{
  "predictions": {
    "PIT_uniformity": "mean -> 0.5, variance -> 1/12, and KS remains small for every continuous marginal",
    "rank_invariance": "For a Gaussian copula, Spearman = (6/pi) asin(rho/2) and Kendall tau = (2/pi) asin(rho), independent of marginal family",
    "dependence_gain": "independent NLL - copula NLL = -0.5 log(1-rho^2), exactly zero at rho=0"
  },
  "derivative_check": {
    "u1": 0.37,
    "u2": 0.61,
    "rho": 0.65,
    "finite_difference": 1.1066858180308614,
    "analytic_density": 1.1066858181022963,
    "relative_error": 6.454845346382531e-11
  },
  "pit_sweep": [
    {
      "marginals": "lognormal/gamma",
      "coordinate": 1,
      "mean": 0.5025708728400309,
      "variance": 0.08379054561705422,
      "ks": 0.006052478091042479
    },
    {
      "marginals": "lognormal/gamma",
      "coordinate": 2,
      "mean": 0.5012865018936924,
      "variance": 0.08274643607100267,
      "ks": 0.00567742735503185
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    {
      "marginals": "gamma/lognormal",
      "coordinate": 1,
      "mean": 0.49874053546245234,
      "variance": 0.08378120625647417,
      "ks": 0.0046018493010879236
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    {
      "marginals": "gamma/lognormal",
      "coordinate": 2,
      "mean": 0.49992784431030524,
      "variance": 0.0841101564193521,
      "ks": 0.0038054650773772236
    },
    {
      "marginals": "lognormal/lognormal",
      "coordinate": 1,
      "mean": 0.5015810764285978,
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      "ks": 0.005338492447773513
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    {
      "marginals": "lognormal/lognormal",
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      "mean": 0.49956384092902556,
      "variance": 0.08249412781279461,
      "ks": 0.005639051777740933
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    {
      "marginals": "gamma/gamma",
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      "mean": 0.4985339964590969,
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      "ks": 0.004378046678996994
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    {
      "marginals": "gamma/gamma",
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      "mean": 0.5004704586413624,
      "variance": 0.083139588762496,
      "ks": 0.0032968037252048554
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  ],
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    {
      "rho": 0.0,
      "predicted_spearman": 0.0,
      "observed_spearman": 0.007992403412745253,
      "abs_error": 0.007992403412745253,
      "predicted_kendall": 0.0,
      "observed_kendall": 0.005298272456811421,
      "kendall_abs_error": 0.005298272456811421
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    {
      "rho": 0.2,
      "predicted_spearman": 0.19130568257555955,
      "observed_spearman": 0.19390582879000365,
      "abs_error": 0.002600146214444099,
      "predicted_kendall": 0.12818843369794988,
      "observed_kendall": 0.13009653491337284,
      "kendall_abs_error": 0.0019081012154229526
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    {
      "rho": 0.5,
      "predicted_spearman": 0.4825837395309974,
      "observed_spearman": 0.48544994498178123,
      "abs_error": 0.0028662054507838097,
      "predicted_kendall": 0.33333333333333337,
      "observed_kendall": 0.335515202880072,
      "kendall_abs_error": 0.0021818695467386573
    },
    {
      "rho": 0.8,
      "predicted_spearman": 0.7859392826067277,
      "observed_spearman": 0.7846116916311324,
      "abs_error": 0.0013275909755953164,
      "predicted_kendall": 0.5903344706017332,
      "observed_kendall": 0.589014330358259,
      "kendall_abs_error": 0.001320140243474155
    }
  ],
  "likelihood_sweep": [
    {
      "rho": 0.0,
      "independent_nll": 2.988177863009571,
      "copula_nll": 2.988177863009571,
      "observed_nll_gain": 0.0,
      "predicted_nll_gain": -0.0,
      "fitted_rho": 0.009830621920879473
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    {
      "rho": 0.2,
      "independent_nll": 2.986484172917755,
      "copula_nll": 2.967029685922538,
      "observed_nll_gain": 0.019454486995217124,
      "predicted_nll_gain": 0.020410997260127583,
      "fitted_rho": 0.19558871441873815
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    {
      "rho": 0.5,
      "independent_nll": 2.990996822875202,
      "copula_nll": 2.844558396185997,
      "observed_nll_gain": 0.14643842668920515,
      "predicted_nll_gain": 0.14384103622589045,
      "fitted_rho": 0.5034209829401081
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    {
      "rho": 0.8,
      "independent_nll": 2.978218673484344,
      "copula_nll": 2.4657172908194975,
      "observed_nll_gain": 0.5125013826648463,
      "predicted_nll_gain": 0.5108256237659908,
      "fitted_rho": 0.8009013610502361
    }
  ],
  "mini_comparison": {
    "dependence": 0.8,
    "baseline_independent_nll": 2.978218673484344,
    "idea_copula_nll": 2.4657172908194975,
    "idea_fitted_rho": 0.8009013610502361
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}