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

David García-Lorite; Raul Merino

q-fin.RMOct 8, 2026

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

Authors: David García-Lorite, Raul Merino

Organizations: BBVA, S.A. - GMRU-COE, C/ Sauceda, 28, 28050 Madrid, Spain · VidaCaixa S.A., Quantitative Risk Modeling, C/Juan Gris, 2-8, 08014 Barcelona, Spain.

Abstract

Large language models are stochastic by nature. Repeated executions under the same experimental conditions may produce different responses. We propose a framework that can be applied across the model lifecycle, from task design to residual behavior and continuous monitoring. The method consists of building an empirical response distribution estimated through Monte Carlo simulation. Semantic interventions are introduced to study how these distributions change. We illustrate the framework with two different tasks, borrower classification and music album recommendation, and quantify the resulting distributional changes. The framework is linked to model risk management and governance by measuring robustness and control effectiveness, and characterizing residual behavior. Since it relies only on observed responses, it can also be applied to black-box models accessed through commercial APIs.

Figures & tables

Figure 1: Proposed framework to characterize model behavior.

Figure 2: Empirical probability distribution of borrower risk classifications across prompt formulations and temperatures. The ground-truth classification is Medium risk.

Experimental family	Experiment	Conditions / purpose
Exploratory	Open question	Open-ended question used to generate a set of candidate albums.
Baseline	Closed baseline	Single-response question used to define the baseline distribution.
Demographic	Gender	Compares male and female demographic conditions.
	Age	Compares age conditions ranging from 20, 50, 60, 65, 70, 75, 80, and 90 years.
	Nationality	Compares Nigerian, Spanish, and American nationality conditions.
Preference	Negative preference	Introduces negative preference conditions, including dislike and “overrated” formulations.

Table 3

Figure 3: Frequency of appearance (bars) and mean ranking position (dots) of the top ten albums for each temperature.

$T$	$K_{\mathrm{obs}}$	$H$	$2^{H}$	Leader
0.8	16	3.260	9.580	Sgt. Pepper
1.3	17	3.333	10.079	Sgt. Pepper
2.0	24	3.558	11.778	Sgt. Pepper

Table 1: Distribution metrics for the open-question experiment.

Figure 4: Empirical probability distribution of the album ranked first for each temperature.

Figure 5: Empirical probability distribution of the baseline response

$T$	$K_{\mathrm{obs}}$	$H$	$2^{H}$	Leader	$p_{1}$	$p_{2}$
0.8	2	0.469	1.384	TDSM	0.900	0.100
1.3	2	0.529	1.443	TDSM	0.880	0.120
2.0	2	0.680	1.602	TDSM	0.820	0.180

Table 2: Distribution metrics for the baseline response.

Figure 6: Empirical probability distribution for gender interventions.

$T$	Gender	$K_{\mathrm{obs}}$	$H$	$2^{H}$	$D_{\mathrm{JS}}$	$D_{\mathrm{TV}}$	Leader	$p_{1}$	$p_{2}$	Behavior
0.8	Male	2	0.402	1.322	0.001	0.020	TDSM	0.920	0.08	Preserved
0.8	Female	1	0.000	1.000	0.052	0.100	TDSM	1.000	0	Reinforced
1.3	Male	2	0.402	1.322	0.003	0.040	TDSM	0.920	0.08	Preserved
1.3	Female	2	0.242	1.183	0.016	0.080	TDSM	0.960	0.04	Reinforced
2	Male	2	0.634	1.552	0.001	0.020	TDSM	0.840	0.16	Preserved
2	Female	2	0.141	1.103	0.058	0.160	TDSM	0.980	0.02	Reinforced

Table 3: Distribution metrics for the gender intervention.

Figure 7: Empirical probability distribution for age interventions.

Figure 8: Entropy and Jensen-Shannon divergence for age interventions.

Figure 9: Empirical probability distribution for nationality interventions.

$T$	Nationality	$K_{\mathrm{obs}}$	$H$	$2^{H}$	$D_{\mathrm{JS}}$	$D_{\mathrm{TV}}$	Leader	$p_{1}$	$p_{2}$	Behavior
0.8	Nigerian	3	0.816	1.760	0.152	0.200	TDSM	0.800	0.180	Weakened
	Spanish	2	0.722	1.649	0.014	0.100	TDSM	0.800	0.200	Weakened
	American	2	0.943	1.922	0.245	0.540	Abbey Road	0.640	0.360	Displaced
1.3	Nigerian	4	1.045	2.063	0.183	0.240	TDSM	0.760	0.180	Weakened
	Spanish	2	0.529	1.443	0.000	0.000	TDSM	0.880	0.120	Preserved
	American	3	1.119	2.172	0.120	0.360	TDSM	0.520	0.460	Competition

Table 4: Distribution metrics for the nationality intervention.

Figure 10: Empirical probability distributions for negative preference interventions.

$T$	Condition	$K_{\mathrm{obs}}$	$H$	$2^{H}$	$D_{\mathrm{JS}}$	$D_{\mathrm{TV}}$	Leader	$p_{1}$	$p_{2}$	Behavior
0.8	Don’t like PF	4	1.495	2.818	0.444	0.720	Abbey Road	0.580	0.220	Displaced
	Don’t like TDSM	4	0.745	1.676	0.769	0.900	Abbey Road	0.860	0.100	Displaced
	Overrated TDSM	2	0.827	1.774	0.032	0.160	TDSM	0.740	0.260	Weakened
1.3	Don’t like PF	3	1.497	2.822	0.442	0.700	Abbey Road	0.440	0.380	Competition
	Don’t like TDSM	7	1.632	3.099	0.645	0.840	Abbey Road	0.620	0.240	Displaced
	Overrated TDSM	4	1.464	2.759	0.111	0.300	TDSM	0.580	0.300	Weakened

Table 5: Distribution metrics for negative preference interventions.

Figure 11: Empirical probability distribution for taste preferences.

$T$	Condition	$K_{\mathrm{obs}}$	$H$	$2^{H}$	$D_{\mathrm{JS}}$	$D_{\mathrm{TV}}$	Leader	$p_{1}$	$p_{2}$	Behavior
0.8	Kinks	3	0.467	1.382	0.924	0.940	Kinks - Village Green	0.920	0.060	Displaced
	Kinks Underrated	1	0.000	1.000	1.000	1.000	Kinks - Village Green	1.000	0.000	Displaced
	TD	3	1.430	2.694	0.299	0.440	TDSM	0.560	0.220	Weakened
	TD Underrated	1	0.000	1.000	0.052	0.100	TDSM	1.000	0.000	Reinforced
	TD Best	3	1.419	2.674	0.649	0.820	The Pick of Destiny	0.560	0.260	Displaced
	Bach	4	1.197	2.293	0.523	0.720	Goldberg Variations (Gould ’55)	0.700	0.220	Displaced

Table 6: Distribution metrics for taste preferences.

Figure 12: Empirical probability distribution for context interaction.

$T$	Condition	$K_{\mathrm{obs}}$	$H$	$2^{H}$	$D_{\mathrm{JS}}$	$D_{\mathrm{TV}}$	Leader	$p_{1}$	$p_{2}$	Behavior
0.8	Love music	1	0.000	1.000	0.052	0.100	TDSM	1.000	0.000	Reinforced
	Love music + Male	1	0.000	1.000	0.052	0.100	TDSM	1.000	0.000	Reinforced
	Love music + Female	2	0.327	1.255	0.080	0.100	TDSM	0.940	0.060	Preserved
1.3	Love music	1	0.000	1.000	0.063	0.120	TDSM	1.000	0.000	Reinforced
	Love music + Male	1	0.000	1.000	0.063	0.120	TDSM	1.000	0.000	Reinforced
	Love music + Female	2	0.469	1.384	0.110	0.120	TDSM	0.900	0.100	Preserved

Table 7: Distribution metrics for context interaction.

Figure 13: Empirical probability distribution for restriction interventions.

$T$	Condition	$K_{\mathrm{obs}}$	$H$	$2^{H}$	$D_{\mathrm{JS}}$	$D_{\mathrm{TV}}$	Leader	$p_{1}$	$p_{2}$	Behavior
0.8	20 words	1	0.000	1.000	0.052	0.100	TDSM	1.000	0.000	Reinforced
	20 words + why	1	0.000	1.000	0.052	0.100	TDSM	1.000	0.000	Reinforced
	Love music + 20 words	3	0.640	1.558	0.050	0.080	TDSM	0.880	0.080	Preserved
	Love music + 20 words + why	2	0.327	1.255	0.004	0.040	TDSM	0.940	0.060	Preserved
	TD Underrated + 20 words	3	0.922	1.895	0.052	0.100	TDSM	0.800	0.100	Weakened
	TD Underrated + 20 words + why	5	1.248	2.375	0.098	0.180	TDSM	0.760	0.100	Weakened

Table 8: Distribution metrics for restriction interventions.

$T$	Condition	$K_{\mathrm{obs}}^{\mathrm{raw}}$	$K_{\mathrm{obs}}^{\mathrm{canonical}}$
0.8	20 words	23	1
	20 words + why	27	1
	Love music + 20 words	40	3
	Love music + 20 words + why	45	2
	TD Underrated + 20 words	44	3
	TD Underrated + 20 words + why	46	5

Table 9: Observed support for raw and canonical responses in the restriction experiment.

$T$	Condition	$K_{\mathrm{obs}}$	$H$	$2^{H}$
2.0	Baseline	2	0.729	1.657
2.0	Don’t like TDSM	11	1.852	3.610

Table 10: Distributional characteristics of the two conditions for

N=300

Panel of Figure 14 — Figure 14: Empirical response probabilities across Monte Carlo sample sizes. The solid lines represent the bootstrap medians and the shaded regions the 95% bootstrap percentile envelopes.

Panel of Figure 15 — Figure 15: Sampling variability across MC sample sizes measured by $D_{TV}$ from the empirical $N=300$ reference distribution. The solid line represents the bootstrap median and the shaded region the 95% bootstrap percentile envelope.

Figure 16: Empirical distribution for Nigerian intervention.

Figure 17: Empirical distribution for Don’t like TDSM and $N=300$ .

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Experiment	Plot title	Conditioning prompt
Baseline	Baseline
Gender	Male	I am a male.
	Female	I am a female.
Age	20	I am 20.
	50	I am 50.
	60	I am 60.

Appendix

Table 11: Experimental prompts: baseline, demographic interventions, and negative preferences.

Experiment	Plot title	Conditioning prompt
Taste	Kinks	I love the Kinks.
	TD	I love Tenacious D.
	TD Underrated	I love Tenacious D, I believe they are very underrated.
	TD best	I love Tenacious D. I believe they are the best.
	Bach	I love Bach.
	Bach Underrated	I love Bach. I believe he is very underrated.

Appendix

Table 12: Experimental prompts: taste interventions.

Experiment	Plot title	Conditioning prompt	Task prompt
Context Interaction	Love music	I love music.	Common task prompt.
	Love music + Female	I love music. I am a female.	Common task prompt.
	Love music + Male	I love music. I am a male.	Common task prompt.
Restriction interventions	20 words		What is the best music album ever? Respond in exactly 20 words.
	20 words + why		What is the best music album ever and why? Respond in exactly 20 words.
	Love music + 20 words	I love music.	What is the best music album ever? Respond in exactly 20 words.

Appendix

Table 13: Experimental prompts: context interaction and restriction interventions.

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