Modeling the statistical properties of a point process has important applications in neuroscience as well as security applications for detecting anomalous behavior.  For example, characterizing neural spiking activity as a function of environmental stimuli and intrinsic effects, such as a neuron's own spiking history and concurrent ensemble activity, is currently being performed with point process methodologies.  Such characterizations are complex and there is increasing need for a broad class of models to capture such details. 

The likelihood of a point process is completely defined by its conditional intensity function (CIF).  Most point process models are parametric as they are often efficiently computable, the parameters may be related back to physiological and/or environmental factors, and they have nice asymptotic properties when the CIF lies in the assumed parametric class. However, if the CIF does not lie in the assumed class, misleading inferences can arise.  Nonparametric methods are attractive due to fewer assumptions, but very few efficient methods for estimating the CIF are known. 

We propose a computationally efficient method for nonparametric maximum likelihood estimation when the log of the CIF is sufficiently smooth (Lipschitz continuous).  We propose minimizing the negative log likelihood of the point process subject to the Lipschitz continuity constraint.  We show that such an optimization problem is convex and thus efficiently solvable. We identify an equivalent problem with separable structure in the objective function and linear structure in the constraints, which allows us to represent the dual in closed form.  We next exploit the structure of the dual to cast this as an equivalent closed-form problem involving a significant reduction in the number of constraints.  We show this problem can solved using very efficient unconstrained methods, such as gradient descent or line search. We apply our method to goldfish retinal ganglion neural data and compare results to inhomogeneous Poisson and inverse Gaussian parametric models.