Bits per Spike as a Betting Game: An Interpretable Unit for Held-Out Log-Likelihood in Neural Data Analysis

Abstract

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. A fitted model $Q$ is treated as a player who bets on each upcoming observation at prices set by a baseline model $B$. Under the optimal (Kelly) betting strategy the player's contract function is exactly the likelihood ratio $q/b$, and the expected log-likelihood ratio $L$ is the exponential growth rate of the player's wealth. Because the wealth process is a nonnegative martingale under the null hypothesis that $B$ generated the data, Ville's inequality turns it into an anytime-valid test: the baseline may be rejected at level $\alpha$ as soon as wealth exceeds $1/\alpha$. This yields a simple summary statistic, the time to significance $\tau \Delta = -\Delta \log(\alpha) / L$, which is the amount of held-out recording needed on average to reject the baseline at level $\alpha$. Since $\tau$ is a strictly decreasing function of $L$, it ranks models identically to bits per spike; it is not a new statistic but a more interpretable unit for an existing one, expressed in seconds of recording rather than in bits. We illustrate the construction on head-direction cells recorded in mouse anterior thalamus, where a generalized linear model reaches significance against a homogeneous Poisson baseline in roughly $120$ ms of held-out data for a strongly tuned cell and roughly $11$ s for a moderately tuned cell.