1{
  2  "prediction_checks": {
  3    "P1_mode_separation": {
  4      "prediction": "persistent d=2 gives p(d=2)>0.5 while clean d=0 retains p(d=0)",
  5      "delayed": {
  6        "delay": 2,
  7        "noise": 0.03,
  8        "slope": 0.0007785781526003521,
  9        "mean_KL": 1.3336887651940748,
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 12        "alarm_median": 8.5,
 13        "dominant_median": 4.0,
 14        "final_true_posterior": 0.9605996611612697,
 15        "final_zero_posterior": 0.005663732997422032,
 16        "max_normalization_error": 4.440892098500626e-16
 17      },
 18      "clean": {
 19        "delay": 0,
 20        "noise": 0.03,
 21        "slope": 0.0,
 22        "mean_KL": 0.0,
 23        "cumulative_llr_slope": 0.0,
 24        "slope_over_KL": 0.0,
 25        "alarm_median": null,
 26        "dominant_median": null,
 27        "final_true_posterior": 0.9778142142508726,
 28        "final_zero_posterior": 0.9778142142508726,
 29        "max_normalization_error": 4.440892098500626e-16
 30      }
 31    },
 32    "P2_KL_slope": {
 33      "prediction": "early cumulative innovation log-likelihood slope approximately equals mean innovation KL",
 34      "noise_sweep": [
 35        {
 36          "delay": 2,
 37          "noise": 0.015,
 38          "slope": 0.0019416305413025238,
 39          "mean_KL": 2.6673775303881495,
 40          "cumulative_llr_slope": 2.748694855905076,
 41          "slope_over_KL": 0.9970763092026242,
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 44          "final_true_posterior": 0.9758583872484718,
 45          "final_zero_posterior": 0.0030562652574435455,
 46          "max_normalization_error": 4.440892098500626e-16
 47        },
 48        {
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 50          "noise": 0.03,
 51          "slope": 0.0007785781526003521,
 52          "mean_KL": 1.3336887651940748,
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 58          "final_zero_posterior": 0.005663732997422032,
 59          "max_normalization_error": 4.440892098500626e-16
 60        },
 61        {
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 63          "noise": 0.06,
 64          "slope": 0.0008225903767851935,
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 70          "final_true_posterior": 0.9239204397602936,
 71          "final_zero_posterior": 0.0111765248666157,
 72          "max_normalization_error": 4.440892098500626e-16
 73        },
 74        {
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 84          "final_zero_posterior": 0.0216412733577526,
 85          "max_normalization_error": 4.440892098500626e-16
 86        }
 87      ]
 88    },
 89    "P3_detection_scaling": {
 90      "prediction": "alarm time increases as KL decreases (larger observation noise)",
 91      "noise_sweep": [
 92        {
 93          "delay": 2,
 94          "noise": 0.015,
 95          "slope": 0.002926337182553665,
 96          "mean_KL": 2.6831277090853902,
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100          "dominant_median": 2.0,
101          "final_true_posterior": 0.9777240715163967,
102          "final_zero_posterior": 0.0030562652574435455,
103          "max_normalization_error": 4.440892098500626e-16
104        },
105        {
106          "delay": 2,
107          "noise": 0.03,
108          "slope": 0.0010502536788597526,
109          "mean_KL": 1.3415638545426951,
110          "cumulative_llr_slope": 1.3784283143154599,
111          "slope_over_KL": 0.9995188912097914,
112          "alarm_median": 8.5,
113          "dominant_median": 3.0,
114          "final_true_posterior": 0.9605996611612697,
115          "final_zero_posterior": 0.005663732997422032,
116          "max_normalization_error": 4.440892098500626e-16
117        },
118        {
119          "delay": 2,
120          "noise": 0.06,
121          "slope": 0.0011164307743681623,
122          "mean_KL": 0.6707819272713476,
123          "cumulative_llr_slope": 0.6903288838713548,
124          "slope_over_KL": 1.0003064693660588,
125          "alarm_median": 9.5,
126          "dominant_median": 5.5,
127          "final_true_posterior": 0.9124826162425348,
128          "final_zero_posterior": 0.0111765248666157,
129          "max_normalization_error": 4.440892098500626e-16
130        },
131        {
132          "delay": 2,
133          "noise": 0.12,
134          "slope": 0.001977235648426705,
135          "mean_KL": 0.3353909636356738,
136          "cumulative_llr_slope": 0.3409259316896296,
137          "slope_over_KL": 1.0110868359305787,
138          "alarm_median": 20.5,
139          "dominant_median": 9.0,
140          "final_true_posterior": 0.837563902349591,
141          "final_zero_posterior": 0.022575170891595865,
142          "max_normalization_error": 4.440892098500626e-16
143        }
144      ]
145    },
146    "P4_normalization": {
147      "prediction": "posterior sums to one to numerical precision",
148      "max_abs_sum_error": 4.440892098500626e-16
149    }
150  },
151  "settings": {
152    "a": 0.85,
153    "q": 0.04,
154    "D": 4,
155    "persistence": 0.985,
156    "steps": 1200,
157    "threshold": 0.01,
158    "replicates": 30,
159    "likelihood": "exact Gaussian innovation with known state history"
160  }
161}