MAGIC

Implementation: crl.estimators.magic.MAGICEstimator

Estimand

$V^\pi = \mathbb{E}\left[\sum_{t=0}^{T-1} \gamma^t r_t\right]$.

Assumptions

• Sequential ignorability
• Overlap/positivity
• Markov property

Inputs required

• TrajectoryDataset
• behavior_action_probs for logged actions
• Q-model fit (linear by default in CRL)

Algorithm

MAGIC mixes truncated DR estimators over multiple horizons to reduce variance.

Formula (sketch)

For truncation horizon $m$:

$\hat V_m = \hat V(s_0) + \sum_{t=0}^{m-1} \gamma^t \rho_t\left(r_t + \gamma \hat V(s_{t+1}) - \hat Q(s_t,a_t)\right)$.

MAGIC returns a variance-weighted mixture of $\hat V_m$ across horizons.

Diagnostics

• overlap.support_violations
• ess.ess_ratio
• model.q_model_mse

Uncertainty

• Normal-approximation CI by default.
• Bootstrap CI available via bootstrap=True.

Failure modes

• Sensitive to poor Q-models and heavy-tailed ratios.
• Mixture weights can be unstable in small samples.

Minimal example

from crl.estimators.magic import MAGICEstimator

report = MAGICEstimator(estimand).estimate(dataset)

References

• Thomas & Brunskill (2016)