The Value of Mechanistic Priors in Sequential Decision Making

The gist The hybrid prior reduces cumulative regret by 1.79× relative to standard body-surface-area dosing and 1.85× relative to an uninformed learner, and remains below both under every calibration bias

How it works

The paper introduces the value of mechanistic priors in sequential decision-making by formalizing the mechanistic information of a model as the mutual information between the model’s recommended policy and the true optimal policy, bounded via a centered, occupancy-weighted bias The core idea is that physical structure reduces data required for good decisions This information is quantified by the mechanistic information Rmech = I(π∗; ˆπ), and the regret is governed by what the model leaves unrevealed, measured by residual entropy Hmech = H(µ) − Rmech

The analysis covers two regimes:

1. The asymptotic regime (large N), where matched bounds reveal that Bayesian regret scales with the residual entropy Hmech, delivering a theoretical sample complexity reduction of H(µ)/Hmech compared to an uninformed baseline

2. The clinically relevant burn-in regime (small N), where a lower bound on the penalty incurred by confidently wrong priors is established

Key Theoretical Bounds

The paper establishes several theoretical bounds to characterize the model's worth before experimentation begins

**)&Theorem 3 (Critical bias, rule-out form) defines the critical bias, Bcritµ(s), a closed-form quantity whose inputs are the population spread of the measured parameter and the prior’s entropy, both of which can be obtained from reported population-pharmacokinetic studies This bias gives a necessary condition, checkable before any interaction, for a candidate model to reduce sample complexity by a target factor s The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The working point is Bcritµ at the one-cycle working point s = N/(N − 1) This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq. (7) and inherits its constants This is computable before any trial, allowing a practitioner to determine in advance whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner The certificate inputs and their ceiling C(Bµ) are those of Running Example 2, and every operating point below respects that ceiling The threshold is a statement about residual decision entropy; reading it as a factor-s saving in samples goes through Eq.

1. Bold mechanistic prior integration for sample complexity reduction: The system can utilize a hybrid prior based on a mechanistic model to achieve a theoretical sample complexity reduction of H(µ)/Hmech compared to an uninformed baseline, as stated in the abstract, leading to reduced data requirements for good decisions.

2. Critical bias-based pre-trial model certification: Before experimentation, the system can employ a computable pre-trial model certificate by checking if Bµ ≥ Bcrit µ (s) to determine whether a candidate mechanistic model is accurate enough to be worth deploying within a hybrid learner.

3. Burn-in regime cost estimation: For individual patient applications, the system can establish a lower bound on the penalty incurred by confidently wrong priors using Proposition 2, quantifying the minimum exploration cost that a confidently wrong prior forces, which is computable before any interaction.

4. Distribution shift robustness: The system can be designed to mitigate performance loss from model drift by utilizing physical priors, as priors sensitive to distribution shift can lose half of their mechanistic information under a small distribution shift, motivating physically grounded priors for safety-critical applications.

5. Information-theoretic regret scaling: The system's decision-making process can be optimized to scale its sample complexity according to the ratio ρ(ε) = Nuninf(ε)/Nmech(ε) = Θ H(µ)/Hmech, providing a principled way to trade prior information against interaction samples.

6. Model quality assessment via channel bounds: The system can assess a candidate model's worth by verifying that its mechanistic information is bounded by the measurement channel capacity, satisfying Rmech ≤ C(Bµ), which provides a necessary condition for sample complexity savings.

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 π and the true optimal policy π*, bounded via a centered, occupancy-weighted bias. In the asymptotic regime (large N), matched bounds reveal that Bayesian regret scales with the residual entropy H mech, delivering a theoretical sample complexity reduction of H(μ)/H mech compared to an uninformed baseline. We further provide a computable pre-trial model certificate. 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 on an illustrative in-silico 5-fluorouracil (5-FU) chemotherapy plant whose structure follows published FOLFOX pharmacokinetics. The hybrid prior reduces cumulative regret by 1.79 times relative to standard body-surface-area dosing and 1.85 times relative to an uninformed learner, and remains below both under every calibration bias tested. Finally, we show that priors sensitive to distribution shift can lose half of their mechanistic information under a small distribution shift, motivating physically grounded priors for safety-critical applications.