IPI: Imputation-Powered Inference

• IPI is a statistical framework that combines imputation, powered by machine learning or probabilistic models, with bias correction to address incomplete or blockwise-missing data.
• The method integrates complete-case analysis with flexible imputation to form debiased estimators, ensuring robust inference even under diverse missingness patterns.
• IPI demonstrates superior performance in simulations and applications such as clinical and survey studies, outperforming standard complete-case and doubly robust approaches.

Imputation-Powered Inference (IPI) refers to a suite of statistical methodologies that leverage imputation—typically powered by machine learning, flexible black-box predictive models, or probabilistic techniques—as a means to boost the efficiency, coverage, and applicability of inferential procedures when confronted with incomplete, partially observed, or blockwise-missing data. The essential architecture combines efficient use of all available (possibly imputed) data with bias correction or uncertainty quantification grounded in the fully observed (complete-case) subset, thereby mitigating bias and offering robust, valid statistical inference even in complex missingness regimes (Zhao et al., 17 Sep 2025).

1. Theoretical Foundations and Model-Lean Debiasing

IPI is constructed as a model-lean alternative to classic approaches such as complete-case analysis, doubly robust estimation, or black-box imputation without bias correction. The central paradigm posits an M-estimation problem for parameters θ defined as solutions to expected convex loss function minimization over the fully observed data: $$\theta^* = \arg\min_{\theta} \mathbb{E}[\ell(X;\theta)]$$ When missingness—especially blockwise or non-monotone missingness—affects the data, IPI relies on an imputation function $f(\cdot)$ to fill in partial observations. The core estimator is debiased by exploiting complete-cases, using the identity: $$L_{\mathrm{IPI}}(\theta; \lambda) = \frac{1}{R} \sum_{r=1}^R \left\{ \lambda_r\, \mathbb{P}_{\tilde{N}_r}[ \ell(f(O);\theta) ] + \mathbb{P}_n [ \ell(X;\theta) - \lambda_r \ell(f(O \circ m_r); \theta) ] \right\}$$ Here, λ = (λ₁, …, λ_R) are pattern- or block-specific tuning parameters, $\mathbb{P}_{\tilde{N}_r}$ is the empirical average over pattern-r partial cases, and $\mathbb{P}_n$ is over n complete cases.

The one-step estimator is then constructed: $$\hat\theta_{\mathrm{IPI}, \lambda} = \hat\theta_n - \hat H_n^{-1} \nabla L_{\mathrm{IPI}}(\hat\theta_n; \lambda)$$ where $\hat\theta_n$ is the initial complete-case estimator and $\hat H_n$ the Hessian. Under blockwise missing completely at random (MCAR), this adjustment recovers unbiasedness. Validators and diagnostics for this methodology center on the first-moment MCAR condition, a relaxation where the expected imputed gradient matches (in pattern-specific averages) the observed gradient from the complete-cases, enabling practical testing for the regime in which the IPI estimator is valid (Zhao et al., 17 Sep 2025).

2. Methodological Innovations and Bias Correction

The distinguishing methodological innovation in IPI is the combination of black-box imputation, using possibly misspecified or machine-learning-based methods, with a bias-correction step that calibrates the imputed loss/statistics using the distribution over the complete-case subpopulation. This is particularly effective in settings where blockwise missingness—common in multi-site, multi-modal, or large-scale clinical datasets—renders classical imputation less reliable or too restrictive.

For each missingness pattern, the IPI bias-correction leverages differential expectations and gradients between imputed and observed loss functions. Power-tuned weights λ are estimated by minimizing the variance of the resulting estimator, further improving efficiency. Extensive diagnostics compare the mean imputed gradient (from the partial data, imputed by f) with the reference from complete-cases, formulating test statistics such as: $$T_{\mathrm{IPI}} = \frac{1}{R} \sum_{r=1}^R \hat\lambda_r \left(\mathbb{P}_{\tilde{N}_r} [\nabla \ell(f(O); \hat\theta_n)] - \mathbb{P}_n [\nabla \ell(f(O \circ m_r); \hat\theta_n)]\right)$$ with the convergence of $\sqrt{n} T_{\mathrm{IPI}}$ to Normality used as a diagnostic for violation of first-moment MCAR.

3. Comparative Advantages and Limitations

Relative to complete-case analysis, which discards all cases with any missingness and thus suffers substantial loss of efficiency and potentially bias in the presence of non-random missingness, IPI harnesses partially observed data to expand effective sample size. Compared to doubly robust (DR) or augmented inverse probability weighted (AIPW) estimators, IPI sidesteps the need for correctly specified propensity models, which can be prohibitive in high-dimensional or block-missing settings due to the exponential growth of missingness patterns and the computational cost of DR estimators.

Simultaneously, IPI is robust to mis-specification of the imputer—as long as the bias correction leverages a sufficient number of complete-cases and the first-moment MCAR holds. However, the method assumes access to a non-trivial set of complete samples per missingness pattern to reliably debias statistical functionals, which may not be met in regimes with extreme or rare patterns. Further, the performance depends on the calibration and diagnostic checking of the first-moment MCAR assumption; if this fails, the correction term may not provide nominal coverage. In rare-pattern settings or extreme high dimensionality, IPI may require pooling or regularization of patterns to stabilize inference (Zhao et al., 17 Sep 2025).

4. Simulation Studies and Applications

Controlled simulation studies using factor-model and multivariate normal data with artificially imposed blockwise missingness demonstrate that IPI recovers the correct coverage and achieves subpopulation efficiency far exceeding complete-case analysis. Naïve single imputation (treating imputed data as observed) and standard doubly robust methods exhibit undercoverage and bias in these settings, particularly as the proportion of missingness and the number of missing patterns increases.

Clinical and survey applications further illustrate the applicability of IPI:

• In the American Community Survey (ACS), IPI provides efficient, valid inference for the effect of schooling on income, improving confidence interval tightness over both complete-case and naive imputations.
• In allergen chip challenge data, IPI (and the cross-fitted variant "CIPI") enables valid inference on regression coefficients for rhinitis and asthma with missing immunoglobulin response data, using diagnosis-based pattern aggregation to ensure the first-moment MCAR diagnostic holds.

In both simulated and real-world settings, practical diagnostics—based on empirical moment comparison and test statistics—identify when the method is reliably applied. If the p-value for the diagnostic test is low, users are alerted to possible bias due to unmet assumptions.

5. Connections to Broader IPI Literature

IPI synthesizes and substantially extends earlier work on imputation-based inference:

• The prediction-powered inference (PPI) connection is explicit; both frameworks employ two-stage estimators—a model prediction or imputation "augmented" by a bias correction computed on gold-standard data (Angelopoulos et al., 2023, Angelopoulos et al., 2023).
• Inverse probability weighting (IPW) extensions of PPI (HT or Hájék corrections) from design-based survey sampling are directly analogous, with IPI accommodating blockwise informatively-missing patterns via the black-box imputer and generalized debiasing (Datta et al., 13 Aug 2025).
• Model-lean approaches contrast with fully parametric/multiple imputation or DR inference. When DR procedures become intractable with increasing patterns or dimensions, IPI remains computationally scalable and empirically validated.

6. Future Research and Diagnostic Advances

Proposed future extensions of IPI include:

• Reducing reliance on auxiliary fully observed features in high-dimensional blockwise missing designs, for example by developing imputer architectures regularized for rare pattern settings.
• Improving the efficiency of standard error estimation, potentially through more efficient bootstrap procedures or analytic formulae tailored to specific loss functions or imputation methods.
• Extending diagnostics for the first-moment MCAR condition—for example, via hierarchical or pooled statistical tests—to increase the practical reach of IPI methods into settings with a high number of missingness patterns but limited complete-case data per pattern.
• Exploring the generalization of IPI principles to non-MCAR, including MNAR regimes, possibly through partial identification or robust bounding methods.

7. Summary Table: Comparison of IPI with Alternatives

Method	Leverages Partial Data	Theoretical Validity	Computational Scalability	Diagnostic Tools
Complete-case	No	Yes (MCAR/MAR)	High	N/A
DR/AIPW	Yes	Yes (if models ok)	Low (many patterns/dims)	Some pattern overlap
Naive Imputation	Yes	No	High	None
IPI	Yes	Yes (MCAR/1st-mom)	High	Empirical moments, MCAR diagnostics

IPI occupies a unique methodological position—leveraging all observed data (via imputation), correcting for estimator bias using the complete-case subpopulation, and enabling diagnostic validation of its moment assumptions. It significantly increases inferential efficiency in block-missing, high-dimensional, and multi-modal empirical contexts, and is robust under broad missingness patterns when the provided diagnostics pass (Zhao et al., 17 Sep 2025).