Artificial and Approximate Likelihoods

DMS 9626266 Mykland该项目寻求将似然方法的使用扩展到半参数和非参数情况。重要的问题包括如何定义和评估似然比和R-STAR统计的准确性。到目前为止，特别令人感兴趣的是对偶似然的发展，对偶象的Bartlett恒等式的发展，以及允许导出鞅的渐近展开的嵌入技术的发展。目前，研究人员正在通过考虑近似似然的准则函数，将该理论扩展到包括非鞅情况。这涵盖了更广泛的数据分析问题。最好描述这一方法可以涵盖哪些类型的推理，以及在执行基于这种方法的程序时，应当使用哪些对可能性推理的修正。这项研究既涉及现有的程序(如经验过程和点过程的“可能性”)，也涉及从人工可能性观点产生的新结构。特别是，正在调查“你自己设计的可能性”，特别是基于重新抽样的标准函数。作为该项目的一部分，正在研究不确定情况下金融工程和投资问题的影响。商界和政府的政策制定者都面临着在不确定的情况下做出决定的必要性。例如，公司在投资新工厂时，对其产品当前和未来的市场状况知之甚少。作为另一个例子，有关环境的法规往往试图影响如此复杂的系统，以至于即使有最好的模型和科学研究，一个人的行动的影响也存在巨大的不确定性。在这种情况下的决策不仅需要估计和预测，而且还需要最准确地量化这些估计可能与实际数字相距多远。这个项目是关于这样做的一种新技术，一种极大地提高这种评估的可靠性的技术。它基于一种统计学理论(“可能性推断”)，该理论最早是在20世纪20年代在英国发展起来的，但直到最近几年，该理论才被开放给政策制定者经常面临的更复杂、更模糊的系统。

DMS 9626266 Mykland The project seeks to extend the use of likelihood methods to semi- and nonparametric situations. Important questions include how to define and assess the accuracy of the likelihood ratio and R-star statistics. Of particular interest so far has been the development of the dual likelihood, of Bartlett identities for martingales, and of embedding techniques which permit the derivation of asymptotic expansions for martingales. Currently, the investigators are expanding the theory to cover non-martingale situations, by considering criterion functions which are approximately likelihoods. This covers a much broader spectrum of data analysis problems. It is desirable to describe what types of inference can be covered by this, and what corrections over likelihood inference that ought to be used when carrying out procedures based on this approach. The study concerns both existing procedures (such as empirical and point process "likelihoods"), and at new constructions which arise from the artificial likelihood point of view. In particular, the "design-your-own likelihood" is being investigated, with particular reference to resampling based criterion functions. The implications for problems in financial engineering and investment under uncertainty are being studied as part of the project. Policy makers in both business and goverment are faced with the need to take decisions under uncertainty. Firms invest in new plants, for instance, with imperfect knowledge of current and future market conditions for their products. Regulations concerning the environment, as another example, often try to affect systems that are so complex that even with the best models and scientific studies, there is tremendous uncertainty about the effects of one's actions. Decisions in such circumstances not only require estimates and predictions, but also a maximally accurate quantification of how far away such estimates are likely to be from the actual figures. This project is about a new technology for doing this, one that substantially improves the reliability of such assessments. It is based on a statistical theory ("likelihood inference") first developed in Britain in the 1920s, but which has only in the last few years been opened up to the more complex and vaguely specified systems often faced by policy makers.