The Value of Mechanistic Priors in Sequential Decision Making

TL;DR

This paper quantifies the value of mechanistic priors in sequential decision-making, showing they reduce data needs and improve efficiency, especially in early decision regimes, with applications in pharmacokinetics.

Contribution

It introduces the concept of mechanistic information and provides theoretical bounds for sample efficiency gains in decision models using physical priors.

Findings

Bayesian regret scales with residual entropy in asymptotic regimes.

Hybrid priors significantly improve sample efficiency in burn-in regimes.

LLMs can lose mechanistic information, highlighting the importance of physical priors.

Abstract

Hybrid mechanistic models, physical priors with learned residuals, promise to reduce the data required for good decisions, but have no computable criterion to test this. We characterize the value of mechanistic priors in sequential decision-making within both asymptotic and burn-in regimes. To formalize this, we introduce the mechanistic information of a model -- the mutual information between the model's recommended policy $\hat{\pi}$ and the true optimal policy $\pi^*$ -- quantified via an occupancy-weighted bias $B_\mu$. In the asymptotic regime (large $N$), matched bounds reveal that Bayesian regret scales with the residual entropy $H_{\mathrm{mech}}$, delivering a theoretical sample complexity reduction of $H(\mu)/H_{\mathrm{mech}}$ compared to an uninformed baseline. Furthermore, we provide a model certificate to determine empirical sample efficiency. Complementarily, in the clinically relevant burn-in regime (small $N$), we establish a lower bound on the penalty incurred by confidently wrong priors. We demonstrate both the asymptotic and burn-in bounds across 5-fluorouracil (5-FU) dosing simulations motivated by published FOLFOX pharmacokinetic data, where a hybrid prior yields large sample-efficiency gains in the burn-in regime. Finally, we contrast these grounded models with LLM priors, demonstrating that LLMs can suffer severe losses in mechanistic information, thereby motivating the exclusive use of physically-grounded priors for safety-critical applications.