Computer Science & Data Science · Information theory

Kullback-Leibler divergence (relative entropy)

Q: What does Kullback-Leibler divergence (relative entropy) describe?

The relation D_{KL}(p\|q)=\sum_i p_i\ln\frac{p_i}{q_i} summarizes KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For p=(0.5,0.5) and q=(0.9,0.1), DKL=0.5ln(0.5/0.9)+0.5ln(0.5/0.1)≈0.511 nats., symbols are connected to concrete values and the result is interpreted in context. Validity limit: For discrete distributions, sum pᵢ ln(pᵢ/qᵢ). It is nonnegative but asymmetric and not a metric. If pᵢ>0 and qᵢ=0, it diverges; terms with pᵢ=0 are zero by continuity. Natural logs Source: SciPy API Reference, scipy.stats.entropy (relative entropy).

KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.

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Formula

D_{K L} (p ∥ q) = i \sum p_{i} ln \frac{p _{i}}{q _{i}}

nats with natural logarithms; dimensionless numerical value

Variables & units

Variables & units
Symbol	Meaning	Unit
$pᵢ$	True probability of outcome i	dimensionless
$qᵢ$	Comparison or model probability for the same outcome	dimensionless
DKL(p‖q)	Relative entropy of q with respect to p	nat

LaTeX

D_{KL}(p\|q)=\sum_i p_i\ln\frac{p_i}{q_i}

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Derivation & background

For discrete distributions, sum p_i ln(p_i/q_i). It is nonnegative but asymmetric and not a metric. If p_i>0 and q_i=0, it diverges; terms with p_i=0 are zero by continuity. Natural logs give nats; log base two gives bits. Cross-entropy and Shannon entropy relate through H(p,q)=H(p)+DKL(p‖q).

Exam blueprint

Validity range

Derivation steps

KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.

1Read the relation from its definition: D_KL(p\|q)=\sum_i p_i\ln\frac{p_i}{q_i}.
2Identify the input, requested output and assumptions of the relation.
3For p=(0.5,0.5) and q=(0.9,0.1), DKL=0.5ln(0.5/0.9)+0.5ln(0.5/0.1)≈0.511 nats.

Rearrangements

Defining relation

D_{KL}(p\|q)=\sum_i p_i\ln\frac{p_i}{q_i}

The equation states the defining relation within its stated domain.

Task variant

Use the example for Kullback-Leibler divergence (relative entropy) to explain which inputs determine the requested output.

The example connects concrete inputs to its result.
KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.
The relation applies only under its stated assumptions.

Common mistakes

Applying the relation to a different case without checking its stated assumptions.

Check the validity range before substituting values: For discrete distributions, sum p_i ln(p_i/q_i). It is nonnegative but asymmetric and not a metric. If p_i>0 and q_i=0, it diverges; terms with p_i=0 are zero by continuity. Natural logs.

Exam context

Compare probability models, quantify information loss and evaluate distributional objectives in machine learning.

Formula cluster

Information theory

KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.

H(X) = −Σ p_i log₂(p_i)

Shannon entropy

I(X;Y) = H(X) − H(X|Y)

Mutual information of discrete variables

H(p,q) = −Σ p_i ln(q_i)

Cross-entropy (classification loss)

Worked example

For p=(0.5,0.5) and q=(0.9,0.1), DKL=0.5ln(0.5/0.9)+0.5ln(0.5/0.1)≈0.511 nats.

Applications

Compare probability models, quantify information loss and evaluate distributional objectives in machine learning.

How do you calculate with Kullback-Leibler divergence (relative entropy)?

Here is how to work through a typical Kullback-Leibler divergence (relative entropy) (DKL(p‖q) = Σ pᵢ ln(pᵢ/qᵢ)) task step by step:

1
Task
Use the example for Kullback-Leibler divergence (relative entropy) to explain which inputs determine the requested output.
Solution path
1. The example connects concrete inputs to its result.
2. KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.
3. The relation applies only under its stated assumptions.

Frequently asked questions about Kullback-Leibler divergence (relative entropy)

What does Kullback-Leibler divergence (relative entropy) describe?+

The relation D_KL(p\|q)=\sum_i p_i\ln\frac{p_i}{q_i} summarizes KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For p=(0.5,0.5) and q=(0.9,0.1), DKL=0.5ln(0.5/0.9)+0.5ln(0.5/0.1)≈0.511 nats., symbols are connected to concrete values and the result is interpreted in context. Validity limit: For discrete distributions, sum p_i ln(p_i/q_i). It is nonnegative but asymmetric and not a metric. If p_i>0 and q_i=0, it diverges; terms with p_i=0 are zero by continuity. Natural logs Source: SciPy API Reference, scipy.stats.entropy (relative entropy).

How should I work through a calculation?+

First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example For p=(0.5,0.5) and q=(0.9,0.1), DKL=0.5ln(0.5/0.9)+0.5ln(0.5/0.1)≈0.511 nats. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: For discrete distributions, sum p_i ln(p_i/q_i). It is nonnegative but asymmetric and not a metric. If p_i>0 and q_i=0, it diverges; terms with p_i=0 are zero by continuity. Natural logs Source: SciPy API Reference, scipy.stats.entropy (relative entropy).

Which assumption must I keep in mind?+

The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: For discrete distributions, sum p_i ln(p_i/q_i). It is nonnegative but asymmetric and not a metric. If p_i>0 and q_i=0, it diverges; terms with p_i=0 are zero by continuity. Natural logs Source: SciPy API Reference, scipy.stats.entropy (relative entropy).

Which units and quantities are involved?+

The stated units are: nats with natural logarithms; dimensionless numerical value. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: For discrete distributions, sum p_i ln(p_i/q_i). It is nonnegative but asymmetric and not a metric. If p_i>0 and q_i=0, it diverges; terms with p_i=0 are zero by continuity. Natural logs Source: SciPy API Reference, scipy.stats.entropy (relative entropy).

When is comparison with shannon-entropie and mutuelle-information-diskret useful?+

Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source SciPy API Reference, scipy.stats.entropy (relative entropy). supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: For discrete distributions, sum p_i ln(p_i/q_i). It is nonnegative but asymmetric and not a metric. If p_i>0 and q_i=0, it diverges; terms with p_i=0 are zero by continuity. Natural logs Source: SciPy API Reference, scipy.stats.entropy (relative entropy).

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]SciPy API Reference, scipy.stats.entropy (relative entropy).

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Kullback-Leibler divergence (relative entropy)

KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.

AdvancedUniversity/collegeBachelor foundationsSubject specialization

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Symbol

Meaning

Unit

pᵢ

True probability of outcome i

dimensionless

qᵢ

Comparison or model probability for the same outcome

dimensionless

DKL(p‖q)

Relative entropy of q with respect to p

nat

Exam blueprint

Validity range

Derivation steps

KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.

1Read the relation from its definition: D_KL(p\|q)=\sum_i p_i\ln\frac{p_i}{q_i}.
2Identify the input, requested output and assumptions of the relation.
3For p=(0.5,0.5) and q=(0.9,0.1), DKL=0.5ln(0.5/0.9)+0.5ln(0.5/0.1)≈0.511 nats.

How do you calculate with Kullback-Leibler divergence (relative entropy)?

Here is how to work through a typical Kullback-Leibler divergence (relative entropy) (DKL(p‖q) = Σ pᵢ ln(pᵢ/qᵢ)) task step by step:

Task

Use the example for Kullback-Leibler divergence (relative entropy) to explain which inputs determine the requested output.

Solution path

The example connects concrete inputs to its result.
KL divergence measures expected information loss when q is assumed instead of the true discrete distribution p.
The relation applies only under its stated assumptions.

Frequently asked questions about Kullback-Leibler divergence (relative entropy)

What does Kullback-Leibler divergence (relative entropy) describe?+

How should I work through a calculation?+

Which assumption must I keep in mind?+

Which units and quantities are involved?+

When is comparison with shannon-entropie and mutuelle-information-diskret useful?+

Retain Kullback-Leibler divergence (relative entropy) for exams

Create a curated FSRS exam set for DKL(p‖q) = Σ pᵢ ln(pᵢ/qᵢ): formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential