Decision Theory under Partial Identification

• The topic defines decision theory under ambiguity where parameters are set-identified rather than point-identified.
• It integrates classical minimax and Bayesian principles with robust optimization and information design to address worst-case risk scenarios.
• These methods apply to real-world problems like causal inference, policy learning, and treatment assignment using randomization and threshold rules.

Statistical decision theory under partial identification is concerned with the design and analysis of optimal decision rules when the parameter governing the data-generating process is only set-identified. This area synthesizes classical minimax and Bayesian principles with robust optimization, information structures, and modern methods for learning under ambiguity. Partial identification arises when the observable data and maintained assumptions do not point-identify the underlying structural parameters, but rather confine them to a set consistent with the observed empirical or experimental evidence. The associated decision-theoretic problems span individual treatment assignment, policy learning, causal inference, and information design.

1. Foundations: Decision Theory and Partial Identification

The foundational setting comprises a parameter (or state) space $\Theta$, an action set $A$, and a loss (or negative welfare) function $L(a,\theta)$ for $a \in A$ and $\theta \in \Theta$. Partial identification occurs when the data and maintained assumptions induce a set-valued mapping $\mathcal{I}(P)$ from the observed (possibly infinite-sample) data law $P$ to an identified set $\Theta_I(P) \subseteq \Theta$. The practitioner is thus confronted with ambiguity: given $P$, multiple $\theta \in \Theta_I(P)$ are observationally indistinguishable.

In the classic Wald framework, decision rules are functions (possibly randomized) mapping observed data to actions. When $\Theta_I(P)$ is a singleton, the standard risk analysis applies. In contrast, partial identification necessitates robust criteria, typically minimax or minimax-regret, whereby performance is evaluated under the worst-case scenario over the identified set $\Theta_I(P)$ (Manski, 2022, Olea et al., 2023, Yata, 2021).

A concrete structure is provided by models with finite (or compact) action space and partial identification via observable moments, conditional linear programs, or other constraints flowing from incomplete identification (Ben-Michael, 13 Jun 2025, Kitagawa et al., 2023).

2. Robust and Minimax Formulations

The central robust criteria in partial identification are:

• Minimax risk:

$$\min_{\delta} \sup_{\theta \in \Theta_I(P)} r(\delta, \theta)$$

where $\delta$ is a decision rule, and $r(\delta, \theta)$ the risk.

• Minimax regret:

$$\min_{\delta} \sup_{\theta \in \Theta_I(P)} \mathrm{Regret}(\delta, \theta)$$

with regret defined as the risk gap to the oracle (perfect-information) action at $\theta$ (Cui, 2021, Yata, 2021, Olea et al., 2023).

Randomization (in the form of mixed or stochastic decision rules) emerges as an unavoidable phenomenon. Under partial identification, randomized rules can uniquely minimize worst-case risk or regret, and strictly outperform all deterministic rules for certain loss functions and model classes (Manski, 2022, Kitagawa et al., 2023). The two-action lemma characterizes the optimal mixing probabilities in discrete settings.

In general, optimal decision rules under minimax / minimax-regret criteria depend on the structure of the identified set. For convex, centrosymmetric sets with linear or piecewise-linear loss, solutions often reduce to threshold rules, but substantial ambiguity (e.g., large identified sets relative to sampling error) can force mixed or piecewise-linear rules with explicit randomization over nontrivial regions of the sample space (Qiu et al., 25 Jan 2026, Olea et al., 2023, Yata, 2021).

3. Characterization and Implementation of Optimal Rules

The explicit characterization of minimax-optimal rules relies on representing set identification via moment inequalities—or in econometric contexts, as conditional linear programs. For scalar or low-dimensional problems, the optimal rule is often a threshold or a piecewise-linear function of a sufficient statistic (e.g., the plug-in estimator or its functional), possibly with randomization in certain ambiguity regimes.

For instance, in the case of binary treatment choice with parameter $\theta$ set-identified in $[\mu - k, \mu + k]$ and $Y \sim N(\mu, \sigma^2)$, the minimax-regret rule is (Qiu et al., 25 Jan 2026):

• If $k \leq \sigma \sqrt{\pi/2}$, use the nonrandomized threshold $d^*(y) = 1\{y > 0\}$.
• For $k > \sigma \sqrt{\pi/2}$, a linear rule on $y \in [-k^*, k^*]$ is optimal:

$$d_{lin}^*(y) = \begin{cases} 0 & \text{if } y < -k^* \ \frac{y + k^*}{2k^*} & \text{if } -k^* \leq y \leq k^* \ 1 & \text{if } y > k^* \end{cases}$$

where $k^*$ satisfies an equilibrium equation. This structure generalizes to higher-dimensional and more general loss settings under convexity and symmetry assumptions (Olea et al., 2023, Yata, 2021).

In complex models or for policy learning with covariate-dependent bounds, minimax or maximin value functions can be computed via empirical risk minimization using lower bounds constructed from CLPs, possibly regularized for computational tractability. The resulting policy aligns with the worst-case-bounds principle, maximizing lower-bound expected value over all feasible policies (Ben-Michael, 13 Jun 2025, D'Adamo, 2021).

4. Information Structures, Robust Implementation, and Support Priors

Information design under partial identification posits that the ambiguity set itself can be shaped by strategic withholding or disclosure of information. The “prior-free” framework analyzes robust implementability of actions in games where the receiver observes only signals, and the sender controls information release (Rosenthal, 23 Nov 2025):

• The revealed information induces an identified set of compatible state distributions.
• The set of actions that can be robustly implemented (i.e., are max-min optimal for some information structure) is characterized by the existence of a “supporting prior”: a distribution over states that (i) makes the action optimal, and (ii) is supported by the information structure in a certain geometric sense.
• All robustly implementable actions can be induced by withholding at most one-dimensional information—the information design counterpart to “almost fully informative” experiments sufficing for robust policy.

This framework generalizes classical minimax-type uncertainty (fixed ambiguity sets) by allowing the ambiguity set to be endogenously generated through the experiment or information structure itself (Rosenthal, 23 Nov 2025).

5. Randomization, Multiplicity, and Asymptotics

When ambiguity is severe, minimax-regret problems admit infinitely many optimal rules, typically randomizing on sets of positive measure in the data or signal space (Olea et al., 2023, Fernández et al., 2024). Among these, rules minimizing the region or frequency of randomization are of particular interest (“least-randomizing” rules). This multiplicity reflects the absence of differentiating power in the data over portions of the identified set.

In local asymptotic regimes with partial identification, the optimal rule generally differs from the plug-in solution. The asymptotically optimal rule (locally asymptotically minimax, LAM) includes an order $1/\sqrt{n}$ adjustment to the plug-in, determined by a saddle-point argument over local alternatives and the directional derivative of the identification bounds (Kido, 2023). This shows that classical delta-method or MSE-optimal rules are suboptimal in non-regular, partially identified settings and that only directional differentiability (not full differentiability) is required for minimax risk expansion.

Hybrid Bayes–minimax and robust Bayes (multiple-prior) analyses introduce a further layer of randomization and can reveal divergence between ex-ante and ex-post optimal rules. In particular, there is no general guarantee of agreement between ex-ante and ex-post robust Bayes rules under partial identification; both may require randomization in equilibria (Fernández et al., 2024, Christensen et al., 2022).

6. Computational Methods and Applications

The computation of minimax or minimax-regret rules under partial identification may be reduced to convex or linear programming when the feasible action set and identified set are finite or can be discretized (Manski, 2022, Ben-Michael, 13 Jun 2025). In practice, it is common to:

• Use plug-in or entropic-regularized CLP estimators for bounds and policy values (Ben-Michael, 13 Jun 2025).
• Employ finite-sample or asymptotic bootstrap to approximate minimax risk in hybrid Bayes–minimax schemes (Christensen et al., 2022).
• Approximate saddle-point equilibria by fictitious play, adversarial learning, or gradient-based methods, especially for large-scale or complex parameter spaces (Loh, 2024).

Applications include individualized treatment choice under instrumental variable or selection bounds (Cui, 2021), regression discontinuity and eligibility cutoff policies (Yata, 2021, Qiu et al., 25 Jan 2026), causal inference under unmeasured confounding (D'Adamo, 2021), pricing under unobserved heterogeneity (Christensen et al., 2022), and information design for robust causal recommendations (Rosenthal, 23 Nov 2025).

Empirical relevance is highlighted in studies where the minimax/regret-optimal policy, learned in a partially identified setting, differs substantially from the plug-in or naive rule. For example, under strong partial identification, “no-action” or randomized treatment rules may be minimax-optimal even when the data suggest otherwise (Yata, 2021, Olea et al., 2023).

7. Open Problems and Research Directions

Despite tractable solutions in specific canonical models, several directions remain open:

• Extension of closed-form minimax rule characterizations to high-dimensional, nonconvex, or complex identified sets, including multiple moment inequality models (Qiu et al., 25 Jan 2026).
• Scalable algorithms for large-scale minimax statistical games, particularly where the identified set is described via high-dimensional conditional linear programs or non-nested sets.
• Development of coherent asymptotic theory for partial identification, unifying local asymptotic minimax results, admissibility, and efficiency notions (Kido, 2023).
• Investigation of Bayes–minimax, quantile-regret, or mean-squared regret rules under set identification and their decision-theoretic properties.
• Adaptive or sequential decision-making under ambiguity, with dynamic updating of the identified set as new information arrives.

The intersection of decision theory and partial identification thus continues to motivate developments at the methodological, computational, and application interfaces—providing concrete tools for robust policy when full identification is impossible.