content

Objective:
  minimize I(X;T) - beta I(T;Y)

Interpretation:
  learn the smallest representation that still solves the task
===============================================================

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7. Applications in Machine Learning

7.1 Information Gain and Feature Selection

If $Y$ is the label and $X_j$ is a feature, then $I(X_j;Y)$ measures how much label uncertainty is reduced by observing that feature.

This is the basis of:

• information gain in decision trees
• feature screening in classical ML
• filter methods that rank features before model fitting

A feature with large MI need not be linearly predictive. That makes MI-based feature selection more general than correlation-based heuristics.

At the same time, pure marginal MI ranking has a limitation: it can prefer redundant features. If two variables each carry almost the same information about the label, ranking them independently may place both near the top even though the second contributes little once the first has already been chosen. This is why feature-selection pipelines often move from marginal MI to conditional MI or sequential forward selection when redundancy matters.

7.2 Contrastive Predictive Coding and Self-Supervised Learning

Contrastive Predictive Coding (CPC) uses a contrastive objective to distinguish a true future or paired view from negative samples. The resulting InfoNCE loss is motivated as a lower bound on mutual information between context and target.

The practical message is:

• representations are learned by preserving information shared across views or across time
• nuisance details that do not survive view changes are implicitly suppressed

This is why mutual-information language became so common in self-supervised learning. It provides a crisp answer to the question: what should the representation retain?

The answer is not "everything." In fact, the best contrastive pipelines depend on carefully chosen augmentations precisely because those augmentations define what counts as nuisance variation. If two crops of the same image are treated as a positive pair, the system is encouraged to keep semantic content that survives the crop and to discard content that does not. Mutual-information thinking makes that design choice explicit.

7.3 CLIP-Style Multimodal Alignment

CLIP trains image and text encoders so matched image-text pairs score highly while mismatched pairs score poorly. The loss is not "exactly MI maximization," but it is strongly MI-inspired:

• matched image-text pairs should preserve shared semantic content
• negatives force the model to distinguish true pairings from independent pairings
• the contrastive setup approximates a lower-bound style dependence objective

This is one reason MI is such a natural language for multimodal learning. It focuses attention on the shared content between modalities rather than on modality-specific detail.

That shared-content view also explains why multimodal alignment is delicate. Image and text are not equivalent channels. Each modality contains information the other does not. A caption may omit background details visible in the image; an image may omit abstract relational or temporal details expressed in text. The training objective therefore tries to preserve the intersection of the two modalities, not their full information content. Mutual-information language is helpful precisely because it keeps that distinction visible.

7.4 Information Bottleneck in Representation Learning

The information bottleneck provides a principled language for representation quality:

• a good representation compresses irrelevant details
• a good representation preserves task-relevant information

Although full IB optimization is often too difficult to apply literally, the idea lives on in many practical heuristics:

• compression regularization
• latent bottlenecks
• denoising objectives
• sparse or low-rate representations

Even when not optimized explicitly, the bottleneck lens helps us ask the right question about learned representations: what information is being preserved, and what information is being discarded?

This is especially relevant when evaluating larger models whose internal states are hard to interpret directly. Asking whether a representation linearly probes well is useful, but incomplete. Asking what information it retains about spurious style, demographic attributes, or nuisance domain variables is often equally important. The bottleneck perspective naturally supports that broader audit of representation quality.

7.5 Active Learning and Bayesian Experimental Design

In active learning, the learner chooses which example to label next. A natural acquisition score is the expected information gain:

$$I(Y;\Theta \mid X,\mathcal{D}),$$

where:

• $X$ is the candidate query
• $Y$ is its unknown label
• $\Theta$ denotes model parameters or latent predictive state
• $\mathcal{D}$ is the current dataset

This quantity asks:

"If I label this example, how much will I reduce uncertainty about the model?"

That is exactly the logic behind BALD-style acquisition functions and Bayesian experimental design. Mutual information is the mathematically clean bridge between uncertainty estimation and data acquisition.

The same logic appears in scientific experimentation and agentic AI systems. If an agent can choose which document to read, which environment state to inspect, or which tool call to make next, a good strategy is often the action that is expected to reduce posterior uncertainty the most. In that sense, active learning is not a narrow subfield; it is a prototype for information-seeking behavior more broadly.

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8. Common Mistakes

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9. Exercises

1. Exercise 1 (*): Tabular Mutual Information Given a $2\times 2$ joint probability table, compute $I(X;Y)$ directly from the definition and verify symmetry.

2. Exercise 2 (*): Zero Correlation, Positive MI Construct a nonlinear example with zero covariance but positive mutual information, and explain why the difference arises.

3. Exercise 3 (*): MI as KL Divergence Starting from $I(X;Y)=H(X)+H(Y)-H(X,Y)$, derive the KL form $D_{\mathrm{KL}}(p_{XY} \Vert p_X p_Y)$.

4. Exercise 4 (): Chain Rule and Conditional MI** Prove a chain rule such as $I(X,Z;Y)=I(X;Y)+I(Z;Y\mid X)$, then interpret it in a feature-selection setting.

5. Exercise 5 (): Binary Symmetric Channel** Compute the mutual information of a BSC numerically as a function of flip probability $\varepsilon$, and recover the formula $1-h_2(\varepsilon)$ for uniform input.

6. Exercise 6 (): Estimating MI from Samples** Generate dependent and independent synthetic datasets, estimate MI using a simple classical estimator, and analyze estimator bias and variance.

7. Exercise 7 (*): InfoNCE Lower Bound** Derive the $\log K - \mathcal{L}_{\mathrm{InfoNCE}}$ lower-bound form and explain what changes as the number of negatives grows.

8. Exercise 8 (*): MI in Modern ML** Choose one of feature selection, CLIP-style alignment, information bottleneck, or active learning and formulate the exact mutual-information quantity that the system is trying to maximize or preserve.

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10. Why This Matters for AI (2026 Perspective)

Mutual information matters in 2026 because AI systems are increasingly evaluated not just by prediction accuracy, but by what their internal representations preserve, what uncertainty they expose, and how efficiently they align multiple views of the world. MI is one of the few quantities that speaks naturally to all of those concerns at once.

It also matters because the field is more careful now about which information should be preserved. In trustworthy AI, retaining spurious correlations can be harmful. In retrieval and multimodal alignment, preserving semantically shared content is valuable. In privacy-preserving learning, we may want to maximize task-relevant MI while minimizing leakage about sensitive attributes. Mutual information does not solve those trade-offs automatically, but it gives us a precise language for stating them.

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11. Conceptual Bridge

Mutual information sits naturally between the first two sections of this chapter. From 01-Entropy it inherits the language of uncertainty, surprise, chain rules, and conditional structure. From 02-KL-Divergence it inherits the divergence perspective that turns dependence into the gap between a joint law and the product of marginals. In that sense, mutual information is the first major quantity in the chapter that combines entropy and KL into a single object.

It also points forward. 04-Cross-Entropy turns the distribution-matching viewpoint into concrete training losses for classifiers and language models. 05-Fisher-Information studies local sensitivity and statistical geometry, which complements mutual information's global dependence view. Outside this chapter, mutual information connects to Bayesian experimental design in Statistics, data processing and Markov structure in Probability Theory, and contrastive learning, active learning, and multimodal alignment in later ML-specific chapters.

The deeper conceptual message is this: entropy asks how uncertain a variable is, KL divergence asks how mismatched two distributions are, and mutual information asks how much one variable helps explain another. Those are not three unrelated ideas. They are three different lenses on the same central problem in AI: how information is represented, transmitted, compressed, and used for prediction.

If you keep only one mental picture from this section, let it be this: mutual information is the currency of relevance. Whenever we ask whether an input, embedding, sensor, token, prompt, or latent state is useful, we are implicitly asking whether it reduces uncertainty about something we care about. The math of mutual information turns that intuition into a reusable quantitative tool.

MUTUAL INFORMATION IN THE CURRICULUM
===============================================================

Entropy (Section 01)
  uncertainty of one variable
         |
         v
KL Divergence (Section 02)
  mismatch between distributions
         |
         v
Mutual Information (Section 03)
  shared information between variables
         |
         +--> Cross-Entropy (Section 04)
         |      training under a target distribution
         |
         +--> Fisher Information (Section 05)
                local sensitivity and information geometry

Broader ML links:
  decision trees
  contrastive learning
  CLIP-style alignment
  active learning
  information bottleneck
===============================================================

References

1. Shannon, C. E. (1948). "A Mathematical Theory of Communication." Bell System Technical Journal. PDF
2. Cover, T. M., and Thomas, J. A. (2006). Elements of Information Theory (2nd ed.). Wiley.
3. MacKay, D. J. C. (2003). Information Theory, Inference, and Learning Algorithms. Cambridge University Press.
4. Duchi, J. C. Statistics 311 / EE377 Lecture Notes. Online
5. Tishby, N., Pereira, F. C., and Bialek, W. (1999). "The Information Bottleneck Method." PDF
6. van den Oord, A., Li, Y., and Vinyals, O. (2018). "Representation Learning with Contrastive Predictive Coding." arXiv
7. Belghazi, M. I., Baratin, A., Rajeshwar, S., Ozair, S., Bengio, Y., Courville, A., and Hjelm, D. (2018). "Mutual Information Neural Estimation." PMLR
8. Radford, A. et al. (2021). "Learning Transferable Visual Models From Natural Language Supervision." arXiv
9. Fano, R. M. (1961). Transmission of Information. MIT Press.
10. McGill, W. J. (1954). "Multivariate Information Transmission." Psychometrika.