Entropy Calculator – From Shannon Entropy to KL Divergence
Entropy is a central quantity in information theory, statistics and machine learning. It measures uncertainty in a distribution, the amount of information in a message and the ideal code length for lossless compression. This Entropy Calculator on MyTimeCalculator brings together Shannon entropy, binary entropy, joint and conditional entropy, mutual information, cross-entropy and KL divergence in one place, with inputs based on probability vectors and matrices.
Throughout this article p denotes a discrete probability distribution p = (p₁, …, pₙ) with pᵢ ≥ 0 and Σ pᵢ = 1. The symbols log₂, ln and log₁₀ denote logarithms to base 2, base e and base 10 respectively. By convention, terms with pᵢ = 0 contribute 0 to sums of the form pᵢ log pᵢ.
Shannon Entropy H(p)
For a discrete distribution p = (p₁, …, pₙ), Shannon entropy in base b is defined by
Hb(p) = − Σi = 1n pᵢ logb pᵢ.
Common choices of base are
- Base 2: H(p) in bits, using log₂.
- Base e: H(p) in nats, using ln.
- Base 10: H(p) in Hartleys, using log₁₀.
The bases are related by conversion factors. If He(p) is entropy in nats, then
- H2(p) = He(p) / ln 2, in bits
- H10(p) = He(p) / ln 10, in base-10 units
For a uniform distribution over n outcomes, pᵢ = 1/n, the entropy in bits is H(p) = log₂ n. This is the maximum possible entropy among all distributions on n outcomes.
Normalized entropy and redundancy
To compare distributions with different alphabet sizes it is common to define the normalized entropy
Hnorm(p) = H2(p) / log₂ n,
and the redundancy
R(p) = 1 − Hnorm(p).
Normalized entropy lies between 0 and 1. A value near 1 means the distribution is close to uniform, while a value near 0 means the mass is highly concentrated on a few outcomes. Redundancy measures how far the distribution is from uniform in terms of potential compressibility.
Binary Entropy Function H₂(p)
For a Bernoulli random variable that takes value 1 with probability p and 0 with probability 1 − p, the entropy in bits is given by the binary entropy function
H₂(p) = − p log₂ p − (1 − p) log₂(1 − p), for 0 ≤ p ≤ 1,
with the usual convention that terms with p = 0 or p = 1 contribute 0. This function achieves its maximum value of 1 bit at p = 0.5 and drops to 0 bits at p = 0 and p = 1. In nats and base-10 units the values are obtained by multiplying H₂(p) by the appropriate conversion factors.
Joint Entropy, Marginals and Conditional Entropy
Let X and Y be discrete random variables with joint distribution P(X = xᵢ, Y = yⱼ) = pij. Arrange these probabilities in a matrix P with rows indexed by i and columns indexed by j. The joint entropy is
H(X, Y) = − Σi Σj pij log₂ pij,
with terms pij log pij set to 0 whenever pij = 0.
The marginal distributions are
- P(X = xᵢ) = pi· = Σj pij
- P(Y = yⱼ) = p·j = Σi pij
and the entropy of each variable is
- H(X) = − Σi pi· log₂ pi·
- H(Y) = − Σj p·j log₂ p·j
Conditional entropy measures the remaining uncertainty in one variable given knowledge of the other. Two useful formulas are
- H(Y|X) = H(X, Y) − H(X)
- H(X|Y) = H(X, Y) − H(Y)
These can also be expressed using conditional probabilities P(Y|X) and P(X|Y), but subtracting entropies from the joint distribution is often more numerically stable and easier to compute.
Mutual Information I(X;Y)
Mutual information measures how much knowing one variable reduces uncertainty about the other. In terms of entropies it can be written as
I(X;Y) = H(X) + H(Y) − H(X, Y).
An equivalent expression uses the joint and marginal distributions directly:
I(X;Y) = Σi Σj pij log₂ ( pij / (pi· p·j) ),
where terms with pij = 0 contribute 0. Mutual information is always non-negative and equals 0 if and only if X and Y are independent, in which case pij = pi· p·j for all i and j.
Cross-Entropy H(p, q) and KL Divergence D(p‖q)
When comparing two distributions p and q on the same set of outcomes, cross-entropy and KL divergence quantify mismatch and coding inefficiency. The cross-entropy of p relative to q in bits is
H(p, q) = − Σi pᵢ log₂ qᵢ,
assuming qᵢ > 0 whenever pᵢ > 0. This quantity represents the expected number of bits needed to encode outcomes drawn from p using an optimal code designed for q.
KL divergence (relative entropy) from p to q is
D(p‖q) = Σi pᵢ log₂ (pᵢ / qᵢ) = H(p, q) − H(p).
KL divergence is always non-negative and equals 0 if and only if p and q are identical on their support. If there exists an i for which pᵢ > 0 but qᵢ = 0, then D(p‖q) is infinite because log(pᵢ / qᵢ) is undefined, reflecting that a code based on q cannot represent data drawn from p without infinite cost.
Entropy in Different Bases
The formulas above are often written with a generic logarithm base b. If Hb(p) is entropy in base b and He(p) is entropy using the natural logarithm, then
Hb(p) = He(p) / ln b.
This applies equally to entropy, cross-entropy, KL divergence and mutual information. The calculator computes entropy in bits as the primary value and derives nats and base-10 units through these base conversion formulas.
How to Use the Entropy Calculator
The tabs are organized around common information-theoretic questions:
- Shannon Entropy tab: Enter a list of probabilities or counts. The calculator normalizes them, computes H(p) in bits, nats and base 10, and reports normalized entropy and redundancy. It also generates an entropy table with per-symbol contributions pᵢ(−log₂ pᵢ).
- Binary Entropy tab: Enter a single probability p to evaluate H₂(p). This is useful for coin flips, yes/no questions and Bernoulli variables in general.
- Joint & Mutual Information tab: Enter a joint distribution matrix for P(X,Y). The tool automatically normalizes the entries, computes H(X), H(Y), H(X,Y), conditional entropies H(Y|X) and H(X|Y), and mutual information I(X;Y).
- Cross-Entropy & KL Divergence tab: Enter two aligned lists representing p (data) and q (model). The calculator normalizes both, computes H(p), H(q), H(p, q) and KL divergences D(p‖q) and D(q‖p).
All results are intended for learning, debugging information-theoretic formulas and quick analytic checks. For large-scale statistical modeling or training machine learning models, use these outputs as a conceptual guide alongside specialized software and full numerical libraries.
Entropy & Information FAQs
Frequently Asked Questions About Entropy and Information Measures
Key points to help you interpret entropy, cross-entropy, KL divergence and mutual information in practical contexts.
By convention 0 log 0 is defined to be 0, which matches the limiting behavior of p log p as p approaches 0 from above. This allows entropy sums to exclude zero-probability outcomes without changing the result and avoids undefined logarithms at p = 0.
No. Mutual information is always greater than or equal to zero, and equals zero if and only if the two variables are independent. Numerical rounding can produce tiny negative values in computations, but these are artifacts of floating point arithmetic and should be interpreted as zero.
A large KL divergence means that the model distribution q is a poor match for the data distribution p. In coding terms it corresponds to many extra bits per symbol when encoding data from p with a code designed for q. In machine learning it often signals underfitting, misspecification or severe mismatch between model and data.
When labels or targets define a distribution p and the model outputs a distribution q, minimizing cross-entropy H(p, q) with respect to q is equivalent to minimizing KL divergence D(p‖q) because H(p) is constant. This is why cross-entropy loss is widely used in classification and probabilistic modeling.
Entropy measures uncertainty or unpredictability according to a specific probability model. A process can have high entropy if its outcomes are well mixed and unpredictable under that model. Real-world randomness also depends on how accurate the model is, the presence of structure and what information is already known.