In Bayesian inference the prior model provides a valuable opportunity to incorporate domain expertise into our inferences. Unfortunately this opportunity often becomes a contentious issue in many fields, and this potential value is lost in the debate. In this case study I will discuss the challenges of building prior models that capture meaningful domain expertise and some practical strategies for ameliorating those challenges as much as possible.
Choosing informative priors in Bayesian regression models: a simulation study and tutorial using Stan and R
BackgroundBayesian regression models provide a robust framework for complex data analysis, which is particularly advantageous in scenarios with small sample sizes, common in psychology or medical research. However, specifying appropriate prior distributions that incorporate existing knowledge to regularize model parameters remains a challenge for many researchers. This can lead to unstable or implausible estimates. This study aims to demonstrate the impact of different prior distributions on regression models and to provide a practical guide for choosing and justifying informative priors to produce more stable and credible results.MethodsThe study involved two parts. First, a simulation study was conducted to systematically assess the sensitivity of Bayesian linear regression models to prior specification. We systematically varied the sample size, prior location, and prior scale to observe their impact on posterior estimates for a known true effect size. Second, a case–control study using real-world patient data (N = 526) demonstrated the practical application of choosing informative priors. Bayesian logistic regression models were used to analyze the relationship between severe dementia and fall incidence, comparing results from priors based on existing literature (“believer”), conservative priors (“agnostic”), and priors assuming an opposite effect (“skeptical”).ResultsThe simulation study showed that strongly informative priors had a substantial influence on posterior estimates, particularly for smaller sample sizes. As the sample size increased, the influence of the data increased, and the estimates converged toward the true effect. In the case–control study, a standard frequentist logistic regression produced an odds ratio of 8.87 with a very wide and unstable confidence interval (1.66–165.19), likely due to data sparsity. In contrast, a Bayesian model using a moderately informative “believer” prior derived from existing research yielded a more stable and plausible odds ratio of 4.01 with a substantially narrower credible interval (1.99–8.78).ConclusionCareful and transparent specification of informative priors is a critical tool in Bayesian analysis, especially when data are sparse. By incorporating justified evidence-based assumptions, researchers can regularize models to prevent implausible outcomes and produce more stable, interpretable, and credible results. This approach enhances the robustness of statistical inference in fields where small sample sizes are a frequent challenge.
The Wisdom of Individuals: Exploring People's Knowledge About Everyday Events Using Iterated Learning
Abstract Determining the knowledge that guides human judgments is fundamental to understanding how people reason, make decisions, and form predictions. We use an experimental procedure called ‘‘iterated learning,’’ in which the responses that people give on one trial are used to generate the data they see on the next, to pinpoint the knowledge that informs people's predictions about everyday events (e.g., predicting the total box office gross of a movie from its current take). In particular, we use this method to discriminate between two models of human judgments: a simple Bayesian model ( Griffiths & Tenenbaum, 2006 ) and a recently proposed alternative model that assumes people store only a few instances of each type of event in memory (Min K ; Mozer, Pashler, & Homaei, 2008 ). Although testing these models using standard experimental procedures is difficult due to differences in the number of free parameters and the need to make assumptions about the knowledge of individual learners, we show that the two models make very different predictions about the outcome of iterated learning. The results of an experiment using this methodology provide a rich picture of how much people know about the distributions of everyday quantities, and they are inconsistent with the predictions of the Min K model. The results suggest that accurate predictions about everyday events reflect relatively sophisticated knowledge on the part of individuals.
Bayesian Thinking in Everyday Life
More than 200 years ago, Thomas Bayes came up with a brilliant idea that has helped shape the world today, called Bayes Theorem. This…
A Bayesian Approach To Analysing Training Data Attribution In Deep Learning
Training data attribution (TDA) techniques find influential training data for the model's prediction on the test data of interest. They approximate the impact of down- or up-weighting a particular training sample. While conceptually useful, they are hardly applicable to deep models in practice, particularly because of their sensitivity to different model initialisation. In this paper, we introduce a Bayesian perspective on the TDA task, where the learned model is treated as a Bayesian posterior and the TDA estimates as random variables. From this novel viewpoint, we observe that the influence of an individual training sample is often overshadowed by the noise stemming from model initialisation and SGD batch composition. Based on this observation, we argue that TDA can only be reliably used for explaining deep model predictions that are consistently influenced by certain training data, independent of other noise factors. Our experiments demonstrate the rarity of such noise-independent training-test data pairs but confirm their existence. We recommend that future researchers and practitioners trust TDA estimates only in such cases. Further, we find a disagreement between ground truth and estimated TDA distributions and encourage future work to study this gap. Code is provided at https://github.com/ElisaNguyen/bayesian-tda.
A Bayesian Approach To Analysing Training Data Attribution In Deep Learning
Training data attribution (TDA) techniques find influential training data for the model's prediction on the test data of interest. They approximate the impact of down- or up-weighting a particular training sample. While conceptually useful, they are hardly applicable to deep models in practice, particularly because of their sensitivity to different model initialisation. In this paper, we introduce a Bayesian perspective on the TDA task, where the learned model is treated as a Bayesian posterior and the TDA estimates as random variables. From this novel viewpoint, we observe that the influence of an individual training sample is often overshadowed by the noise stemming from model initialisation and SGD batch composition. Based on this observation, we argue that TDA can only be reliably used for explaining deep model predictions that are consistently influenced by certain training data, independent of other noise factors. Our experiments demonstrate the rarity of such noise-independent training-test data pairs but confirm their existence. We recommend that future researchers and practitioners trust TDA estimates only in such cases. Further, we find a disagreement between ground truth and estimated TDA distributions and encourage future work to study this gap. Code is provided at https://github.com/ElisaNguyen/bayesian-tda.
Reasoning Under Uncertainty: Exploring Probabilistic Reasoning Capabilities of LLMs
Despite widespread success in language understanding and generation, large language models (LLMs) exhibit unclear and often inconsistent behavior when faced with tasks that require probabilistic reasoning. In this work, we present the first comprehensive study of the reasoning capabilities of LLMs over explicit discrete probability distributions. Given observations from a probability distribution, we evaluate models on three carefully designed tasks, mode identification, maximum likelihood estimation, and sample generation, by prompting them to provide responses to queries about either the joint distribution or its conditionals. These tasks thus probe a range of probabilistic skills, including frequency analysis, marginalization, and generative behavior. Through comprehensive empirical evaluations, we demonstrate that there exists a clear performance gap between smaller and larger models, with the latter demonstrating stronger inference and surprising capabilities in sample generation. Furthermore, our investigations reveal notable limitations, including sensitivity to variations in the notation utilized to represent probabilistic outcomes and performance degradation of over 60% as context length increases. Together, our results provide a detailed understanding of the probabilistic reasoning abilities of LLMs and identify key directions for future improvement.
Reasoning Under Uncertainty: Exploring Probabilistic Reasoning Capabilities of LLMs
Despite widespread success in language understanding and generation, large language models (LLMs) exhibit unclear and often inconsistent behavior when faced with tasks that require probabilistic reasoning. In this work, we present the first comprehensive study of the reasoning capabilities of LLMs over explicit discrete probability distributions. Given observations from a probability distribution, we evaluate models on three carefully designed tasks, mode identification, maximum likelihood estimation, and sample generation, by prompting them to provide responses to queries about either the joint distribution or its conditionals. These tasks thus probe a range of probabilistic skills, including frequency analysis, marginalization, and generative behavior. Through comprehensive empirical evaluations, we demonstrate that there exists a clear performance gap between smaller and larger models, with the latter demonstrating stronger inference and surprising capabilities in sample generation. Furthermore, our investigations reveal notable limitations, including sensitivity to variations in the notation utilized to represent probabilistic outcomes and performance degradation of over 60% as context length increases. Together, our results provide a detailed understanding of the probabilistic reasoning abilities of LLMs and identify key directions for future improvement.
Ground the model, or it invents the evidence
Trusted, curated, subject-specific models are needed for academia
Eliciting Beliefs with Random Generation Tasks
Elicitation methods, such as asking people to produce the deciles of a distribution, are standard practices in policy or applied statistics. Similarly, much of cognitive science and psychology focuses on determining people's people's beliefs or latent traits through questionnaires or judgment tasks. However, these approaches often only capture a rough outline of what people know and are usually limited to point estimates of people's beliefs. Here, we present a novel experimental paradigm that allows us to access people's beliefs and how variable these beliefs are. Our task is based on an established random generation paradigm in which participants produce quantities from a particular domain as randomly as possible. We hypothesize that due to the minds' general-purpose mechanisms for probabilistic inferences, these random sequences represent the participants' underlying prior beliefs. We show that our method can infer participants' beliefs for a wide range of numeric quantities at comparable accuracy as an established elicitation method. Moreover, these inferred beliefs are consistent with individual participants' generalization and inference patterns in a subsequent conditional prediction task. We then extend our approach to non-numeric belief elicitation, highlighting how our method can go beyond numeric elicitation and provide insight into complex beliefs that are challenging to assess experimentally. Empirically, our results highlight that people know the rough shapes of environmental distributions, and these beliefs guide inference and generalization. Moreover, using our novel approach, we also show that people know the fine details of environmental distributions. Finally, our experimental results show that random generation paradigms can be a useful tool for cognitive scientists, psychologists, and applied statisticians.
Hierarchical Modeling
Hierarchical modeling is a powerful technique for modeling heterogeneity and, consequently, it is becoming increasingly ubiquitous in contemporary applied statistics. Unfortunately that ubiquitous application has not brought with it an equivalently ubiquitous understanding for how awkward these models can be to fit in practice. In this case study we dive deep into hierarchical models, from their theoretical motivations to their inherent degeneracies and the strategies needed to ensure robust computation. We'll learn not only how to use hierarchical models but also how to use them robustly.
Crafting a good (reasoning) model
A recent talk I gave on model training, reasoning, and the next frontier.
Prompting Science Report 4: Playing Pretend: Expert Personas Don't Improve Factual Accuracy
<span> <p><span>This is the fourth in a series of short reports that help business, education, and policy leaders understand the technical details of working with AI through rigorous testing. Here, we ask whether assigning personas to models improves performance on difficult objective multiple-choice questions. We study both domain-specific expert personas and low-knowledge personas, evaluating six models on GPQA Diamond (Rein et al. 2024) and MMLU-Pro (Wang et al. 2024), graduate-level questions spanning science, engineering, and law. </span></p> <p><span>We tested three approaches:</span></p> <p><span>• In-Domain Experts: Assigning the model an expert persona (“you are a physics expert”) matched to the problem type (physics problems) had no significant impact on performance (with the exception of the Gemini 2.0 Flash model).</span></p> <p><span> • Off-Domain Experts (Domain-Mismatched): Assigning the model an expert persona (“you are a physics expert”) not matched to the problem type (law problems) resulted in marginal differences.</span></p> <p><span> • Low-Knowledge Personas: We assigned the model negative capability personas (layperson, young child, toddler), which were generally harmful to benchmark accuracy. </span></p> <p><span>Across both benchmarks, persona prompts generally did not improve accuracy relative to a no-persona baseline. Expert personas showed no consistent benefit across models, with few exceptions. Domain-mismatched expert personas sometimes degraded performance. Low-knowledge personas often reduced accuracy. These results are about the accuracy of answers only; personas may serve other purposes (such as altering the tone of outputs), beyond improving factual performance.</span></p></span>
Economic futures for LLM inference
Inference providers will look for other lines of business.
Open Models Inference for Coding · Umans AI
Hosted Kimi K3, GLM 5.2, and DeepSeek V4 Flash. Pay per token, on infrastructure we own.
Language and Experience: A Computational Model of Social Learning in Complex Tasks
The ability to combine linguistic guidance from others with direct experience is central to human development, enabling safe and rapid learning in new environments. How do people integrate these two sources of knowledge, and how might AI systems? We present a computational framework that models social learning as joint probabilistic inference over structured, executable world models given sensorimotor and linguistic data. We make this possible by turning a pretrained language model into a probabilistic model of how humans share advice conditioned on their beliefs, allowing our agents both to generate advice for others and to interpret linguistic input as evidence during Bayesian inference. Using behavioral experiments and simulations across 10 video games, we show how linguistic guidance can shape exploration and accelerate learning by reducing risky interactions and speeding up key discoveries in both humans and models. We further explore how knowledge can accumulate across generations through iterated learning experiments and demonstrate successful knowledge transfer between humans and models -- revealing how structured, language-compatible representations might enable human-machine collaborative learning.