1{
  2  "predictions": {
  3    "PIT_uniformity": "mean -> 0.5, variance -> 1/12, and KS remains small for every continuous marginal",
  4    "rank_invariance": "For a Gaussian copula, Spearman = (6/pi) asin(rho/2) and Kendall tau = (2/pi) asin(rho), independent of marginal family",
  5    "dependence_gain": "independent NLL - copula NLL = -0.5 log(1-rho^2), exactly zero at rho=0"
  6  },
  7  "derivative_check": {
  8    "u1": 0.37,
  9    "u2": 0.61,
 10    "rho": 0.65,
 11    "finite_difference": 1.1066858180308614,
 12    "analytic_density": 1.1066858181022963,
 13    "relative_error": 6.454845346382531e-11
 14  },
 15  "pit_sweep": [
 16    {
 17      "marginals": "lognormal/gamma",
 18      "coordinate": 1,
 19      "mean": 0.5025708728400309,
 20      "variance": 0.08379054561705422,
 21      "ks": 0.006052478091042479
 22    },
 23    {
 24      "marginals": "lognormal/gamma",
 25      "coordinate": 2,
 26      "mean": 0.5012865018936924,
 27      "variance": 0.08274643607100267,
 28      "ks": 0.00567742735503185
 29    },
 30    {
 31      "marginals": "gamma/lognormal",
 32      "coordinate": 1,
 33      "mean": 0.49874053546245234,
 34      "variance": 0.08378120625647417,
 35      "ks": 0.0046018493010879236
 36    },
 37    {
 38      "marginals": "gamma/lognormal",
 39      "coordinate": 2,
 40      "mean": 0.49992784431030524,
 41      "variance": 0.0841101564193521,
 42      "ks": 0.0038054650773772236
 43    },
 44    {
 45      "marginals": "lognormal/lognormal",
 46      "coordinate": 1,
 47      "mean": 0.5015810764285978,
 48      "variance": 0.08274240998874416,
 49      "ks": 0.005338492447773513
 50    },
 51    {
 52      "marginals": "lognormal/lognormal",
 53      "coordinate": 2,
 54      "mean": 0.49956384092902556,
 55      "variance": 0.08249412781279461,
 56      "ks": 0.005639051777740933
 57    },
 58    {
 59      "marginals": "gamma/gamma",
 60      "coordinate": 1,
 61      "mean": 0.4985339964590969,
 62      "variance": 0.08283447024967824,
 63      "ks": 0.004378046678996994
 64    },
 65    {
 66      "marginals": "gamma/gamma",
 67      "coordinate": 2,
 68      "mean": 0.5004704586413624,
 69      "variance": 0.083139588762496,
 70      "ks": 0.0032968037252048554
 71    }
 72  ],
 73  "dependence_sweep": [
 74    {
 75      "rho": 0.0,
 76      "predicted_spearman": 0.0,
 77      "observed_spearman": 0.007992403412745253,
 78      "abs_error": 0.007992403412745253,
 79      "predicted_kendall": 0.0,
 80      "observed_kendall": 0.005298272456811421,
 81      "kendall_abs_error": 0.005298272456811421
 82    },
 83    {
 84      "rho": 0.2,
 85      "predicted_spearman": 0.19130568257555955,
 86      "observed_spearman": 0.19390582879000365,
 87      "abs_error": 0.002600146214444099,
 88      "predicted_kendall": 0.12818843369794988,
 89      "observed_kendall": 0.13009653491337284,
 90      "kendall_abs_error": 0.0019081012154229526
 91    },
 92    {
 93      "rho": 0.5,
 94      "predicted_spearman": 0.4825837395309974,
 95      "observed_spearman": 0.48544994498178123,
 96      "abs_error": 0.0028662054507838097,
 97      "predicted_kendall": 0.33333333333333337,
 98      "observed_kendall": 0.335515202880072,
 99      "kendall_abs_error": 0.0021818695467386573
100    },
101    {
102      "rho": 0.8,
103      "predicted_spearman": 0.7859392826067277,
104      "observed_spearman": 0.7846116916311324,
105      "abs_error": 0.0013275909755953164,
106      "predicted_kendall": 0.5903344706017332,
107      "observed_kendall": 0.589014330358259,
108      "kendall_abs_error": 0.001320140243474155
109    }
110  ],
111  "likelihood_sweep": [
112    {
113      "rho": 0.0,
114      "independent_nll": 2.988177863009571,
115      "copula_nll": 2.988177863009571,
116      "observed_nll_gain": 0.0,
117      "predicted_nll_gain": -0.0,
118      "fitted_rho": 0.009830621920879473
119    },
120    {
121      "rho": 0.2,
122      "independent_nll": 2.986484172917755,
123      "copula_nll": 2.967029685922538,
124      "observed_nll_gain": 0.019454486995217124,
125      "predicted_nll_gain": 0.020410997260127583,
126      "fitted_rho": 0.19558871441873815
127    },
128    {
129      "rho": 0.5,
130      "independent_nll": 2.990996822875202,
131      "copula_nll": 2.844558396185997,
132      "observed_nll_gain": 0.14643842668920515,
133      "predicted_nll_gain": 0.14384103622589045,
134      "fitted_rho": 0.5034209829401081
135    },
136    {
137      "rho": 0.8,
138      "independent_nll": 2.978218673484344,
139      "copula_nll": 2.4657172908194975,
140      "observed_nll_gain": 0.5125013826648463,
141      "predicted_nll_gain": 0.5108256237659908,
142      "fitted_rho": 0.8009013610502361
143    }
144  ],
145  "mini_comparison": {
146    "dependence": 0.8,
147    "baseline_independent_nll": 2.978218673484344,
148    "idea_copula_nll": 2.4657172908194975,
149    "idea_fitted_rho": 0.8009013610502361
150  }
151}