{
  "handle": "JensenShannonDivergence",
  "ref": "JensenShannonDivergence#b3b4",
  "stub": "b3b4",
  "semaId": "sema:JensenShannonDivergence#mh:SHA-256:b3b4314fee746b1e97b133372bcdb1d9dd1a3a98f90fd19ca19bb47dfef04a6b",
  "layer": "Infrastructure",
  "category": "Primitives",
  "gloss": "A symmetric, bounded distance between two distributions, built from their mixture",
  "mechanism": "The Kullback-Leibler divergence D_KL(P||Q) = sum_i p_i log(p_i/q_i) measures the information lost when Q approximates P, but it is asymmetric and infinite wherever q_i = 0 and p_i > 0, which empirical histograms with unseen states often produce. The Jensen-Shannon divergence fixes this by comparing both distributions to their mixture M = (P+Q)/2: D_JS = 1/2 D_KL(P||M) + 1/2 D_KL(Q||M). Equivalently D_JS = {{shannon_entropy}}(M) - (H(P)+H(Q))/2. With log base 2 it is symmetric, always finite and bounded in [0, 1], and zero only when P = Q. Typical use: compare an empirical distribution with a null-model distribution over the same state space, so the value reads as distance from the baseline.",
  "invariants": [
    "D_JS(P,Q) = D_JS(Q,P)",
    "0 <= D_JS <= 1 with log base 2",
    "D_JS = 0 if and only if P = Q",
    "Both distributions are defined over the same state space"
  ],
  "failureModes": [
    "Comparing distributions built on different state spaces or binnings",
    "A non-monotonic response: distance from a central reference rises for deviations in either direction (e.g. attraction and repulsion both raise D_JS from a uniform baseline), so the value alone does not say which way the system deviates",
    "Small samples inflate D_JS between two draws of the same process"
  ],
  "ring": 0,
  "tier": 0,
  "aliases": {
    "shannon_entropy": "ShannonEntropy"
  },
  "buildsOn": [
    {
      "handle": "ShannonEntropy",
      "type": "references",
      "alias": "shannon_entropy",
      "url": "/knowledge-graph/ShannonEntropy/",
      "json": "/knowledge-graph/ShannonEntropy.json"
    }
  ],
  "usedBy": [
    {
      "handle": "NullModelBaseline",
      "type": "composes_with",
      "alias": "jensen_shannon_divergence",
      "url": "/knowledge-graph/NullModelBaseline/",
      "json": "/knowledge-graph/NullModelBaseline.json"
    },
    {
      "handle": "LocalSpatialStateDivergence",
      "type": "composes_with",
      "alias": "jensen_shannon_divergence",
      "url": "/knowledge-graph/LocalSpatialStateDivergence/",
      "json": "/knowledge-graph/LocalSpatialStateDivergence.json"
    }
  ],
  "sources": {
    "research": [],
    "works": [
      {
        "kind": "thesis",
        "label": "Kevin Do, PhD thesis on ant collective behaviour (NCSU, 2026)",
        "note": "Ch. 1.3.3, Eqs. 1.10-1.11; Ch. 3.1.1, Eqs. 3.2-3.3, Fig. 3.3",
        "url": ""
      }
    ],
    "homePage": false
  },
  "url": "/knowledge-graph/JensenShannonDivergence/",
  "graphUrl": "/knowledge-graph#JensenShannonDivergence",
  "directory": "/knowledge-graph/patterns.json"
}