Evaluating point-process likelihoods

We recently discussed two recent papers proposing improvements to the commonly used discrete approximation of the log-likelihood for a point process (Paninski, 2004). The likelihood is written as
ll(T|t_{1,..,N(T)}) = \sum_{i = 1}^{N(T)} \log( \lambda(t_i|\mathcal{H}_{t_i})) - \int_0^T \lambda(t|\mathcal{H}_{t}) dt
where t_i are the spike times and \lambda(t|\mathcal{H}_t) is the conditional intensity function (CIF) of the process at time t given the preceding spikes. Typically, the integral in this equation cannot be evaluated in closed form. The standard approximation computes the function by binning along a regular lattice with bins size \delta
ll(T|t_{1,..,N(T)}) \approx \sum_{i = 1}^{l} \Delta N_i \log(\lambda_i \delta) - \lambda_i\delta
where \Delta N_i is the number of spikes in the ith bin. Both papers demonstrate that smarter approximations to the integral are better for point-process statistics than naïvely binning spike train data.

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