I am trying to understand under what circumstances the log likelihood function of a point process concave. Assume that the process can be defined by a conditional intensity function and that the log likelihood function exists. Is there a general theory or good reference for understanding when the log likelihood function is concave (and hence you can reliably find the MLE)?

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#### Best Answer

**This depends on the parameterization of the conditional intensity function**. The (regular) point process likelihood is given by,

$L = left[ prod_i lambda^ast(t_i) right] expleft(-int lambda^ast(u) ,mathrm{d}u right) $

with the conditional intensity function $lambda^ast(t)$ (from Daley & Vere-Jones, 2002). The main problem is that it is a functional; it is a function of realization history. Therefore, a finite or nonparametric parameterization is needed to practically estimate it.

A successful class of estimators assume a linear form of dependence from the history. In the neuroscience community, it is (unfortunately) known as GLM due to its resemblance to (Poisson) generalized linear models when time is discretized. Here the conditional likelihood is parametrized as $lambda^ast(t) = f(mathbf{h}^top mathbf{r})$ where $mathbf{r}$ is the discretized finite history of the process, $mathbf{h}$ is the (finite vector) parameter, and $f(cdot)$ is a (pointwise) nonlinear function. **It can be shown that when $f$ is convex and log-concave, the log-likelihood is concave.** See (Paninski 2004).

- Paninski, L. (2004). Maximum likelihood estimation of cascade point-process neural encoding models. Network: Comput. Neural Syst., 15(04):243-262.

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