New Horizons in Statistical Decision Theory

统计模型(实验)的经典度量理论已经扩展到渐近等价范式，允许分类和关联本质上无限维的和不适定的问题。渐近等价理论正在成为统计学中一个公认的研究领域。从一个模型向另一个模型传递最优过程的理论可能性允许更好地理解渐近推理的概念性。这一领域仍在蓬勃发展，在基于高维数据的渐近推理背景下出现了新的问题。像这样的现代统计概念也被整合到新兴的量子统计领域，该领域是在量子工程技术突破的背景下发展起来的。对量子统计模型的分析指向一个潜在的非对易统计决策理论，该理论与算符代数、量子信息和量子概率有关。通过将重点放在统计学和数学物理交界处这一日益增长的研究领域，该项目将产生促进不同科学界之间互动的副作用。渐近等价理论中未被探索的主题之一是非参数模型中附加观测的影响。Le Cam证明了在正则参数情形下，如果附加观测值的数目比原始样本量小一个数量级，则附加观测值在渐近信息意义下是可以忽略的，但对于更大的参数空间，这个临界阈值似乎更低。在过去，通过附加观测的推理已被应用于证明谱密度与白噪声模型的等价性，并且可能再次被证明对与高维矩阵估计有关的问题有用。在一个相关的主题中，进一步完善像Pinsker界这样的尖锐的非参数风险界是有意义的，它推动了等价理论的发展。在这里，研究程序旨在证实关于尖锐自适应非参数检验的这类界的一个猜想，这可能是对自适应估计结果的补充。在对称量子假设检验或两个量子态之间的区分问题中，该程序侧重于与错误概率的指数衰减率有关的量子切尔诺夫界的应用。在这方面，出现了几个新的问题，如量子光学中可实现的接收器的界限的可达性，以及量子马尔可夫链给出的辨别状态的变体。一个特别感兴趣的领域是量子统计模型的局部渐近正态。文献中已经提出了这种理论的一些元素；这些概念将本着Le Cam经典理论的精神进一步发展，首先集中在高斯平稳序列的量子模拟上。

The classical metric theory of statistical models (experiments) has been extended towards an asymptotic equivalence paradigm, allowing to classify and relate problems which are essentially infinite dimensional and ill-posed. Asymptotic equivalence theory is emerging as a recognizable research area in statistics. The theoretical possibility to carry over optimal procedures from one model to another allows a better conceptual understanding of asymptotic inference. This area is still under vigorous development, and new problems arise in the context of asymptotic inference based on high-dimensional data. Modern statistical concepts like these are also being integrated into the emerging field of quantum statistics, which is developing on the background of technological breakthroughs in quantum engineering. The analysis of quantum statistical models points towards an underlying non-commutative statistical decision theory with connections to operator algebra, quantum information and quantum probability. By focusing on this growing research area at the interface of Statistics and Mathematical Physics, the project will have the side effect of fostering interaction between the different scientific communities. One of the unexplored topics in asymptotic equivalence theory is the impact of additional observations in nonparametric models. Le Cam showed that in the regular parametric case, additional observations are negligible in an asymptotic information sense if their number is one order of magnitude below the original sample size, but for larger parameter spaces this critical threshold seems to be lower. Reasoning via additional observations has been applied in the past to prove equivalence of spectral density to white noise models, and may prove useful again for problems related to estimation of high-dimensional matrices. In a related topic, it is of interest to further refine sharp nonparametric risk bounds like the Pinsker bound, which have motivated the development of equivalence theory. Here the research program aims at confirming a conjecture about a bound of this type for sharp adaptive nonparametric testing, a possible complement to results in adaptive estimation. In the problem of symmetric quantum hypothesis testing, or discrimination between two quantum states, the program focuses on applications of the quantum Chernoff bound pertaining to the exponential rate of decay of the error probability. In that connection, several new problems appear, such as attainability of the bound by realizable receivers in quantum optics, and variants for discriminating states given by quantum Markov chains. An area of particular interest is local asymptotic normality for quantum statistical models. Some elements of such a theory have already been put forward in the literature; these notions will be further developed in the spirit of Le Cam's classical theory, focusing at first on the quantum analog of Gaussian stationary sequences.