Sequential multi-class diagnosis, also known as multi-hypothesis testing, is a classical sequential decision problem with broad applications. However, the optimal solution remains, in general, unknown as the dynamic program suffers from the curse of dimensionality in the posterior belief space. We consider a class of practical problems in which the observation distributions associated with different classes are related through exponential tilting, and show that the reachable beliefs could be restricted on, or near, a set of low-dimensional, time-dependent manifolds with closed-form expressions. This sparsity is driven by the low dimensionality of the observation distributions (which is intuitive) as well as by specific structural interrelations among them (which is less intuitive). We use a matrix factorization approach to uncover the potential low dimensionality hidden in high-dimensional beliefs and reconstruct the beliefs using a diagnostic statistic in lower dimension. For common univariate distributions, e.g., normal, binomial, and Poisson, the belief reconstruction is exact, and the optimal policies can be efficiently computed for a large number of classes. We also characterize the structure of the optimal policy in the reduced dimension. For multivariate distributions, we propose a low-rank matrix approximation scheme that works well when the beliefs are near the low-dimensional manifolds. The optimal policy significantly outperforms the state-of-the-art heuristic policy in quick diagnosis with noisy data. (forthcoming in Operations Research)
翻译:连续的多级诊断,也称为多假测试,是典型的、具有广泛应用的顺序决定问题。然而,由于动态程序在后视信仰空间中受到多元性的诅咒,因此,最佳解决方案一般仍不为人所知。我们考虑了一系列实际问题,其中与不同类别相关的观测分布通过指数倾斜相关联,并表明可实现的信念可能限制在或接近于一组具有封闭式表达式的低维、依赖时间的元件。这种偏狭性是由观测分布(不直观的)的低维度驱动的,以及它们之间具体的结构性相互关系(不直观的)。我们采用矩阵化的因子化方法来发现高维信仰中隐藏的潜在低维度,并利用较低维度的诊断性统计来重建信仰。对于常见的单向分布,例如正常的、二元的和普瓦森,信仰的重建是准确的,而最佳的政策政策可以有效地为接近于一个类的精度的精度分析性分类而进行精确的统计分析。我们用矩阵模型来分析高度的模型来分析高度政策结构,在最优度的模型中提出最优度的模型。