Poisson spike-train likelihood fitting ($\ell ={\sum }_i\log \lambda (t_i)-\int \lambda (t)dt$)

For a stimulus-conditioned Inhomogeneous Poisson process, the event log-likelihood is the displayed expression. A causal filter expansion $\lambda (t)=\exp [b+{\sum }_a{w}_a(s\ast b_a)(t)]$ can be fitted by maximum likelihood estimation, with a smoothness or size penalty and held-out validation. The integral term penalizes predicted spikes that did not occur; fitting only stimulus values at observed spikes omits this essential part.