{
  "handle": "MutualInformation",
  "mechanism": "The substrate information-theoretic measure of dependence between two random variables, defined generally as the Kullback-Leibler divergence between their joint distribution and the product of their marginals. When the relevant {{entropy}} terms are finite, I(X; Y) = H(X) + H(Y) − H(X, Y), equivalently the reduction in one variable's entropy from knowing the other. Zero mutual information indicates statistical independence; a positive value indicates dependence, but its unnormalized magnitude has no universal scale. The property exists whether or not anyone estimates it. Distinct from normalized correlation coefficients, whose sensitivity depends on the selected method, and from mere co-occurrence.",
  "gloss": "Information-theoretic dependence between two variables",
  "invariants": [
    "Non-negativity: I(X; Y) ≥ 0 for all random variables X, Y.",
    "Symmetry: I(X; Y) = I(Y; X).",
    "Zero iff independent: I(X; Y) = 0 if and only if X and Y are statistically independent.",
    "Discrete entropy bound: for discrete X and Y with finite entropies, I(X; Y) ≤ min(H(X), H(Y))."
  ],
  "_meta": {
    "tier": 1,
    "ring": 0,
    "path": [
      "Physics",
      "Primitives"
    ],
    "supersedes": [
      "sema:MutualInformation#mh:SHA-256:03a0a79a3c3433e42aa63a97157c7b376adf0aa3d06c28103b63229a458dcb07",
      "sema:MutualInformation#mh:SHA-256:40369b10e6cf188b12f1922452d4ebc9184b8e398a765dd857bb4efd0a13589b"
    ]
  },
  "sema_id": "sema:MutualInformation#mh:SHA-256:e714024fcd78de043d4d86021d907157d7f44f7a9c719b1d760f5810dd762a8b",
  "sema_ref": "MutualInformation#e714",
  "sema_stub": "e714",
  "failure_modes": [
    "Estimator conflation: reading a finite-sample estimate of zero as proof of independence. Zero-iff-independent is a property of the population quantity, not of an estimate."
  ],
  "dependencies": {
    "references": {
      "entropy": "sema:Entropy#mh:SHA-256:d5f7482e6c06a1e48eadbf28e6dca0cc0eb176bf75c70e7e467284f656d20bae"
    }
  },
  "sema_layer": "Physics",
  "sema_category": "Primitives",
  "used_by": {}
}