{
  "equiprobable_XOR": {
    "joint": [
      {
        "X": 0,
        "Y": 0,
        "Z": 0,
        "count": 1,
        "probability": "1/4"
      },
      {
        "X": 0,
        "Y": 1,
        "Z": 1,
        "count": 1,
        "probability": "1/4"
      },
      {
        "X": 1,
        "Y": 0,
        "Z": 1,
        "count": 1,
        "probability": "1/4"
      },
      {
        "X": 1,
        "Y": 1,
        "Z": 0,
        "count": 1,
        "probability": "1/4"
      }
    ],
    "MI_X_Y_bits": 0.0,
    "MI_XZ_Y_bits": 1.0,
    "MI_Z_Y_bits": 0.0,
    "MI_terms": [
      {
        "x": [
          0
        ],
        "y": [
          0
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      },
      {
        "x": [
          0
        ],
        "y": [
          1
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      },
      {
        "x": [
          1
        ],
        "y": [
          1
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      },
      {
        "x": [
          1
        ],
        "y": [
          0
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      }
    ],
    "conditional_slices": [
      {
        "z": 0,
        "mass": "1/2",
        "conditional_MI_bits": 1.0
      },
      {
        "z": 1,
        "mass": "1/2",
        "conditional_MI_bits": 1.0
      },
      {
        "z": 2,
        "mass": "0",
        "status": "zero_mass_skipped_no_conditional_distribution"
      }
    ],
    "CMI_X_Y_given_Z_bits": 1.0,
    "independent_entropy_CMI_bits": 1.0,
    "independent_chain_rule_CMI_bits": 1.0,
    "independent_joint_ratio_CMI_bits": 1.0
  },
  "independent_three_bits": {
    "joint": [
      {
        "X": 0,
        "Y": 0,
        "Z": 0,
        "count": 1,
        "probability": "1/8"
      },
      {
        "X": 0,
        "Y": 0,
        "Z": 1,
        "count": 1,
        "probability": "1/8"
      },
      {
        "X": 0,
        "Y": 1,
        "Z": 0,
        "count": 1,
        "probability": "1/8"
      },
      {
        "X": 0,
        "Y": 1,
        "Z": 1,
        "count": 1,
        "probability": "1/8"
      },
      {
        "X": 1,
        "Y": 0,
        "Z": 0,
        "count": 1,
        "probability": "1/8"
      },
      {
        "X": 1,
        "Y": 0,
        "Z": 1,
        "count": 1,
        "probability": "1/8"
      },
      {
        "X": 1,
        "Y": 1,
        "Z": 0,
        "count": 1,
        "probability": "1/8"
      },
      {
        "X": 1,
        "Y": 1,
        "Z": 1,
        "count": 1,
        "probability": "1/8"
      }
    ],
    "MI_X_Y_bits": 0.0,
    "MI_XZ_Y_bits": 0.0,
    "MI_Z_Y_bits": 0.0,
    "MI_terms": [
      {
        "x": [
          0
        ],
        "y": [
          0
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      },
      {
        "x": [
          0
        ],
        "y": [
          1
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      },
      {
        "x": [
          1
        ],
        "y": [
          0
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      },
      {
        "x": [
          1
        ],
        "y": [
          1
        ],
        "p": "1/4",
        "ratio_to_product_marginals": "1",
        "term_bits": 0.0
      }
    ],
    "conditional_slices": [
      {
        "z": 0,
        "mass": "1/2",
        "conditional_MI_bits": 0.0
      },
      {
        "z": 1,
        "mass": "1/2",
        "conditional_MI_bits": 0.0
      }
    ],
    "CMI_X_Y_given_Z_bits": 0.0,
    "independent_entropy_CMI_bits": 0.0,
    "independent_chain_rule_CMI_bits": 0.0,
    "independent_joint_ratio_CMI_bits": 0.0
  },
  "nonlinear_X_squared_zero_correlations": {
    "analysis": {
      "joint": [
        {
          "X": -1,
          "Y": 1,
          "Z": 0,
          "count": 1,
          "probability": "1/3"
        },
        {
          "X": 0,
          "Y": 0,
          "Z": 0,
          "count": 1,
          "probability": "1/3"
        },
        {
          "X": 1,
          "Y": 1,
          "Z": 0,
          "count": 1,
          "probability": "1/3"
        }
      ],
      "MI_X_Y_bits": 0.9182958340544893,
      "MI_XZ_Y_bits": 0.9182958340544893,
      "MI_Z_Y_bits": 0.0,
      "MI_terms": [
        {
          "x": [
            -1
          ],
          "y": [
            1
          ],
          "p": "1/3",
          "ratio_to_product_marginals": "3/2",
          "term_bits": 0.1949875002403854
        },
        {
          "x": [
            0
          ],
          "y": [
            0
          ],
          "p": "1/3",
          "ratio_to_product_marginals": "3",
          "term_bits": 0.5283208335737186
        },
        {
          "x": [
            1
          ],
          "y": [
            1
          ],
          "p": "1/3",
          "ratio_to_product_marginals": "3/2",
          "term_bits": 0.1949875002403854
        }
      ],
      "conditional_slices": [
        {
          "z": 0,
          "mass": "1",
          "conditional_MI_bits": 0.9182958340544893
        }
      ],
      "CMI_X_Y_given_Z_bits": 0.9182958340544893,
      "independent_entropy_CMI_bits": 0.9182958340544896,
      "independent_chain_rule_CMI_bits": 0.9182958340544893,
      "independent_joint_ratio_CMI_bits": 0.9182958340544893
    },
    "ordinary_covariance": "0",
    "tied_rank_covariance": "0",
    "nonzero_variances_so_correlations_zero": true
  },
  "all_eight_binary_category_relabelings": 8,
  "invalid_cases_rejected": [
    "negative",
    "bool_count",
    "fractional_count",
    "empty",
    "zero_total",
    "bad_shape",
    "mismatched_supplied_marginal_totals"
  ],
  "units": "log2 bits; sklearn mutual_info_score uses natural-log nats and is not executed",
  "scope": "complete authored finite counted joint law, no population estimation/pvalue/causality/feature-ranking advice; zero-mass conditional values skipped explicitly; Cornell Remark11.1 item2 first displayed line has an apparent typo and is not copied, independent entropy identity uses H(X,Y|Z)"
}