Entropy

Entropy

When to Use

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

Decision Tree

1. 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

2. 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"

3. 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

4. Differential Entropy (Continuous)

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

5. 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 & 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.