{
  "schemaVersion": 1,
  "lessonId": "probability-04",
  "contentHash": "6376255cbf8b00058c317b070f37c405c6d1a3d223f63852cccd9c0bcb42a727",
  "rendererHash": "a1bd5b648f4731ae5d5f0a24811c2540d1a0a132149ce589d17c4113509cbedf",
  "kind": "executed",
  "generatedAt": "2026-10-20T10:46:22.399640+00:00",
  "provenance": "recorded_cpu_computation",
  "environment": {
    "backend": "cpu",
    "deviceCount": 1,
    "jax": "0.9.2",
    "numpy": "2.4.4",
    "matplotlib": "3.10.8"
  },
  "stdout": "VI mean: [ 0.99999994 -0.99999994] variances: [0.36000016 0.36000016] remaining KL: 0.5108256340026855\nsynthetic nominal-95% interval coverage: 0.2222222238779068\nstandardized residuals: [12.  5.  0. -3. -4. -3.  0.  5. 12.]\n",
  "data": {
    "kind": "bar",
    "x": [
      0,
      1,
      2
    ],
    "labels": [
      "weight 1",
      "weight 2",
      "sum of weights"
    ],
    "xlabel": "quantity",
    "ylabel": "variance (squared parameter units)",
    "series": [
      {
        "label": "exact target",
        "y": [
          1.0,
          1.0,
          3.6
        ]
      },
      {
        "label": "optimized diagonal q",
        "y": [
          0.3600001633167267,
          0.3600001633167267,
          0.7200003266334534
        ]
      }
    ]
  },
  "diagram": {
    "title": "The covariance changes uncertainty in a sum",
    "prediction": "Can matching each coordinate's mean prove a correct distribution for their sum?",
    "reading": "Both analytic matrices have unit marginal variances. Purple adds covariance \\(0.8\\), stretching along the shared diagonal; green has zero covariance. These are unit Mahalanobis-distance contours, not 95% regions or fitted samples. Sum variances are \\(3.6\\) and \\(2\\), respectively.",
    "alt": "Both analytic matrices have unit marginal variances. Purple adds covariance \\(0.8\\), stretching along the shared diagonal; green has zero covariance. These are unit Mahalanobis-distance contours, not 95% regions or fitted samples. Sum variances are \\(3.6\\) and \\(2\\), respectively.",
    "scope": "Architecture and dataflow mechanism diagram.",
    "height": 5.8,
    "panel": {
      "kind": "covariance"
    }
  }
}