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

Jul 16, 2026

Value-of-information (VOI) analysis is usually performed under a single probability measure, but real evidence often identifies only a set of measures. Under such imprecise information, VOI needs new formulations. This work defines a rule-specific VOI by fixing a decision rule for acting under imprecision and quantifies the information’s worth to that decision maker. It also derives a fixed-measure envelope evaluating the classical VOI over all admissible precise measures and studies endpoints for perfect, partial, and sample information.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages39

Reading Time1 h 38 min

Introduction

Value-of-information under imprecision and credal sets

Rule-specific VOI versus fixed-measure VOI envelopes

Consequences for perfect, partial, and sample information

arXiv :2607 .06570v1 [ stat .ML] 26 Jun 2026 Value of Information under Imprecise Probabilities: Decision Rule-Specific Values and Fixed-Measure Envelopes on a Credal Set Rowan Iskandara,b a Medtronic Trading Sàrl, Tolochenaz, VD, Switzerland b Center for Evidence Synthesis in Health, Brown University School of Public Health, Providence, RI, USA Abstract 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 rulespecific VOI that fixes a decision rule for acting under imprecision (such as Γ-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 Γ-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 Email address: [rowan.iskandar@gmail.com](rowan.iskandar@gmail.com) (Rowan Iskandar) admissible measure from conclusions that depend on one unidentified choice of measure. Keywords: Imprecise probability, Credal sets, Value of information, Lower previsions, Γ-maximin, Probability bounds analysis, Decision making under uncertainty 1. Introduction A decision-maker often must decide before the evidence is complete. Such a decision-making situation, often formulated as a decision model, is fraught with uncertainty. The uncertainty may induce loss to the decision-maker due to choosing a suboptimal decision. Value-of-information (VOI) analysis then asks how much the decision could improve if some or all of that uncertainty were resolved before the choice is made [1, 2, 3] . This question has a clear answer when a single probability measure is accepted as the basis for characterizing uncertainty. However, there are situations when the evidence does not single out one such measure. For example, the results of two evidence syntheses may both be defensible yet disagree. Committing to one probability measure then makes the VOI analysis results look more precise than the evidence warrants. One way to relax this restriction is to consider every plausible measure in the analysis, instead of choosing one. A credal set P is a set of probability measures, each compatible with the evidence [4, 5] . Such sets naturally arise from interval or bound constraints, partially specified moments, weakly identified dependence, robust classes of priors, or a finite set of acceptable model structures [6, 7, 8] . This has a direct consequence for VOI. A single measure plays two roles: it ranks actions by their expected net benefit, and it determines the distribution over which the VOI is computed. In contrast, a credal set determines neither a single ranking of the actions nora single VOI value. We must then distinguish how much the information is worth for the decision once a rule for acting under several measures is fixed, from

FAQs

Why does VOI require different formulations under imprecise probabilities?

When evidence does not pin down a single probability measure and instead allows a set of measures (a credal set), the classical VOI computed under one measure becomes overly specific. The decision maker must account for both action ranking and the VOI computation across admissible measures.

What is rule-specific value of information (e.g., Γ-maximin) used for?

Rule-specific VOI fixes a decision rule for acting under imprecision and measures the gain from observing information when the decision can adapt to that observation. It answers what the information is worth to someone who follows that specific rule.

How do fixed-measure VOI envelopes relate to rule-specific VOI endpoints?

The fixed-measure envelope gives the lowest and highest classical VOI values across measures compatible with the evidence. The rule-specific VOI is not guaranteed to match an envelope endpoint and can even lie outside the whole envelope, so reporting both supports robustness-aware interpretation.

Jul 16, 2026

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