Bits per Spike as a Betting Game: An interpretable unit for held-out log-likelihood in neural data analysis

Held-out log-likelihood is the standard currency for comparing statistical models of neural spike trains, and is often reported as bits per spike relative to a homogeneous Poisson baseline. The units of this metric are difficult to reason about: it is rarely obvious whether an improvement of, say, 0.34 bits per spike is a large effect or a negligible one. This note develops an interpretation of held-out log-likelihood borrowed from game-theoretic statistics. If $ℒ$ denotes a model's expected log-likelihood ratio to the homogeneous Poisson baseline, then we show that $-log\left(\alpha \right)/ℒ$ represents the number of held-out time bins of recording needed to reject the baseline at level $\alpha$ under a particular null hypothesis testing procedure based on a betting game. Thus, a simple re-scaling of the heldout log-likelihood yields an intuitive metric of model performance in units of recording time which answers "how much heldout data would I need, on average, to disprove the null." We illustrate the construction on head-direction cells recorded in mouse anterior thalamus, where a generalized linear model reaches significance in roughly 120 ms of held-out data for a strongly tuned cell and roughly 11 s for a moderately tuned cell.