In most commonly used ranking systems, some level of underlying transitivity is assumed. If transitivity exists in a system then information about pairwise comparisons can be translated to other linked pairs. For example, if typically A beats B and B beats C, this could inform us about the expected outcome between A and C. We show that in the seminal Bradley-Terry model knowing the probabilities of A beating B and B beating C completely defines the probability of A beating C, with these probabilities determined by individual skill levels of A, B and C. Users of this model tend not to investigate the validity of this transitive assumption, nor that some skill levels may not be statistically significantly different from each other; the latter leading to false conclusions about rankings. We provide a novel extension to the Bradley-Terry model, which accounts for both of these features: the intransitive relationships between pairs of objects are dealt with through interaction terms that are specific to each pair; and by partitioning the $n$ skills into $A+1\leq n$ distinct clusters, any differences in the objects' skills become significant, given appropriate $A$. With $n$ competitors there are $n(n-1)/2$ interactions, so even in multiple round robin competitions this gives too many parameters to efficiently estimate. Therefore we separately cluster the $n(n-1)/2$ values of intransitivity into $K$ clusters, giving $(A,K)$ estimatable values respectively, typically with $A+K<n$. 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. The model is shown to have an improved fit out of sample in both simulated data and when applied to American League baseball data.
翻译:在最常用的排名系统中,可以假定某种程度的中转性。如果在一个系统中存在中转性,那么关于对称比较的信息可以被翻译成其他相联配对。例如,如果通常A打B和B打C,这可以告诉我们A和C的预期结果。我们显示,在了解A打B和B打C概率的原始布拉德利-Terriy模型中,可以完全确定A打C的概率,这些概率由A、B和C的个人技能水平确定。这个模型的用户往往不会调查这一过渡性假设的有效性,或者某些技能水平可能不会在统计上与对方有显著差异;例如,如果典型的Beat B和B打C的概率,这可以告诉我们A-Terriy模型的预期结果。 在每对模型中,一对相对相对的不透明关系通过互动术语来处理;通过将美元技能分成美元(美元+1美元)的数值跳升至美元(美元),这个模型的用户往往不会调查该对象的技能差异变得很明显, 以美元为美元为美元。