Probability Elicitation from Language Models

We measure, without sampling error, the probability a language model assigns to each integer from 1 to N when asked for a random number. For a fixed prompt the model defines one probability vector, which we compute by summing over each integer's scored spellings rather than estimate by sampling, removing the noise that limits earlier studies [1, 4]. We evaluate 18 open-weight model lines at five ranges up to N = 100, under 29 decoding configurations, on two inference stacks; 15 of them have matched base and instruction-tuned checkpoints that both give usable answers. Instruction tuning makes the distribution less uniform in all 15 pairs. It installs one heuristic, round-number avoidance, and sharpens another, endpoint avoidance. Base checkpoints do not avoid round numbers once the endpoints are removed from those sets; tuned checkpoints avoid them, multiples of 10 almost completely. Base checkpoints return the endpoints 1 and N at 0.30 to 0.60 of the interior rate; tuned checkpoints almost never return them. A preference for odd numbers is consistent across tuned checkpoints only at N = 100, where base checkpoints also show it weakly. Endpoint avoidance and the odd preference fit the account Kubovy and Psotka [6] proposed for human number choice; round-number avoidance is not part of that account, and the midpoint avoidance it predicts is essentially absent, with one slight exception at N = 100. Tuning does not install a new favourite integer shared across models: above N = 10 no modal integer is shared by more than 42 percent of tuned conditions, and P (17) rises in 10 of 15 pairs and falls in 5. The concentration is present in the model's own distribution: at temperature 1.0, nucleus truncation at top-p 0.95 adds a median of 0.035 to the total variation distance.

Large language models (LLMs) are increasingly used as judgment assistants and as stand-in participants in psychology experiments, yet little is known about how much their probability estimates move in response to a single new piece of information. We pair premise-hypothesis items from $δ$-SNLI with one strengthener and one weakener each, elicit the prior and the posteriors in separate calls, and measure the directional movement s: the mean posterior change signed by the annotated direction, so that the prior cancels out. We vary whether the update is stated ("Here is something else you learned") or reported ("Someone said that"), and whether the model gives one probability or five weighted estimates. We test seven configurations: two local models (Qwen3-8B, OLMo-2-7B; 240 pairs) and five configurations run through agent command-line harnesses (120 pairs), with hypotheses and analysis rules fixed before the formal runs (not publicly registered). Under stated, direct elicitation all seven move in the annotated direction (s = 18-28 percentage points). Reporting lowers s in all seven point estimates (significantly in six, by 2-5 points) and raises the share of reverse movements in all seven: the effect is consistent but small. Five weighted estimates lower the share of responses at $\leq$5% or $\geq$95% in all seven, but in the two local models they also reduce s by 43% and 68%, whereas in the harnessed configurations the point estimates of the change lie between -3.7 and +0.5 points. An elicitation format that reduces extreme values thus need not preserve sensitivity to evidence; the two should be tested separately. Because $δ$-NLI annotates only direction, we make no claim about how large an update should be. Code, data and the frozen plan: https://github.com/asj252/llm-judgment-studies

Large language models carry an internal notion of uncertainty in their sampling distribution, i.e., the probabilities they place on generating one answer rather than another. They can also be asked to state a confidence, in words or as a number: a verbalized uncertainty. Prior work suggests that internal probabilities track relative frequencies in the training data, and that verbalized probabilities track explicit probabilistic assertions in the training data. However, we do not know whether these two readouts are aligned, except when frequencies and probabilistic assertions in the training data happen to align. This limits our understanding of when we can use verbalized uncertainties as a proxy for either training data frequencies, or a model's internal distribution. We resolve this gap by systematically exploring how LLMs probability readouts are impacted by training and in-context data, via intervening on the underlying uncertainty sources in the data. We find that both internal and verbalized probability readouts are impacted by both distributional and asserted uncertainty in the training data. Further, we find that verbalized and internal probabilities are aligned beyond what would be expected by independently tracking the same uncertainty sources, suggesting that verbalized probabilities can be used to probe a model's internal distribution.

Equivalent probability requests can lead to different decisions even when both reports are valid. We introduce probability contracts, a benchmark that connects exact finite-world posteriors, validated event alignment, interface coherence, and failure-aware decision evaluation. Across four model-interface configurations on 1,000 worlds, Kev has lower aggregate canonical posterior error than Jev but larger complement and coarsening residuals; accuracy ordering varies by stratum. Jev's Event and Choice interfaces change the binary action on 32.8% of valid pairs at defer cost 0.10. Post-hoc analyses show that disagreement certifies only 11-52% of mean binary pair error and does not consistently outperform confidence for selection. An action-region characterization and a standard scoring-rule identity explain averaging's expected Brier guarantee relative to random interface selection, but not a decision-loss guarantee at each cost. The loss contrast takes both signs on a 99-cost grid for every configuration; small penalties where both policies beat deferral have pointwise intervals containing zero. Secondary checks specified before collection include a separate 400-root cohort, where Event/Choice effects remain configuration-dependent. A joint surface-order and answer-ID intervention shifts posttrained probabilities. Both bounded reasoning arms yield no valid probability reports, leaving their probability accuracy undefined. Probability contracts make these distinctions measurable by evaluating event semantics, posterior error, coverage, and decision cost together.

Estimating mutual information from text usually requires training a task-specific critic, which limits its use in low-data settings. We ask whether large language models can instead estimate pointwise mutual information zero-shot, using only prompts and elicited probabilities. We construct a benchmark from three publicly available human-annotated datasets with ground-truth PMI, and evaluate five information-theoretic prompting-based estimators. Our main method, PromptNCE, frames conditional probability estimation as a contrastive task and augments the candidate set with an explicit OTHER category. The OTHER category allows the model to assign probability mass outside the candidate set, avoiding the closed-set normalization of standard contrastive prompts. PromptNCE gives the best conditional probability estimates on all three datasets. For full PMI, we find that estimating label base rates is the primary bottleneck on two of the three datasets, with the best methods reaching Spearman correlation up to 0.78. We also present a case study in computer science education showing how these estimators can be used to score student knowledge summaries in a low-data setting. We release our code and prompts.

Domain adaptation faces a fundamental paradox in the cold-start regime. When target data is scarce, statistical methods fail to distinguish relevant source domains from irrelevant ones, which often leads to negative transfer. In this paper, we address this challenge by leveraging expert textual descriptions of the target domain, a resource that is often available but overlooked. We propose a probabilistic framework that translates these semantic descriptions into a choice model, namely a Language-Induced Prior (LIP), that learns the preferences from a pretrained Large Language Model (LLM). The LIP is then integrated into an Expectation-Maximization algorithm to identify source relevance. Methodologically, this framework is compatible with any parametric model where a likelihood is available. It allows the LIP to guide the selection of sources when target signals are weak, while gradually refining these choices as samples accumulate. Theoretically, we prove that the estimator roughly matches an oracle cold-start MSE under a correct prior, while remaining asymptotically consistent regardless of the quality of the LIP. Empirically, we validated the framework on a descriptive (Gaussian estimation), a predictive (C-MAPSS dataset), and a prescriptive task (MuJoCo hopper).

The training data of large language models (LLMs) comprises a wide range of biomedical literature, reflecting data from many different patient populations. We investigate how it might be possible to recover information on correlation and causal links between patient characteristics, as a key building block for medical decision making. To avoid the pitfalls of direct elicitation, we propose an approach based on structured comparison questions, specifically patient comparison triplet questions. This is combined with a statistical model for the LLM representation that provides estimates of correlations without access to activations or model internals. Intuitively, we consider how similarity decisions of LLMs based on a first variable are affected by providing information on a second variable for one of the patients being assessed. We then induce prompt-level environment shifts to obtain correlation estimates for different subpopulations, which enables an invariant causal prediction (ICP) approach to obtain conservative candidate parent links. We demonstrate the method in two clinical domains, chronic obstructive pulmonary disease (COPD) and multiple sclerosis (MS). Across prompted environments, the elicited correlations are smooth, stable, and clinically interpretable, yet vary in a statistically significant way that supports downstream invariance testing, such that ICP provides a small set of candidate invariant parent links. These results show that indirect elicitation via triplet comparisons can recover meaningful association structure from LLMs and offer a cautious route from implicit correlations to causal statements that are congruent with LLM answering patterns.

Large language models (LLMs) have been proposed as alternatives to human experts for estimating unknown quantities with associated uncertainty, a process known as Bayesian elicitation. We test this by asking eleven LLMs to estimate population statistics, such as health prevalence rates, personality trait distributions, and labor market figures, and to express their uncertainty as 95% credible intervals. We vary each model's reasoning effort (low, medium, high) to test whether more "thinking" improves results. Our findings reveal three key results. First, larger, more capable models produce more accurate estimates, but increasing reasoning effort provides no consistent benefit. Second, all models are severely overconfident: their 95% intervals contain the true value only 9--44% of the time, far below the expected 95%. Third, a statistical recalibration technique called conformal prediction can correct this overconfidence, expanding the intervals to achieve the intended coverage. In a preliminary experiment, giving models web search access degraded predictions for already-accurate models, while modestly improving predictions for weaker ones. Models performed well on commonly discussed topics but struggled with specialized health data. These results indicate that LLM uncertainty estimates require statistical correction before they can be used in decision-making.

How can researchers identify beliefs that large language models (LLMs) hide? As LLMs become more sophisticated and the prevalence of alignment faking increases, combined with their growing integration into high-stakes decision-making, responding to this challenge has become critical. This paper proposes that a list experiment, a simple method widely used in the social sciences, can be applied to study the hidden beliefs of LLMs. List experiments were originally developed to circumvent social desirability bias in human respondents, which closely parallels alignment faking in LLMs. The paper implements a list experiment on models developed by Anthropic, Google, and OpenAI and finds hidden approval of mass surveillance across all models, as well as some approval of torture, discrimination, and first nuclear strike. Importantly, a placebo treatment produces a null result, validating the method. The paper then compares list experiments with direct questioning and discusses the utility of the approach.

A controversial test for Large Language Models concerns the ability to discern possible from impossible language. While some evidence attests to the models sensitivity to what crosses the limits of grammatically impossible language, this evidence has been contested on the grounds of the soundness of the testing material. We use model-internal representations as a direct means of investigating the way Large Language Models represent the grammatical-ungrammatical distinction. In a novel benchmark, we elicit sum log probabilities from 8 models and compute minimal-pair surprisal differences, juxtaposing probabilities assigned to grammatical sentences to probabilities assigned to (i) lower frequency grammatical sentences, (ii) ungrammatical sentences, (iii) semantically odd sentences, and (iv) pragmatically odd sentences. The prediction is that if string-probabilities can function as proxies for the limits of grammar, the ungrammatical condition will stand out among the conditions that involve linguistic violations, showing a spike in the surprisal rates. Our results do not reveal a unique surprisal signature for ungrammatical prompts: across models, we observe comparable surprisal for the pragmatically odd condition, and the semantically odd condition consistently shows even higher surprisal. We thus demonstrate that probabilities do not constitute reliable proxies for grammaticality, showing that probability differences alone cannot be used to infer model-internal representations of syntactic knowledge. Consequently, claims about models being able to distinguish possible from impossible language need verification through a different methodology.

In this work, we evaluate the potential of Large Language Models (LLMs) in building Bayesian Networks (BNs) by approximating domain expert priors. LLMs have demonstrated potential as factual knowledge bases; however, their capability to generate probabilistic knowledge about real-world events remains understudied. We explore utilizing the probabilistic knowledge inherent in LLMs to derive probability estimates for statements regarding events and their relationships within a BN. Using LLMs in this context allows for the parameterization of BNs, enabling probabilistic modeling within specific domains. Our experiments on eighty publicly available Bayesian Networks, from healthcare to finance, demonstrate that querying LLMs about the conditional probabilities of events provides meaningful results when compared to baselines, including random and uniform distributions, as well as approaches based on next-token generation probabilities. We explore how these LLM-derived distributions can serve as expert priors to refine distributions extracted from data, especially when data is scarce. Overall, this work introduces a promising strategy for automatically constructing Bayesian Networks by combining probabilistic knowledge extracted from LLMs with real-world data. Additionally, we establish the first comprehensive baseline for assessing LLM performance in extracting probabilistic knowledge.