Abstract
<title>Abstract</title> <p>Interventional SHAP depends on the background covariate distribution and can therefore change when a fixed prediction model is transferred from a source population to a target population. A large source sample does not eliminate the resulting background bias, while direct target-background estimation can be unstable when few target covariates are available. We formulate target-background interventional SHAP as a two-sample functional under covariate shift and propagate uncertainty in the target distribution to a case-verification action. In a Gaussian exponential-tilt model with a fixed linear predictor, we derive an exact posterior distribution for target SHAP, an exact importance-weighted variance involving the overlap factor Γ = Eₛ(w₀²), the corresponding effective-sample-size limit, and an exact source-to-target attribution bias bound. We also characterize when squared SHAP contributions represent prediction-fidelity loss and identify the covariance residual that breaks this link. The actual finite-computation action is selected using finite posterior and background budgets, then evaluated with independent posterior draws and a Dvoretzky–Kiefer–Wolfowitz quantile correction. Across 1,200 simulated datasets, Bayesian exponential tilting attained RMSE 0.1044 and 95% repeated-sampling coverage 0.9539; source-background RMSE rose from 0.1688 to 0.6892 as Γ increased from 1.1 to 5. Selective acceptance reduced mean realized opportunity loss from 0.0350 at full coverage to 0.0081 at 25% coverage. A nonlinear Gaussian-shift experiment supported external applicability, whereas a semi-synthetic diabetes experiment showed that Gaussian misspecification can make direct target estimation preferable. The results establish an auditable special-case theory and delimit the conditions needed for broader posterior and action-risk guarantees.</p>