Likelihood and Cost as Path Integrals
本文研究了时间序列的最优估计、预测和控制，提出通过直接最大化似然而非最小二乘来获得最优解，并将风险敏感准则下的问题转化为路径积分形式，推导了随机最大值原理。
SUMMARY We consider the optimal estimation, prediction and control etc. of time series. The essence of a path integral characterization is considered to be that optimal estimates and decisions etc. are obtained by the free extremization of some form. In Sections 2–5 it is shown for some stock situations that this property is ensured by the direct maximization of likelihood (rather than by appeal to the formally equivalent but operationally contrasting least square principle). In Sections 6–9 a risk-sensitive criterion is taken, in that the criterion is an exponential function of cost rather than cost itself. It is shown that this class of problems admits a natural path integral treatment, in which extremization of stress replaces separate extremization of likelihood and cost. A stochastic maximum principle is derived. For the linear–quadratic–Gaussian (LQG) case, results are exact and depend on proof of a certainty equivalence principle. In the non-LQG case they are valid as large deviation approximations, depending on a formalism going back to Bartlett. Finally, in Sections 10 and 11 we note the efficacy (if not indeed the necessity) of direct likelihood arguments in two other time series areas: recursive estimation and structural models.