The Geometry of Ignorance: LLMs Know When to Temper Bayesian Priors

What does a language model predict when it has few clues? The answer lurks in its unembedding geometry: a single direction of the unembedding matrix encodes the unigram distribution of the training corpus, which serves as the Bayesian prior the model falls back on when uncertain. This structure --- which we term the \emph{direction of ignorance} --- appears in all four model families examined (\texttt{Llama}, \texttt{Qwen}, \texttt{Gemma}, and \texttt{Pythia}), ranging from 0.4B to 405B parameters. Projecting the final prediction state onto this direction yields a per-token \emph{prior loading factor} $λ$, which, empirically, declines steadily as the context becomes more informative. Formally, the same projection decomposes the prediction state into two orthogonal vectors that correspond exactly to the two factors of a tempered Bayesian update: a unigram prior raised to the exponent $λ$ and a context-driven likelihood. This geometric-probabilistic interpretation calibrates $λ$, making it meaningfully comparable across model sizes and families, with larger models generally exhibiting lower prior reliance in the high-context limit. Finally, we show that the direction of ignorance is causally active: raising or lowering $λ$ at the final prediction state steers the prediction toward or away from the unigram prior in KL divergence.

Harnessing the Potential of Optimizing Data Mixtures via Bayesian Domain Reweighting

The performance of Large Language Models (LLMs) is fundamentally influenced by the distributional composition of multi-domain pre-training data. While manual heuristics were prevalent in early models, they increasingly fail to capture the intricate synergies between domains as data complexity grows. To overcome the issue, a dominant approach seeks to fit a proxy function mapping between domain weights and their corresponding validation losses, and then find the optimal domain weights to minimize validation losses. These methods rely on strong structural assumptions, such as rank invariance or scaling laws, which are often violated, resulting in non-negligible estimation bias. A promising approach is to directly optimize the weighting scheme from data. However, it suffers from unstable optimization trajectory and prohibitive computational overhead, limiting its potential to search better domain weights configurations. This paper presents a Bayesian domain weighting method to infer the weights from a Dirichlet distribution via introducing Gamma prior information learned from observations. Experimental results demonstrate that proposed method could achieve stable and efficient domain weights learning, and identifies optimal mixtures while consuming substantially less data than search-based function-fitting methods, revitalizing optimization-based domain weighting for large-scale applications.

What does a Bayes-filtered transformer believe? A predictive Monte Carlo approach

A Bayes-filtered transformer (BFT) is a transformer trained on sequences that are generated in two steps: first a latent task is drawn from a prior, then observations are drawn conditional on that task. Trained under autoregressive log loss, the BFT's next-token prediction, in the idealized limit, is the Bayesian posterior predictive distribution (PPD) induced by that prior and that conditional law. In practice the trained BFT is only an approximation of this ideal PPD, raising an interpretive question: what prior and posterior over the latent task has the trained BFT actually internalized? Existing work answers this question by comparing the trained BFT's predictions against the predictions of various "reference" posteriors, each standing in for a different candidate algorithm or computation the BFT might be implementing. This prediction-space comparison is fragile: different posteriors can share the same posterior-mean predictions. We use predictive Monte Carlo (PMC) as a general interpretability tool for any BFT: using only next-token generation, PMC returns an approximation to the implicit prior and posterior over the latent task, answering the interpretive question directly in latent space. We apply PMC to three stylized task families spanning 0-Markov and 1-Markov exchangeability. The phenomena previously reported in these settings remain visible in latent space. Code is available at https://github.com/afiq-aswadi/bft-pmc

BayesPO: Bayesian Prompt Optimization via Parallel-Tempered Gradient-Guided Discrete MCMC

Prompt optimization adapts large language models (LLMs) without updating model parameters, but many automatic prompt optimizers remain heuristic search procedures over candidate instructions. This paper studies prompt optimization as Bayesian posterior sampling over discrete prompt tokens. We define a posterior distribution by combining a task likelihood term, which rewards prompts that explain input-output examples, with a language-model prior, which favors fluent instructions. This converts prompt optimization into an energy-based posterior sampling problem, for which gradients can be used to guide discrete Markov chain Monte Carlo (MCMC) proposals over vocabulary tokens. We refer to our framework as BayesPO, short for Bayesian Prompt Optimization. In this paper, BayesPO is instantiated with Markov chain Monte Carlo: it uses a Metropolis-Hastings corrected Gibbs-with-Langevin (GwL) proposal and integrates parallel tempering for global exploration of rugged LLM-induced energy landscapes. The concrete sampler further adapts the GwL sampler to the practical constraints of non-weight-tied LLM embeddings. Experiments with Qwen2.5 models show that the sampler discovers semantically meaningful prompts on diagnostic tasks, that parallel tempering helps escape a local optimum in a poetry completion task, and that post-optimizing APE prompts on 24 instruction-induction subtasks improves average accuracy from 60.04% to 63.23%. The study also reveals two main limitations: energy minimization may overfit small optimization sets, and the current sampler remains computationally expensive. These findings position Bayesian prompt sampling as a principled post-optimization tool and point to a promising direction for probabilistic prompt optimization.

We Hebben Een Serieus Translatie: Modeling Intercomprehension as Probabilistic Inference

Intercomprehension refers to partial intelligibility of an unfamiliar language (L2) by a speaker of a related language (L1). How is this zero-shot cross-language comprehension possible? In this work, we extend past work on algorithmic models of noisy-channel inference to model intercomprehension in a Bayesian framework. The model uses an LM in L1 only for scoring latent hypotheses about the translations of observed L2 utterances, and a general-purpose noise model to infer a mapping between L2 and L1 words based on either form-based similarity or symbolic rules. We then conduct a human behavioral experiment, eliciting inferences for utterances in Dutch, Italian, and Ukrainian from speakers of English, Spanish, and Russian, respectively. Our full model shows a closer alignment to the distribution of human intercomprehension performance than ablations, and also compares favorably to zero-shot prompting of much larger models. These results provide a cognitively plausible computational model of intercomprehension, and highlight the flexible inferences made by comprehenders under wide uncertainty in real-world cross-language scenarios. We share our code publicly.

Bayesian Sparse Low-Rank Adaptation for Large Language Model Uncertainty Estimation

Large language models (LLMs) exhibit remarkable reasoning capabilities, but their task-specific fine-tuning is notoriously plagued by overconfidence, severely hindering trustworthy deployment. We propose Data-Adaptive Lower-Rank Adaptation (DALorRA), a simple and effective variational Bayesian sparse framework that shifts the paradigm of uncertainty quantification from the dense parameter space to the lightweight rank level of low-rank adaptation (LoRA). With the insight that LoRA essentially aggregates multiple rank-one components that may provide superfluous model capacity, DALorRA imposes stochastic masking on rank dimensions, enabling Bayesian regularization of model capacity during training and ensemble-like calibration during inference. Extensive experiments demonstrate DALorRA's excellent calibration of LLMs without compromising reasoning accuracy.

Beyond Skepticism: Evaluating LLMs Pedagogical Intent Reasoning with the Adaptive Pedagogical Vigilance Framework

The capacity of Large Language Models (LLMs) to reason about pedagogical intent within instructional communication remains underexplored, particularly in educational domains such as translation pedagogy. To address this, we propose the \textbf{Adaptive Pedagogical Vigilance (APV)} framework, a novel computational formalism that reframes communicative vigilance as an adaptive mechanism for optimizing learning through intent inference. APV formalizes the problem via a Bayesian Pedagogical Intent Inference Engine (PIIE), which models how instructors select content to maximize pedagogical utility and how vigilant learners should inversely reason about latent instructional configurations -- encompassing genre, stance, and incentives. We evaluate APV through a three-tier hierarchy: distinguishing instructional genre, reasoning about structured pedagogical setups, and generalizing to authentic educational discourse. Experiments on leading LLMs (e.g., GPT-4o, Claude 3.5) show that APV substantially improves model vigilance. It achieves the strongest discrimination between pedagogical and exposure-based content, correlates highly with human judgments ($r=0.958$), and maintains robust performance on naturalistic data where baseline methods degrade. This work establishes a unified framework for assessing and enhancing LLMs' understanding of pedagogical motives, advancing the development of more reliable AI-assisted learning systems.

BaRA: Bayesian Adaptive Rank Allocation for Parameter-Efficient Fine-Tuning

While Low-rank adaptation (LoRA) enables highly efficient fine-tuning by constraining task-specific updates to fixed low-rank subspaces, this rigid design limits representational flexibility and often results in overconfident predictions and miscalibrated uncertainty, especially in low-data regimes. Recent Bayesian LoRA variants improve uncertainty estimation by modeling posterior distributions over adaptation parameters. However, these approaches typically rely on fixed or heuristically determined ranks, overlooking the inherently context-dependent nature of adaptation capacity. In this paper, we propose BaRA, a Bayesian Adaptive Rank Allocation framework for parameter-efficient fine-tuning. Drawing inspiration from probabilistic topic models, BaRA dynamically allocates adaptation capacity by activating a sparse, context-dependent subset of disentangled latent factors, enabling instance-wise variation in effective rank. This Bayesian formulation provides principled, data-driven capacity control, mitigating over-parameterization while preserving expressiveness. Beyond the modeling contribution, we provide a complexity-theoretic generalization analysis showing that the generalization gap of BaRA depends on the learned joint effective rank $\bar{s}_{Φ,θ}$ induced by the global-local gate, rather than the maximum rank $r$. This result explains why sparse adaptive rank allocation can reduce the effective hypothesis complexity while preserving input-dependent expressiveness. Extensive experiments on diverse natural language benchmarks demonstrate that BaRA consistently improves predictive performance, robustness, and uncertainty calibration compared to standard LoRA and existing Bayesian LoRA variants.

Why It Hurts: Identifying the Drivers of Negative Thoughts in Emotional Support Conversations

Large Language Models (LLMs) are increasingly used for emotional support tasks, such as negative thought reframing. This task relies on modifying cognitive appraisals, the subjective interpretation of events that elicit negative emotions, which is typically conceptualized along multiple discrete dimensions. Current LLM-based frameworks model cognitive appraisal by exhaustively evaluating all possible dimensions, but they fail to account for the varying saliency of these dimensions across different contexts. In this work, we investigate a vital yet overlooked question: "Can LLMs infer the salient appraisal dimensions from emotional support conversations?" To address this question, we introduce the AppraiSal benchmark, containing 996 emotional support conversations with human-annotated mental states, including salient cognitive appraisal dimensions. Furthermore, we propose PRISM, a multi-agent probabilistic framework grounded in Bayesian Inverse Planning, designed to improve LLMs' ability to identify context-specific appraisal dimensions. Experimental results show that PRISM brings improvements to LLMs across various sizes, particularly in identifying the most salient appraisal dimensions.

Distilling Bayesian Belief States into Language Models for Auditable Negotiation

Negotiation agents must infer what their counterpart values, update those beliefs over dialogue turns, and choose actions under uncertainty. End-to-end large language models (LLMs) can imitate negotiation dialogue, but their opponent beliefs are usually implicit and difficult to inspect. We propose BOND (Bayesian Opponent-belief Negotiation Distillation), a framework for auditable negotiation. BOND consists of an LLM-based Bayesian teacher that scores dialogue contexts against the six possible opponent priority orderings, updates a posterior over those orderings, and uses the posterior for menu-based decision making, as well as a smaller 8B student language model that emits both negotiation actions and normalized posterior beliefs as tagged text. In the CaSiNo negotiation dataset, BOND outperforms the state-of-the-art and achieves mean Brier score 0.085 over opponent-priority posteriors. The distilled student preserves much of this belief signal, achieving Brier 0.114, below the uniform six-ordering reference of 5/36, approximately 0.139. Compared with a 70B structured-CoT baseline, the significantly smaller 8B student model yields substantially better elicited posterior calibration. We further showcase auditability through posterior trajectories, belief-versus-policy error decomposition, and posterior-prefix interventions. These diagnostics reveal that distillation preserves a scoreable belief report more strongly than causal belief-conditioned control, making weak belief-action coupling visible, not hidden.