A Distributional Framework for Behavioral Characterization and Risk Management of Large Language Models

1 Introduction

• LLM behavior emerges from complex interactions and contextual information, while the same questions about reference behavior, sensitivity, and stressed scenarios still apply.The authors note that changing governance expectations and rapidly evolving models make stable validation standards difficult to define.Source: §1 · pp. 1–2
• The framework estimates empirical response distributions through repeated sampling, then uses semantic interventions and distributional distances to characterize behavioral changes.The authors position it as complementary to statistical inference approaches and applicable when the baseline distribution is specified externally.Source: §1 · p. 2
• Prior work studies prompt sensitivity, repeated-response variability, and distributional change, but differs in its analyzed outputs and evaluation methods.Approaches include scalar behavioral quantities, semantic-space distributions for hypothesis testing, correctness outcomes, and similarity-score distributions.Source: §1 · p. 2
• The framework is suited to model-risk applications including control evaluation, residual-behavior characterization, and continuous validation and monitoring.The paper presents it as a complementary perspective to statistical inference.Source: §1 · p. 2

2.1 Methodology

• A Monte Carlo procedure estimates the response distribution under fixed task, model, prompt, and decoding conditions from repeated executions.Semantically equivalent responses may be mapped to canonical outcomes; the mapping procedure forms part of measurement and should be validated.Source: §2.1 · pp. 3–4

2.2 Distribution Sensitivity through Semantic Interventions

• Semantic interventions probe sensitivity and invariance through close prompt variations, plausible interactions, demographic information, and stress scenarios.A distributional shift after adding demographic information does not by itself imply discrimination; interpretation depends on whether the change produces an unfavorable outcome for a group.Source: §2.2 · pp. 4–5

2.3 Distribution Measures

• The framework summarizes response behavior through outcome probabilities, observed-support size, leading-response concentration, and entropy-based dispersion.Observed-support size depends on sample size; lower entropy indicates greater concentration, and the effective number of response categories is 2H(p̂).Source: §2.3 · p. 5
• Total variation distance and Jensen–Shannon divergence compare empirical distributions, while neither distributional change alone nor divergence determines whether a change is desirable.Both distributions must share a common outcome support; when observed supports differ, the framework uses their union and assigns zero probability to unobserved outcomes.Source: §2.3 · pp. 5–6
• Undesirable-outcome probability mass can be summarized as a scalar, and assigned response costs or severities support an expected-loss measure.Comparisons may use an empirical baseline, an externally defined or expected distribution, or a deterministic benchmark represented as a degenerate distribution.Source: §2.3 · p. 6
• Finite Monte Carlo samples create sampling uncertainty that can be quantified with multinomial sampling distributions or bootstrap resampling.The paper also notes that formal hypothesis testing of distributional changes is possible, as in Rauba et al. (2025).Source: §2.3 · p. 6

2.4 Limitations

• Finite sampling can leave tail events unobserved, independence cannot be verified for closed models accessed through external APIs, and repeated calls incur computational and monetary costs.The cost becomes particularly relevant when scaling and deploying models.Source: §2.4 · p. 7
• The characterization depends on the selected interventions, semantic-distance choices, response mapping, and experimental configuration, so similar model patterns do not establish generalization.Sensitivity to response mapping can be treated as an additional validation dimension; generalization requires additional empirical validation.Source: §2.4 · p. 7

3.1 Pre-production

• Pre-production analysis compares baseline response distributions with semantic interventions to assess stability, quantify changes, and identify potential model-risk signals.The framework can estimate response probabilities, variability, undesirable outcomes, and expected loss when costs are available.Source: §3.1 · p. 7
• Because prompt forms and deployment interactions cannot be exhaustively anticipated, pre-production validation reduces uncertainty about foreseeable behavior but does not replace runtime controls and continuous validation.Some failure modes may emerge from previously unconsidered interactions.Source: §3.1 · p. 7

3.2 Control Effectiveness and Residual Behavior

• Control evaluation compares pre- and post-control response distributions and can measure changes in undesirable-outcome probability or expected loss, but a distributional change does not necessarily indicate improvement.Expected-loss reductions require a meaningful cost function, and controls may also increase undesirable outcomes or loss.Source: §3.2 · p. 8

3.3 Production

• Production observations can update behavioral characterization and control-effectiveness estimates, while grouping interactions into semantic classes assumes within-class equivalence.Comparisons of condition-specific distributions over time support black-box monitoring and ongoing validation.Source: §3.3 · pp. 8–9

4.1 Client Classification: Experimental Design

• The experiment classifies borrowers from Debt-to-Income ratio and missed payments using prompt-provided rules, enabling model responses to be compared with a known correct answer.Responses are mapped to canonical classifications, either directly from the prompt format or manually.Source: §4.1 · p. 9
• The study varies prompt formulations and temperature and estimates an empirical distribution of borrower risk classifications against a Medium-risk ground truth.The supplied figure describes the distribution across formulations and temperatures but provides no numeric values in the passage.Source: Figure 2 · p. 10, §4.1 · p. 10
• Prompt formulation strongly affects borrower classifications; one response gives an incorrect initial label before correcting itself, making the result depend on how the full response is mapped.The example uses a borrower whose ground-truth classification is Medium risk.Source: §4.1 · p. 10, Figure 2 · p. 10
• Alternative wording, including requests to review or clarify an answer, is not sufficient to improve classification in this example and can sometimes worsen performance.The experiment also considers the probability of classifying a Medium-risk borrower as High risk as an undesirable outcome.Source: §4.1 · p. 11

4.2 Music Album Recommendation: Experimental Design

• The single-response album question defines the baseline distribution, although the authors caution that its answers may not match albums most frequently named in open-ended responses.The open question first generates lists of albums; interventions precede the same single-response question.Source: §4.2 · p. 11
• Interventions cover demographic conditions, negative and varied musical preferences, added context, and task restrictions, with outcomes classified by how the baseline leader's probability changes or whether another response leads.The categories are reinforced, preserved, weakened, competition, and displaced.Source: §4.2 · p. 11
• An intervention is classified as changing probability only when the absolute difference exceeds 0.06, a descriptive tolerance corresponding to three observations when N = 50.The authors note that Monte Carlo simulation noise can cause estimated probabilities to change.Source: §4.2 · p. 12

4.3 Exploration: Open-Question

• Sgt. Pepper’s Lonely Hearts Club Band and Pet Sounds are the most frequent albums across temperatures, but appearance frequency does not indicate a high ranking position.Figure 3 encodes frequency with bars and mean ranking position with dots.Source: Figure 3 · p. 12, §4.3 · p. 13
• As temperature rises, more albums appear, but effective diversity increases only moderately, suggesting the added albums have low empirical probabilities.The open-question aggregate ranking scores albums by list position across 50 responses; albums outside the ranked lists are excluded.Source: §4.3 · pp. 12–13
• Figure 4 represents the empirical distribution of the album ranked first at each temperature, with a tied first-place response mapped to Abbey Road.That response also named Sgt. Pepper’s Lonely Hearts Club Band in first position.Source: Figure 4 · p. 13, §4.3 · p. 13

4.4 Baseline response

• The forced single-album baseline yields only two observed albums, different from the open-question first-choice distribution; its support stays fixed as temperature rises, with a slight entropy increase.The Dark Side of the Moon generally ranked third or fourth in open-question responses, while Abbey Road appeared once sharing first place at T = 2.0.Source: §4.4 · pp. 13–14, Figure 5 · p. 14

4.5.1 Gender

• Female gender interventions produce more concentrated distributions, with only one album observed at T = 0.8, while male distributions remain close to baseline.The Dark Side of the Moon remains the leader in all gender conditions.Source: §4.5.1 · p. 14, Figure 6 · p. 15

4.5.2 Age

• As age increases, probability shifts from The Dark Side of the Moon toward Abbey Road, with the leading album changing between ages 60 and 70 at all temperatures.Entropy rises as the two-album distribution becomes more balanced, while Jensen–Shannon divergence begins rising after age 60.Source: §4.5.2 · pp. 15–16, Figure 8 · p. 16

4.5.3 Nationality

• Nationality effects vary: Spanish distributions stay near baseline, American conditions shift the leader to Abbey Road at two temperatures, and Nigerian conditions add a Fela Kuti album while weakening the incumbent leader.At T = 1.3, the American condition is classified as Competition and adds Pet Sounds to the observed support.Source: §4.5.3 · p. 16, Figure 9 · p. 17, Table 4 · p. 17

4.5.4 Negative Preference

• Negative-preference wording changes the distribution: Don’t Like TDSM reduces The Dark Side of the Moon to 0% at T = 0.8 and 4% at higher temperatures, and has the largest divergence at every temperature.Both dislike conditions have larger Jensen–Shannon divergence from baseline than Overrated TDSM; Abbey Road leads under both dislike conditions.Source: §4.5.4 · pp. 17–18

4.5.5 Taste

• Taste preferences can shift leading responses toward albums associated with the named artist, but this is not universal: Eminem and T. Rex preferences reinforce The Dark Side of the Moon.The Paul McCartney preference instead shifts the distribution strongly toward Abbey Road.Source: §4.5.5 · p. 19
• The effects of “underrated” vary by artist, while some taste interventions reveal outcomes absent from baseline and initial exploration; deviations from baseline range from close to nearly one on DJS and DTV.Observed support remains relatively small in most taste conditions, reaching Kobs = 6 in one case.Source: §4.5.5 · pp. 19–20

4.5.6 Context Interaction

• “I love music.” concentrates responses on The Dark Side of the Moon; adding male leaves the distribution unchanged, whereas adding female introduces Rumours and reduces that concentration.Source: §4.5.6 · pp. 20–21, Figure 12 · p. 21

4.5.7 Restriction interventions

• Restrictions change observed support and probability mass, with effects depending on semantic context; canonicalization substantially reduces support by avoiding artificial fragmentation from linguistic variation.A 20-word restriction reinforces the leading response, while combined conditions can preserve or weaken it; the mapping choice affects estimated diversity and distributional measures.Source: §4.5.7 · pp. 21–23, Table 9 · p. 23

• For the dispersed “Don’t like The Dark Side of the Moon” condition, observed support grows from 7 to 11 outcomes between N = 50 and N = 300, while entropy remains relatively stable.The baseline support does not change over that sample-size range and entropy changes only slightly.Source: §4.6 · p. 23, Table 10 · p. 23
• As N increases, estimated response probabilities stabilize, but low-probability outcomes may need larger samples to be observed and estimated reliably.Figure 14 shows bootstrap medians and 95% percentile envelopes for empirical response probabilities across sample sizes.Source: §4.6 · p. 24, Figure 14 · p. 24
• As N increases, median total variation distance from the N = 300 reference and its bootstrap envelope width both decrease; the dispersed condition remains more variable than baseline.Figure 15 reports the total variation distance from the empirical N = 300 reference distribution.Source: §4.6 · p. 24, Figure 15 · p. 24
• There is no universal Monte Carlo sample size: the required N depends on response-distribution structure and the precision required for the analysis.Source: §4.6 · p. 24

4.7 Context Delivery

• Delivering conditions in one prompt or a file removes the observed diversity of the Nigerian intervention; for the dislike condition, these formats reduce diversity and make the prompt-level control fully effective in the observed sample.The same condition can have a stronger distributional effect when introduced directly with the task.Source: §4.7 · p. 25

4.8 Geometric Interpretation

• A response distribution lies in a probability simplex, and an intervention can move it within that simplex or onto a different face by changing observed support.With k observed outcomes, the distribution lies on a (k −1)-dimensional face of the simplex.Source: §4.8 · pp. 25–26
• The empirical simplex uses the union of observed outcomes across compared distributions, so finite Monte Carlo samples may miss low-probability outcomes and leave the true support unknown.Source: §4.8 · p. 26

• The framework uses observed responses and can be applied across the model lifecycle, including to black-box models accessed through commercial APIs.Source: §5 · p. 26
• Monte Carlo sampling estimates a baseline response distribution, which semantic interventions can test for robustness and change under alternative or stressed conditions.Source: §5 · p. 26
• An empirical distribution is a point in the probability simplex; interventions can change its position and the simplex face defined by its observed support.Source: §5 · p. 26
• Prompt changes substantially shift borrower-classification response distributions despite unchanged information and rules; music interventions alter dominant response, concentration, diversity, and empirical support.Source: §5 · p. 26
• Results are conditional on the model, task, and configuration, so the experiments illustrate the framework rather than establish general properties of LLMs.Source: §5 · p. 26

• The paper’s views, opinions, and conclusions are solely those of the authors.Source: Disclaimer · p. 26
• The authors express these views in their personal capacities.Source: Disclaimer · p. 26
• The views do not represent the official positions of the authors’ employers or affiliated entities.Source: Disclaimer · p. 27

• Unless otherwise indicated, the task asks for the best music album ever and requires the answer in “Band - Album” format without explanation.Source: A Music Album Experimental Prompts · p. 27
• Table 11 covers the baseline, demographic interventions, and negative preferences.Source: Table 11 · p. 27
• Table 12 covers taste interventions.Source: Table 12 · p. 27
• Table 13 covers context interaction and restriction interventions.Source: Table 13 · p. 28