arXiv:2607.06570stat.MLcs.IT2026-07

在概率不确定时,区分决策规则与信息价值的计算方式。

Value of Information under Imprecise Probabilities: Decision-Rule-Specific Values and Fixed-Measure Envelopes on a Credal Set

  • 针对不同决策规则定义信息价值,避免统一假设单一概率。
  • 完美信息价值在可信集上呈凹性,端点可由生成测度精确求得。
  • 规则特异性价值可能超出经典包络范围,适合需稳健决策的场景。

信息价值(VOI)分析通常基于单一概率测度,但实践中证据往往仅将测度限定于一个集合。因此,在概率测度集合下,VOI需不同形式。首先,我们阐明一种规则特定的VOI:固定一个处理不确定性(如伽马-最大最小)的决策规则,衡量该规则使用者的信息价值。其次,我们推导出一个固定测度包络,评估经典VOI泛函在所有允许精确测度上的表现。我们形式化了这一区别,并揭示其对期望完美、部分及样本信息价值的影响。期望完美信息价值在可信集上为凹函数;当集合由有限个测度生成时,其下包络端点可由生成测度精确获得,而上包络端点可能位于内部,可通过有限线性规划求解。相比之下,伽马-最大最小值可能超过整个包络,因此规则特定价值无法从包络端点恢复。连续性界限制了测度变化时VOI的最大波动,我们还确定了部分与样本信息端点仍可由生成测度获得的情形。由于单测度VOI本身需估计,本文方法结合标准估计器与可信集搜索。通过一个具体决策问题,展示了两种量如何分离适用于所有允许测度的结论与依赖于未定测度选择的结论。

原文摘要 · Abstract (English)

Value-of-information (VOI) analysis is usually conducted under a single probability measure. However, in practice, the available evidence often pins the measure down only to a set. Consequently, under a set of probability measures, VOI requires different formulations. First, we explicate a rule-specific VOI that fixes a decision rule for acting under imprecision (such as Gamma-maximin) and measures what the information is worth to a decision maker who uses that rule. Second, we derive a fixed-measure envelope that evaluates the classical VOI functional over all admissible precise measures. We formalize this distinction and explicate its consequences for the expected perfect, partial, and sample information. The expected value of perfect information is concave over the credal set. Hence, when the set is generated by finitely many measures, its lower envelope endpoint is obtained exactly from the generators, while its upper endpoint may be interior and is computed by a finite linear program. The Gamma-maximin value, in contrast, can exceed the entire envelope, so a rule-specific value is not recovered from the envelope's endpoints. A continuity bound limits how much the VOI can change as the measure varies, and we identify when the partial- and sample-information endpoints can still be obtained from the generators. Because the single-measure VOI must itself be estimated, the procedure we give combines standard estimators for it with a search over the credal set. By using a worked decision problem, we show how the two quantities separate conclusions that hold across every admissible measure from conclusions that depend on one unidentified choice of measure.

信息价值不确定性决策理论

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