minimum-risk bayesian inference

In Bayesian sequential trials, error rate evaluation relies on computationally expensive Monte Carlo simulations, hindering efficient optimization of sample size and decision thresholds. Method: This paper establishes, for the first time, analytical functional relationships between posterior and posterior predictive probabilities and sample size. Leveraging Bayesian decision theory and asymptotic analysis—combined with numerical fitting and error-rate inversion—the method enables precise error-rate assessment for any sample size using only two simulations, and rapidly identifies optimal design parameters. Contribution/Results: The approach drastically reduces computational cost while achieving error-rate control accuracy comparable to conventional simulation-based methods. In two real-world case studies, it attains exact error-rate calibration and accelerates design optimization by several orders of magnitude. This provides a scalable, verifiable, and highly efficient design paradigm for Bayesian adaptive trials.

This paper addresses selective classification in high-stakes settings, aiming to minimize the rejection (indecision) rate under a user-specified misclassification constraint—potentially stricter than the Bayes optimal error. We propose a threshold-adaptive framework grounded in statistical learning theory and risk-controlling optimization. By constructing tight confidence sets and dynamically adjusting decision boundaries, we establish, for the first time, theoretical guarantees on achieving the optimal rejection rate under stringent misclassification constraints. Our work challenges the conventional belief that the Bayes error is an insurmountable lower bound, and instead derives the fundamental trade-off between misclassification rate and rejection cost. Experiments demonstrate substantial reductions in misclassification—approaching zero—on hard classification tasks, while incurring only negligible rejection rates; the gain in misclassification reduction far outweighs the cost introduced by rejection.

This paper addresses binary treatment choice under partial identification, focusing on optimal policy decisions when external validity is questionable or experimental data for the target population are unavailable. We develop a multi-prior robust Bayesian framework to derive both ex ante and ex post robust decision rules, explicitly distinguishing between randomizable and non-randomizable decision-makers. We establish, for the first time, that these two rules generally differ; moreover, we rigorously prove that randomized interventions can be optimal in both settings—challenging the conventional “no randomization” assumption. Integrating partial identification theory, robust decision analysis, and methods for aggregating evidence across multiple trials, we propose a causally grounded inference principle that balances theoretical rigor with practical policy implementation. Our approach substantially enhances decision robustness and reliability in settings involving external generalization and synthesis of multi-center randomized controlled trials.

This paper investigates statistical decision-making for treatment choice under partial identification, focusing on theoretical properties and practical challenges of welfare maximization and regret minimization within the Gaussian likelihood framework. We propose the “profile regret” criterion to systematically distinguish decision rules and uniquely characterize the minimax-regret-optimal nonrandomized decision rule—its first such characterization. Key theoretical findings include: (i) discarding certain data may improve minimax welfare performance; and (ii) under strong partial identification, infinitely many randomized rules achieve minimax regret optimality. The method is validated across canonical settings—including experimental estimate aggregation, LATE extrapolation, and omitted-variable bias correction—demonstrating robustness and practical applicability. Our work establishes a novel theoretical foundation and provides implementable tools for robust policy decision-making under partial identification.

This paper investigates asymptotically optimal discrete decision-making under partial identification: when the payoff function depends on an incompletely identified parameter θ, while only an auxiliary parameter P—subject to known constraints linking it to θ—is observable, how can we construct a decision rule conditional on P that minimizes maximum risk or regret? Methodologically, it establishes, for the first time, a systematic statistical theory of optimal decision-making under partial identification, grounded in the limit experiment framework and integrating bootstrap and Bayesian inference techniques, applicable to both parametric and semiparametric models. Theoretically, the proposed rule is proven to strictly dominate conventional plug-in or worst-case substitution rules in asymptotic efficiency. Empirically, it substantially reduces worst-case regret and demonstrates both robustness and practical feasibility in applications such as treatment choice and optimal pricing.

This study addresses the challenge that posterior approximations often fail to preserve downstream decision-making performance by proposing a decision sufficiency framework grounded in Kullback-Leibler divergence contraction. Methodologically, local decision directions are ranked via a generalized eigenvalue problem, thereby decoupling decision coverage from information efficiency. Furthermore, the geometric relationship between the objective loss and baseline approximations is analytically characterized through quadratic local limits on Bayesian action fibers, leveraging Hessian and Fisher information metrics. Ultimately, this work establishes decision-relative criteria for evaluating and designing families of posterior approximations, enabling precise quantification of the decision sufficiency distance.

This work addresses a fundamental limitation in conventional Bayesian experimental design, which relies on prior-to-posterior uncertainty reduction and yields an intractable objective that is doubly hard to evaluate and poorly aligned with downstream tasks. By reframing the problem through decision theory, the authors formulate it as optimizing the expected future loss (EFL) of downstream actions, thereby reducing the objective to a singly intractable form that obviates explicit posterior or marginal likelihood computation. They introduce a stochastic gradient method that jointly optimizes both the experimental design and the action policy, requiring only samples from the joint parameter–data model and evaluations of the loss function. This approach naturally accommodates implicit modeling and task-specific customization, demonstrating marked improvements over existing methods in both optimization efficiency and task adaptability.

This work addresses the stochastic shortest path (SSP) problem—a class of undiscounted, infinite-horizon Markov decision processes with absorbing states—by introducing the first Bayesian framework that directly models the posterior distribution over the optimal action-value function $Q^*$. The approach constructs an exact posterior grounded in the Bellman optimality equation, incorporates a manifold density to capture the intrinsic structure of $Q^*$, and analyzes its identifiability under a Gaussian-relaxed likelihood. In the tabular setting, the posterior probability of optimal actions admits an analytical solution. Empirical evaluations demonstrate that the proposed framework accurately quantifies uncertainty and achieves substantially higher data efficiency than existing Bayesian methods based on temporal-difference learning, particularly on variants of the Deep Sea benchmark.

本文探讨了贝叶斯信号处理的复杂性问题，通过构建计算框架分析最优决策的难解性，并讨论了近似概念下的可处理条件。

This work investigates the minimal input information required to achieve Bayes-optimal prediction in fixed supervised learning tasks. To this end, it introduces the notion of a *Bayes-sufficient representation*—a representation that admits a predictor attaining Bayes-optimal performance—and formalizes the *Bayes-minimal representation* via the Bayes quotient, which precisely characterizes the information that must be preserved. Theoretically, the study establishes a systematic connection among representation learning, loss functions, Bayes-optimal actions, and attribute elicitation theory, revealing fundamental differences in the information requirements across distinct losses. Empirically, through neural network bottleneck analysis, controlled synthetic data, and real-world experiments on iNaturalist, the work validates the sufficiency and minimality of such representations, demonstrates the ability to distinguish redundant from essential information, and confirms that the minimal information needed for optimal prediction is jointly determined by the data distribution and the choice of loss function.