Exchangeability -- in which the distribution of an infinite sequence is invariant to reorderings of its elements -- implies the existence of a simple conditional independence structure that may be leveraged in the design of probabilistic models, efficient inference algorithms, and randomization-based testing procedures. In practice, however, this assumption is too strong an idealization; the distribution typically fails to be exactly invariant to permutations and de Finetti's representation theory does not apply. Thus there is the need for a distributional assumption that is both weak enough to hold in practice, and strong enough to guarantee a useful underlying representation. We introduce a relaxed notion of local exchangeability -- where swapping data associated with nearby covariates causes a bounded change in the distribution. We prove that locally exchangeable processes correspond to independent observations from an underlying measure-valued stochastic process. We thereby show that de Finetti's theorem is robust to perturbation and provide further justification for the Bayesian modelling approach. Using this probabilistic result, we develop three novel statistical procedures for (1) estimating the underlying process via local empirical measures, (2) testing via local randomization, and (3) estimating the canonical premetric of local exchangeability. These three procedures extend the applicability of previous exchangeability-based methods without sacrificing rigorous statistical guarantees. The paper concludes with examples of popular statistical models that exhibit local exchangeability.
翻译:然而,在实践中,这一假设过于理想化;这种分布通常不能完全不易变异,而De Finetti的表述理论则不适用。因此,需要一种分布式假设,这种假设在实践上足够薄弱,足够强大,足以保证有用的基本代表性。 我们引入了一种简单的有条件的可交换性概念,在设计概率模型、高效推算算算算法和随机测试程序时可以利用这种结构。但在实践中,这一假设过于强烈;但实际上,这一假设过于理想化;分配通常不能完全不易变异,而De Finetti的表述理论则不适用。因此,需要一种分布式假设,这种假设既足以维持在实际中足够弱,又足够强大,足以保证有用的基本代表性。我们采用了一种宽松的本地可交换性概念,即与邻近的共变式相关数据互换后导致分配的限制性变化。我们证明,本地可互换程序与基本衡量性观测结果相对一致;我们由此表明,De Finetti的方程式坚固,为Bayesian的当地民众模拟方法提供了进一步的理由。利用这种可比较性的结果,我们制定了三个新的统计程序:(1) 通过地方可比较性评估性评估的可兑换性评估性,通过地方可兑换性评估的统计可兑换性,通过地方可兑换性前的统计性检验性检验性检验性检验性检验性检验性,可以采用前的统计性,这些可兑换性,可以评估前的统计性检验性,可以进行。