Many models require integrals of high-dimensional functions: for instance, to obtain marginal likelihoods. Such integrals may be intractable, or too expensive to compute numerically. Instead, we can use the Laplace approximation (LA). The LA is exact if the function is proportional to a normal density; its effectiveness therefore depends on the function's true shape. Here, we propose the use of the probabilistic numerical framework to develop a diagnostic for the LA and its underlying shape assumptions, modelling the function and its integral as a Gaussian process and devising a "test" by conditioning on a finite number of function values. The test is decidedly non-asymptotic and is not intended as a full substitute for numerical integration - rather, it is simply intended to test the feasibility of the assumptions underpinning the LA with as minimal computation. We discuss approaches to optimize and design the test, apply it to known sample functions, and highlight the challenges of high dimensions.

翻译：许多模型需要高维函数的积分：例如，为了获得边缘似然。这些积分可能难以处理，或者数值计算成本过高。作为替代，我们可以使用拉普拉斯近似（LA）。如果函数与正态密度成比例，则LA是精确的；因此其有效性取决于函数的真实形状。在此，我们提出利用概率数值框架来开发一种针对LA及其基础形状假设的诊断方法，将函数及其积分建模为高斯过程，并通过以有限数量的函数值为条件来设计一种“检验”。该检验明确是非渐近的，并非旨在完全替代数值积分——相反，它仅旨在以尽可能少的计算量测试支撑LA的假设的可行性。我们讨论了优化和设计该检验的方法，将其应用于已知的样本函数，并强调了高维情况下的挑战。