{
  "article_slug": "missing-modality-identifiability",
  "figures": [
    {
      "id": "private-factor-risk",
      "title": "Missing private information creates irreducible risk",
      "kind": "analytical_curve",
      "x_label": "Private-factor coefficient alpha",
      "y_label": "Bayes MSE (dimensionless)",
      "x": {
        "name": "Private-factor coefficient alpha",
        "values": [
          0,
          0.25,
          0.5,
          0.75,
          1,
          1.5,
          2,
          3
        ]
      },
      "series": [
        {
          "name": "Only shared S observed",
          "values": [
            0,
            0.0625,
            0.25,
            0.5625,
            1,
            2.25,
            4,
            9
          ]
        },
        {
          "name": "Both S and P observed",
          "values": [
            0,
            0,
            0,
            0,
            0,
            0,
            0,
            0
          ]
        }
      ],
      "caption": "Constructed independent standard-normal S,P and targetY=S+alpha P. Missing P gives conditional variance alpha^2. A finite symmetric moment check verifies the MSE identity; no neural or clinical data."
    },
    {
      "id": "selection-ambiguity",
      "title": "Same observed cases, different missing outcomes",
      "kind": "analytical_curve",
      "x_label": "Population positive prevalence p",
      "y_label": "Positive rate",
      "x": {
        "name": "Population positive prevalence p",
        "values": [
          0.25,
          0.275,
          0.3,
          0.325,
          0.35,
          0.375,
          0.4,
          0.42500000000000004,
          0.45,
          0.475,
          0.5,
          0.525,
          0.55,
          0.575,
          0.6000000000000001,
          0.625,
          0.65,
          0.675,
          0.7,
          0.7250000000000001,
          0.75
        ]
      },
      "series": [
        {
          "name": "Observed-case positive rate",
          "values": [
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5
          ]
        },
        {
          "name": "Missing-case positive rate",
          "values": [
            0.0,
            0.050000000000000024,
            0.09999999999999998,
            0.15000000000000005,
            0.19999999999999998,
            0.25,
            0.30000000000000004,
            0.3500000000000001,
            0.39999999999999997,
            0.45,
            0.5,
            0.55,
            0.6,
            0.6499999999999999,
            0.7000000000000002,
            0.75,
            0.8,
            0.8500000000000001,
            0.8999999999999998,
            0.9500000000000001,
            1.0
          ]
        }
      ],
      "caption": "Constructed binary selection family with pa=(1-p)b=1/4. Observation rate and observed positive rate remain1/2; missing positive rate2p-1/2 ranges0..1. Outcomes for missing cases are assumed unavailable."
    }
  ]
}