We address the statistical inference of a time-dependent rate of events in the framework of Bayesian field theory. This maps the problem to a Langevin equation which, beyond the local linear regime taken as reference, involves nonlinearities and an explicit dependence on the local shape of the maximum likelihood curve. We study the corresponding impacts in a perturbative expansion, formulating a scaling hypothesis for the order of shape corrections. We find that the pure nonlinearities dominate the mean and skewness. Crucially, we uncover that the leading correction to the variance is driven by noise propagation from the signal's effective curvature. We test the derived expansion with numerical simulations and illustrate its applicability on real neural spike data.

翻译：我们在贝叶斯场论的框架下，研究时间依赖事件速率的统计推断问题。该问题可映射至一个朗之万方程，该方程不仅包含作为参考的局部线性区域，还涉及非线性项以及对最大似然曲线局部形态的显式依赖。我们通过微扰展开研究相应影响，并针对形态修正的阶次提出标度假设。研究发现，纯非线性项主导了均值和偏度。关键的是，我们发现方差的主要修正项由信号有效曲率的噪声传播所驱动。我们通过数值模拟验证了所推导的展开式，并在真实神经脉冲数据上展示了其适用性。