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  "Description": "Solves the horse race problem in both directions. Infers\nrelative ability from win probabilities (or betting dividends)\nfor contests whose entrants share a common performance\ndistribution, with dead heats handled exactly, by the lattice\nfixed-point algorithm of Cotton (2021)\n<doi:10.1137/19M1276261>. Also computes all N win probabilities\nof a factor-structured Gaussian race (multinomial probit with\nlow-rank-plus-diagonal covariance) in one shared-lattice pass\nof O(Q*N*L) operations, and inverts observed shares to\nabilities by a damped Newton method with analytic slopes. A\ndependency-free base-R port of the reference python package\n'winning'. Performances are times: lowest wins.",
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      "page": "abilities_from_probabilities_factor",
      "title": "Calibrate abilities from observed shares",
      "topics": [
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    },
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      "title": "Block, nested and tree races",
      "topics": [
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        "block_race_jacobian",
        "block_race_probabilities",
        "nested_race_jacobian",
        "nested_race_probabilities",
        "tree_race_jacobian",
        "tree_race_probabilities"
      ]
    },
    {
      "page": "calibration",
      "title": "The horse race problem: ability from win probabilities, and back",
      "topics": [
        "ability_implied_dividends",
        "ability_implied_state_prices",
        "dividend_implied_ability",
        "solve_for_implied_offsets",
        "state_price_implied_ability"
      ]
    },
    {
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      "topics": [
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    },
    {
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      "title": "Polish a race onto linear constraints",
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        "polish_race",
        "race_jacobian"
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    },
    {
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      "title": "Convert between dividends and probabilities",
      "topics": [
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        "prices_from_dividends"
      ]
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      "page": "race_probabilities",
      "title": "The general race: one API, distributions and correlation as parameters",
      "topics": [
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        "calibrate_abilities",
        "race_probabilities"
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    },
    {
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      "title": "Skew-normal performance density on a symmetric lattice",
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    },
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        "winner_of_many"
      ]
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        "Factor",
        "Independent",
        "Nested",
        "structures",
        "Tree"
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        "tree_from_linkage"
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