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. If L denotes a model's expected log-likelihood ratio to the homogeneous Poisson baseline, then we show that −log(α) / L represents the number of heldout time bins of recording needed to reject the baseline at level α under a particular null hypothesis testing procedure based on a betting game. Thus, a simple re-scaling of the log-likelihood yields an intuitive metric of model performance in units of recording time which answers "how much held out 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.
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BibTeX
@misc{williams2026bits,
title = {{Bits per Spike as a Betting Game: An Interpretable Unit for Held-Out Log-Likelihood in Neural Data Analysis}},
author = {Alex H. Williams},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2607.28779},
eprint = {2607.28779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.28779},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bits per Spike as a Betting Game: An Interpretable Unit for Held-Out Log-Likelihood in Neural Data Analysis
AU - Alex H. Williams
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2607.28779
UR - https://arxiv.org/abs/2607.28779
AB - 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(α) / L represents the number of heldout time bins of recording needed to reject the baseline at level α under a particular null hypothesis testing procedure based on a betting game. Thus, a simple re-scaling of the log-likelihood yields an intuitive metric of model performance in units of recording time which answers "how much held out 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.
ER -