Artificial and Approximate Likelihoods

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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.