We show that MID is a maximum-likelihood estimator for the parameters of a Iinear—nonlinear—Poisson (LNP) model, and that the empirical single-spike information corresponds to the normalized log-likelihood under a Poisson model.

It is straightforward to show that empirical Bernoulli information equals the LNB model log-likelihood per spike plus a constant:

We will now unpack the details of this estimate in order to show its relationship to the LNP model log-likelihood .

This allows us to directly relate the empirical single-spike information (Equation 8) With the LNP model log-likelihood , normalized by the spike count as follows:

Where Llnpwo, D) denotes the Poisson log-likelihood under a “null” model in Which spike rate

This equivalence follows from the fact that the plugin estimate for the single-spike information [26], the quantity that MID optimizes, is equal to a normalized Poisson log-likelihood .

To show this, we derive the mutual information between the stimulus and a Bernoulli distributed spike count, and show that this quantity is closely related to the log-likelihood under a linear-nonlin-ear-Bernoulli encoding model.

Thus, the log-likelihood for projection matrix K, having already maximized With respect to the nonlinearities by using their plugin estimates, is

pluginComparison With the LNC model log-likelihood (Equation 30) reveals that: where H [r] = — 2133" logNVU) is the plug-in estimate for the marginal entropy of the ob-served spike countsobserved.

To do so, we approximate the marginal likelihood (10) by the eXpectation of the complete data log-likelihood

A derivation of the expected hidden log-likelihood and the maximum likelihood estimates is given in ‘Estimating the hidden signal’ of 81 Text.

To assess the impact that pathway disruption has on the cell population level, we ran the simulations on a standard NEM using the log-likelihood model introduced in [23].

The first column gives the log-likelihood for each model, showing that the true network is much less likely than the inferred networks.

Minimization of Eq 4 is equivalent to maximizing the log-likelihood given by

Model statistics With respect to the objective function, the log-likelihood and the AIC are summarised in 81 Table.

The values of the best fit and the average over 100 fits are given for the objective value (WRSS), the log-likelihood (ln(L( p ) )) and the Akaike Information Criterion (AIC), as defined in Materials and Methods.