channel-capacity skill
Problem-solving strategies for channel capacity in information theory
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Install the channel-capacity skill
A skill is a folder. Copy it into your agent's skills folder and the agent loads it when the task matches its description.
git clone --depth 1 https://github.com/parcadei/Continuous-Claude-v3.git /tmp/Continuous-Claude-v3 mkdir -p ~/.claude/skills cp -r /tmp/Continuous-Claude-v3/.claude/skills/math/information-theory/channel-capacity ~/.claude/skills/channel-capacity
available in every project
In the Claude apps, zip the folder and upload it from the Skills settings. The folder on GitHub
The instructions your agent would load
SKILL.md as published, without the frontmatter. Read it on GitHub
Channel Capacity
When to Use
Use this skill when working on channel-capacity problems in information theory.
Decision Tree
- Mutual Information
- I(X;Y) = H(X) + H(Y) - H(X,Y)
- I(X;Y) = H(X) - H(X|Y) = H(Y) - H(Y|X)
- Symmetric: I(X;Y) = I(Y;X)
- scipy.stats.entropy(p) + scipy.stats.entropy(q) - joint_entropy
- Channel Model
- Input X, output Y, channel P(Y|X)
- Channel matrix: rows = inputs, columns = outputs
- Element (i,j) = P(Y=j | X=i)
- Channel Capacity
- C = max_{p(x)} I(X;Y)
- Maximize over input distribution
- Achieved by capacity-achieving distribution
- Common Channels
- Blahut-Arimoto Algorithm
- Iterative algorithm to compute capacity
- Alternates between optimizing p(x) and p(y|x)
- Converges to capacity
- z3solve.py prove "capacityupper_bound"
Tool Commands
ScipyMutualInfo
uv run python -c "from scipy.stats import entropy; p = [0.5, 0.5]; q = [0.6, 0.4]; H_X = entropy(p, base=2); H_Y = entropy(q, base=2); print('H(X)=', H_X, 'H(Y)=', H_Y)"SympyBscCapacity
uv run python -m runtime.harness scripts/sympy_compute.py simplify "1 + p*log(p, 2) + (1-p)*log(1-p, 2)"Z3CapacityBound
uv run python -m runtime.harness scripts/z3_solve.py prove "I(X;Y) <= H(X)"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. Using a randomly generated code, Shannon showed that one can send information at any rate below the capacity C* of the channel with an arbitrarily low probability of error. The idea of a randomly generated code is very unusual.
Cognitive Tools Reference
See .claude/skills/math-mode/SKILL.md for full tool documentation.
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