成对比较中非传递性的建模及其在棒球数据中的应用
Modelling intransitivity in pairwise comparisons with application to baseball data
AI总结：
本文提出一种半参数贝叶斯分层模型，通过随机数量的技能层级和非传递性层级灵活刻画成对比较中的非传递性，并在棒球数据上验证其优于 Bradley-Terry 等现有模型的预测性能。
摘要：
经典的 Bradley-Terry 模型表现出传递性，即选手 A 击败 B 和 B 击败 C 的概率可以给出 A 击败 C 的概率，这些概率由每个选手的技能参数决定。这类传递性模型没有考虑每对选手之间不同的比赛策略，而这种差异会导致非传递性。目前已提出多种非传递性参数模型，但它们缺乏灵活性，无法覆盖 n 个选手之间的不同策略，其 O(n^2) 个非传递性取值仅用 O(n) 个参数建模，同时在非传递性结构简单时又不够简约。我们通过将每对选手分配到随机数量的 K 个非传递性层级之一来克服其适应性不足的问题，每个层级代表一种不同的策略。我们对技能参数的新方法是将 n 个选手分配到随机数量的 A<n 个不同技能层级，以提高效率并避免虚假排名。尽管对于 (A,K) 我们可能需要估计多达 O(n^2) 个未知参数，但我们预计在许多实际情境中 A+K<n。使用贝叶斯分层模型，将 (A,K) 视为未知，并通过可逆跳马尔可夫链蒙特卡罗（RJMCMC）算法进行推断。我们的半参数模型在 (A=n-1, K=0) 时退化为 Bradley-Terry 模型，在应用于模拟数据和美国联盟棒球数据的样本外测试中，相对于 Bradley-Terry 模型以及现有的非传递性模型，显示出更好的拟合效果。本文的补充材料可在网上获取。
英文摘要：
The seminal Bradley-Terry model exhibits transitivity, i.e., the property that the probabilities of player A beating B and B beating C give the probability of A beating C, with these probabilities determined by a skill parameter for each player. Such transitive models do not account for different strategies of play between each pair of players, which gives rise to {\it intransitivity}. Various intransitive parametric models have been proposed but they lack the flexibility to cover the different strategies across $n$ players, with the $O(n^2)$ values of intransitivity modelled using O(n) parameters, whilst they are not parsimonious when the intransitivity is simple. We overcome their lack of adaptability by allocating each pair of players to one of a random number of $K$ intransitivity levels, each level representing a different strategy. Our novel approach for the skill parameters involves having the $n$ players allocated to a random number of $A<n$ distinct skill levels, to improve efficiency and avoid false rankings. Although we may have to estimate up to $O(n^2)$ unknown parameters for $(A,K)$ we anticipate that in many practical contexts $A+K < n$. Using a Bayesian hierarchical model, $(A,K)$ are treated as unknown, and inference is conducted via a reversible jump Markov chain Monte Carlo (RJMCMC) algorithm. Our semi-parametric model, which gives the Bradley-Terry model when $(A=n-1, K=0)$, is shown to have an improved fit relative to the Bradley-Terry, and the existing intransitivity models, in out-of-sample testing when applied to simulated and American League baseball data. Supplementary materials for the article are available online.
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