优化惩罚模型惩罚强度的简单数据驱动方法及其在非参数平滑中的应用
A simple data-driven method to optimise the penalty strengths of penalised models and its application to non-parametric smoothing
摘要：
感兴趣的信息通常只能通过模型拟合从数据中提取。当这种模型的函数形式不能从第一原理推导出来时，人们必须在不同的可能模型之间做出选择。在这种情况下，一种常见的方法是通过尽可能减少拟合变量的数量（或模型的灵活性），同时仍然产生可接受的数据拟合，以最大限度地减少模型中的信息损失。通过赤池信息准则（AIC）的模型选择提供了奥卡姆剃刀的这种实现。我们认为，同样的原则可以应用于优化惩罚的最大似然模型的惩罚强度。然而，虽然在典型的应用中，AIC用于从有限的离散的最大似然模型集合中进行选择，但惩罚优化需要从连续的候选模型中进行选择，并且这些模型违反了最大似然条件。我们推导出一个广义的信息准则AICp，包括这种情况下。它自然涉及有效自由参数的概念，这是非常灵活的，可以应用于任何模型，无论是线性或非线性，参数或非参数，并与或没有约束方程的参数。我们表明，广义AICP允许优化的任何处罚强度，而不需要单独的蒙特卡罗模拟。作为一个应用实例，我们讨论了非参数模型中的平滑优化，它在天体物理学中有许多应用，如动力学建模，光谱拟合或引力透镜。
英文摘要：
Information of interest can often only be extracted from data by model fitting. When the functional form of such a model can not be deduced from first principles, one has to make a choice between different possible models. A common approach in such cases is to minimise the information loss in the model by trying to reduce the number of fit variables (or the model flexibility, respectively) as much as possible while still yielding an acceptable fit to the data. Model selection via the Akaike Information Criterion (AIC) provides such an implementation of Occam's razor. We argue that the same principles can be applied to optimise the penalty-strength of a penalised maximum-likelihood model. However, while in typical applications AIC is used to choose from a finite, discrete set of maximum-likelihood models the penalty optimisation requires to select out of a continuum of candidate models and these models violate the maximum-likelihood condition. We derive a generalised information criterion AICp that encompasses this case. It naturally involves the concept of effective free parameters which is very flexible and can be applied to any model, be it linear or non-linear, parametric or non-parametric, and with or without constraint equations on the parameters. We show that the generalised AICp allows an optimisation of any penalty-strength without the need of separate Monte-Carlo simulations. As an example application, we discuss the optimisation of the smoothing in non-parametric models which has many applications in astrophysics, like in dynamical modeling, spectral fitting or gravitational lensing.
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