Entropy

Entropy

When to Use

Use this skill when working on entropy problems in information theory.

Decision Tree

• Shannon Entropy
• H(X) = -sum p(x) log2 p(x)
• Maximum for uniform distribution: H_max = log2(n)
• Minimum = 0 for deterministic (one outcome certain)
• scipy.stats.entropy(p, base=2) for discrete

• Entropy Properties
• Non-negative: H(X) >= 0
• Concave in p
• Chain rule: H(X,Y) = H(X) + H(Y|X)
• z3_solve.py prove "entropy_nonnegative"

• Joint and Conditional Entropy
• H(X,Y) = -sum sum p(x,y) log2 p(x,y)
• H(Y|X) = H(X,Y) - H(X)
• H(Y|X) <= H(Y) with equality iff independent

• Differential Entropy (Continuous)
• h(X) = -integral f(x) log f(x) dx
• Can be negative!
• Gaussian: h(X) = 0.5 * log2(2*pi*e*sigma^2)
• sympy_compute.py integrate "-f(x)*log(f(x))" --var x

• Maximum Entropy Principle
• Given constraints, max entropy distribution is least biased
• Uniform for no constraints
• Exponential for E[X] = mu constraint
• Gaussian for E[X], Var[X] constraints

Tool Commands

Scipy_Entropy

uv run python -c "from scipy.stats import entropy; p = [0.25, 0.25, 0.25, 0.25]; H = entropy(p, base=2); print('Entropy:', H, 'bits')"

Scipy_Kl_Div

uv run python -c "from scipy.stats import entropy; p = [0.5, 0.5]; q = [0.9, 0.1]; kl = entropy(p, q); print('KL divergence:', kl)"

Sympy_Entropy

uv run python -m runtime.harness scripts/sympy_compute.py simplify "-p*log(p, 2) - (1-p)*log(1-p, 2)"

Key Techniques

*From indexed textbooks:*

• [Elements of Information Theory] Elements of Information Theory -- Thomas M_ Cover &amp; Joy A_ Thomas -- 2_, Auflage, New York, NY, 2012 -- Wiley-Interscience -- 9780470303153 -- 2fcfe3e8a16b3aeefeaf9429fcf9a513 -- Anna’s Archive. What is the channel capacity of this channel? This is the multiple\-access channel solved by Liao and Ahlswede.

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.