含相关表现的竞赛的可扩展反演，包括Softmax和多元Probit
Scalable Inversion of Contests with Correlated Performances, Including Softmax and Multinomial Probit
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中文总结 AI 辅助
该研究针对含相关表现的竞赛，提出适用于因子等协方差结构的可扩展反演校准方法，线性复杂度下在n=1e6时仍高精度，可作为Logit的实用替代。
中文摘要 AI 辅助
n个选项上的多元Probit选择概率是高斯正交积分，三十年来一直通过模拟计算，每个选项对应一个昂贵的积分。反演即确定与规定选择概率向量一致的选项吸引力，这一问题更为困难，当n较大时，对于相关竞赛，反演被认为不切实际。然而，对于包含因子、块和层次协方差结构的语法内的族，我们展示了一种校准方法，在n=1,000,000时进行了测试，即使在极端尾部也能以极高精度重现概率。由于现有技术的限制，要进行任何性能比较，我们必须回到小得多的问题。Geweke-Hajivassiliou-Keane模拟器是标准方法（仍然适用于高秩情况），但在n=200时已经慢200倍，其测得的成本大致按n^2.8增长，而我们的方法是线性的。此外，我们的方法适用于合理范围内的任何连续表现分布：由此，Thurstone-Mosteller模型族在现代规模下成为Logit的实用替代方案。
英文摘要
Multinomial probit choice probabilities over n alternatives are Gaussian orthant integrals, computed by simulation for thirty years, one expensive integral per alternative. Inversion, which is to say determining item attractiveness consistent with a prescribed choice probability vector, is even more difficult and has been considered impractical for correlated contests when n is large. Yet here, for families lying within a grammar including factor, block and hierarchical covariance structures, we exhibit a calibration tested at n = 1,000,000 reproducing probabilities to very high accuracy, even in the extreme tail. We must return to much smaller problems for any performance comparison to be possible due to limitations of the prior art. The Geweke-Hajivassiliou-Keane simulator is the standard (and still appropriate for high rank) but is two hundred times slower already at n = 200, and its measured cost grows roughly as n^2.8 while ours is linear. Furthermore our approach applies to any continuous performance distributions within reason: the Thurstone-Mosteller model families thereby become a practical alternative to logit at modern scale.
