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A Hierarchical Imprecise Probability Approach to Reliability Assessment of Large Language Models

This paper introduces HIP-LLM, a hierarchical imprecise probability framework that enhances the reliability assessment of Large Language Models by modeling failure-free operation probabilities across operational profiles and subdomains while explicitly quantifying epistemic uncertainty through posterior reliability envelopes.

Original authors: Robab Aghazadeh-Chakherlou, Qing Guo, Siddartha Khastgir, Peter Popov, Xiaoge Zhang, Xingyu Zhao

Published 2026-01-30

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CC BY 4.0

Original authors: Robab Aghazadeh-Chakherlou, Qing Guo, Siddartha Khastgir, Peter Popov, Xiaoge Zhang, Xingyu Zhao

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have hired a very talented, but somewhat unpredictable, new assistant (a Large Language Model, or LLM) to help you with your work. You want to know: "How likely is it that this assistant will make a mistake on the next 10 tasks I give them?"

Most current ways of testing these assistants are like giving them a single, static pop quiz. You grade the quiz, get a score (like "85%"), and that's it. The problem is, a pop quiz doesn't tell you how they will perform in the real world, where tasks vary, and where you might need them to do 10 things in a row without failing.

This paper introduces a new tool called HIP-LLM. Think of it as a "Reliability Weather Forecast" for AI assistants. Instead of giving you a single number, it gives you a range of possibilities that accounts for what you know, what you don't know, and how you actually use the tool.

Here is how it works, broken down into simple concepts:

1. The "Operational Profile" (The Menu of Tasks)

Imagine you run a restaurant. If you only test your chef on making pizza, you don't know if they can make sushi.

The Problem: Current tests often just look at a fixed list of questions (like a fixed menu).
The HIP-LLM Solution: It asks, "What does the chef actually cook most often?" If 80% of your orders are for pizza and 20% are for sushi, the reliability score should reflect that mix. HIP-LLM uses an Operational Profile (OP) to weigh the tasks based on how you actually use the AI. If you mostly use it for coding, the score focuses on coding reliability, not its ability to write poetry.

2. The "Family Tree" of Skills (Hierarchical Structure)

Imagine the assistant has different "families" of skills.

The Family Tree: "Coding" is a big family. Inside that family, you have "Python" and "C++." These two are cousins; if the assistant is good at Python, it's likely (but not guaranteed) to be decent at C++ because they share similar logic. However, "Coding" and "Law" are like distant relatives; being good at one doesn't tell you much about the other.
The HIP-LLM Solution: It builds a hierarchical map. It understands that skills within a category (like Python and C++) are connected, but skills across categories (like Coding and Law) are independent. This allows it to learn from one area to help estimate the reliability of another, making the prediction smarter.

3. The "Fuzzy Glasses" (Imprecise Probability)

Imagine you are trying to guess the weight of a watermelon.

The Old Way: You might say, "I guess it's exactly 10 lbs." (This is a single, precise guess).
The HIP-LLM Way: You realize you don't have a scale, so you say, "I'm pretty sure it's between 8 and 12 lbs, but I'm not 100% sure."
The Concept: This is called Imprecise Probability. Instead of forcing a single "best guess" about how good the AI is (which can be misleading), HIP-LLM admits uncertainty. It produces a range (or envelope) of possible reliability scores. This range gets tighter as you get more data, but it always acknowledges that your initial guesses (priors) might be slightly off.

4. Predicting the Future (Not Just the Past)

Most tests tell you, "The assistant got 8 out of 10 right today."

The HIP-LLM Solution: It answers, "What is the probability the assistant will get the next 50 tasks right in a row?"
The Analogy: It's the difference between saying, "I drove safely yesterday," and "I have a 95% chance of driving safely for the next 100 miles." This is crucial for high-stakes situations where a single failure matters.

5. Why This Matters (The "Gap" Fillers)

The authors argue that current methods have five big holes:

Static vs. Dynamic: Tests are static quizzes; real life is a dynamic flow of tasks. HIP-LLM handles the flow.
Isolated vs. Connected: Tests treat every task as unrelated. HIP-LLM knows that skills are related (like the family tree).
Description vs. Prediction: Tests describe the past; HIP-LLM predicts the future.
Failure vs. Success: Tests often just count failures. HIP-LLM calculates the probability of a long streak of success.
Ignorance vs. Knowledge: Tests often ignore what experts already know. HIP-LLM lets experts say, "I think it's good at math," and folds that into the math.

The Bottom Line

HIP-LLM is a sophisticated calculator that takes test results, mixes them with how you actually use the AI, and accounts for what you already know (and what you don't). Instead of giving you a single, potentially misleading grade, it gives you a confidence envelope: "Based on what we know, there is a 90% chance this AI will succeed on your next 20 tasks, provided you use it mostly for [Your Specific Task]."

It turns a simple "score" into a risk-aware reliability report, helping you decide if an AI is safe enough to use for your specific needs.

Technical Summary: A Hierarchical Imprecise Probability Approach to Reliability Assessment of Large Language Models

1. Problem Statement

The paper addresses the lack of rigorous, standardized reliability assessment methods for Large Language Models (LLMs). While LLMs are increasingly deployed in safety-critical and high-stakes domains, current evaluation practices rely heavily on benchmark-based descriptive statistics (e.g., point-estimate accuracy scores). The authors identify five critical research gaps in existing literature:

Static vs. Dynamic Use: Benchmarks use fixed datasets that fail to capture the dynamic nature of real-world user behaviors and Operational Profiles (OPs).
Independence vs. Hierarchy: Existing benchmarks often treat tasks in isolation or average them, ignoring the hierarchical dependencies between subdomains (e.g., Python vs. C++ coding) and the independence of high-level domains (e.g., coding vs. law).
Description vs. Inference: Current methods report descriptive statistics rather than performing statistical inference to quantify uncertainty about future performance over a population of inputs.
Failure Probability vs. Future Reliability: Prior works focus on the probability of failure for a single task, neglecting the broader, standardized definition of reliability: the probability of failure-free operation over a specified number of future tasks.
Non-informative vs. Informative Priors: Existing Bayesian approaches often use non-informative priors, failing to incorporate valuable (albeit potentially vague) prior knowledge from domain experts.

The core problem is to develop a framework that models LLM reliability as a probabilistic quantity dependent on an Operational Profile, accounts for hierarchical task dependencies, and quantifies uncertainty arising from both data and prior knowledge.

2. Methodology: HIP-LLM

The authors propose HIP-LLM (Hierarchical Imprecise Probability for Large Language Models), a framework grounded in software reliability engineering and robust Bayesian analysis.

2.1. Formal Definition of Reliability

Adhering to the ANSI definition of software reliability, HIP-LLM defines LLM reliability ( $R$ ) as the probability of failure-free operation over a specified number of future tasks ( $n$ ) under a given Operational Profile ( $\pi$ ).

Operational Profile (OP): A probability distribution over the input space (task types) reflecting real-world usage frequencies.
i.i.d. Assumption: The framework assumes tasks are independent and identically distributed (i.i.d.) Bernoulli trials. This aligns with "reset" scenarios (e.g., new chat sessions without context memory) common in offline benchmarking, distinguishing it from agentic workflows with long-term memory.

2.2. Hierarchical Structure

The model structures the LLM's capability space into a hierarchy:

Independent Domains: High-level domains (e.g., Coding, Reasoning) are treated as statistically independent.
Dependent Subdomains: Within a domain, subdomains (e.g., Python coding, Data Science coding) are modeled as dependent, sharing latent competencies.
Partial Pooling: Information flows from subdomains to domains and up to the system level. Subdomains with sparse data borrow strength from related subdomains within the same domain.

2.3. Imprecise Probability for Epistemic Uncertainty

To address the difficulty of selecting a single precise prior (which encodes epistemic uncertainty), HIP-LLM employs Imprecise Probability.

Instead of fixing hyperparameters (e.g., $\alpha, \beta$ for Beta priors) to point values, the framework specifies them as intervals (credal sets).
This results in posterior envelopes (lower and upper bounds) for reliability metrics rather than single posterior distributions.
This approach captures uncertainty about the prior knowledge itself, providing a range of plausible reliability estimates consistent with the data and the assessor's vague beliefs.

2.4. Mathematical Formulation

The inference proceeds hierarchically:

Subdomain Level: Observed successes ( $C_{ij}$ ) and trials ( $N_{ij}$ ) update subdomain reliabilities ( $\theta_{ij}$ ) using a Binomial likelihood and a Beta prior parameterized by domain-level hyperparameters ( $\mu_i, \nu_i$ ).
Domain Level: Subdomain reliabilities are aggregated using OP weights ( $\Omega_{ij}$ ) to form domain reliability ( $p_i$ ).
System Level: Domain reliabilities are aggregated using domain-level OP weights ( $W_i$ ) to form overall LLM reliability ( $p_L$ ).
Future Reliability: The framework derives the probability of $n_F$ consecutive failure-free runs ( $R(n_F)$ ) by raising the non-failure probability to the power of $n_F$ .

Theorems 1–6 provide the mathematical derivation for the posterior distributions (or CDF envelopes) at each level, handling the intractability of weighted sums of dependent Beta variables via numerical integration and Monte Carlo sampling.

3. Key Contributions

Formal Definition: The paper formally defines LLM reliability in accordance with established software reliability standards, distinguishing it from safety, security, and accuracy.
HIP-LLM Framework: It introduces a hierarchical imprecise probability model that simultaneously addresses:
- Explicit modeling of Operational Profiles (OPs).
- Hierarchical dependencies between subdomains and independence between domains.
- Principled uncertainty quantification (both aleatory and epistemic) via posterior envelopes.
- Incorporation of informative (imprecise) prior knowledge.
Public Repository: The authors release a public repository containing experimental data, source code, and models.

4. Experimental Results

The authors evaluated HIP-LLM on four benchmark datasets (MBPP, DS-1000, BoolQ, RACE-H) across four LLMs (GPT-4o, GPT-4o-mini, Claude Sonnet 4.5, Claude Haiku 3.5).

Effectiveness (RQ1): HIP-LLM successfully generated posterior CDF envelopes at subdomain, domain, and system levels. It revealed nuanced differences between models that point-estimate benchmarks obscured (e.g., distinguishing between models with similar mean accuracies but different uncertainty profiles).
Sensitivity to Hyperparameters (RQ2): The analysis showed that hyperparameters controlling prior expectations ( $a, b$ ) shift the location of the reliability envelope, while those controlling confidence/pooling strength ( $c, d$ ) primarily affect the width (uncertainty) of the envelope.
Sensitivity to OPs (RQ3): The framework demonstrated that reliability estimates are sensitive to OP variations, particularly for models with uneven performance across subdomains (e.g., GPT-4o's reasoning performance varied significantly between BoolQ and RACE-H, leading to high sensitivity to task mix).
Predictability (RQ4): The model successfully predicted the decay of reliability as the number of required future failure-free tasks ( $n_F$ ) increased, providing a tool for risk-aware decision-making based on operational horizons.
Comparison to Baselines (RQ5): In synthetic experiments with known ground truth, HIP-LLM produced robust bounds. In low-data regimes, it provided wider, more conservative intervals reflecting prior uncertainty, whereas data-dominated regimes collapsed these intervals to match frequentist estimates.
Robustness (RQ6, RQ7): The framework showed moderate sensitivity to alternative success definitions (Pass@1 vs. Pass@3) and remained stable under simulated memory-induced dependencies, with dependence primarily causing controlled shifts in the envelopes rather than qualitative changes.
Scalability (RQ8): Computational cost was found to be dominated by subdomain-level inference, scaling linearly with the number of domains and hyperparameter configurations.

5. Significance and Claims

The paper claims that HIP-LLM offers a more nuanced and standardized reliability characterization than existing benchmark-based or state-of-the-art approaches. Its significance lies in:

Shifting the Paradigm: Moving from descriptive benchmark scores to statistical inference about future operational reliability.
User-Centricity: Explicitly incorporating the Operational Profile, acknowledging that reliability is context-dependent.
Handling Uncertainty: Providing a mechanism to embed and propagate epistemic uncertainty (vague prior knowledge) through the assessment, resulting in "envelopes" rather than overconfident point estimates.
Decision Support: Enabling stakeholders to make informed decisions (e.g., model selection, risk assessment) based on the probability of failure-free runs over specific future operational horizons.

The authors maintain a modest stance, noting that the framework assumes "reset" (i.i.d.) scenarios and that accurately defining "failure" and estimating real-world OPs remain open challenges requiring future work. They position HIP-LLM as a foundational step toward principled reliability assessment for generative AI.