Measuring and making DNA is central to modern biology and biomedicine.

This study provides a normative theory for how Bayesian causal inference can be implemented in neural circuits. In both cognitive processes such as causal reasoning and perceptual inference such ascue integration, the nervous systems need to choose different models representing the underlying causal structures when making inferences on external stimuli.

Prediction markets are often described as mechanisms that ``aggregate information'' into prices, yet the mapping from dispersed private information to observed market histories is typically noisy, endogenous, and shaped by heterogeneous and strategic participation. This paper formulates prediction markets as Bayesian inverse problems in which the unknown event outcome \(Y\in\{0,1\}\) is inferred from an observed history of market-implied probabilities and traded volumes. We introduce a mechanism-agnostic observation model in log-odds space in which price increments conditional on volume arise from a latent mixture of trader types. The resulting likelihood class encompasses informed and uninformed trading, heavy-tailed microstructure noise, and adversarial or manipulative flow, while requiring only price and volume as observables. Within this framework we define posterior uncertainty quantification for \(Y\), provide identifiability and well-posedness criteria in terms of Kullback--Leibler separation between outcome-conditional increment laws, and derive posterior concentration statements and finite-sample error bounds under general regularity assumptions. We further study stability of posterior odds to perturbations of the observed price--volume path and define realized and expected information gain via the posterior-vs-prior KL divergence and mutual information. The inverse-problem formulation yields explicit diagnostics for regimes in which market histories are informative and stable versus regimes in which inference is ill-posed due to type-composition confounding or outcome--nuisance symmetries. Extensive experiments on synthetic data validate our theoretical predictions regarding posterior concentration rates and identifiability thresholds.

Neural Information Processing Systems http://nips.cc/

Measuring and making DNA is central to modern biology and biomedicine.

Strategic randomization is a key principle in game theory, yet it remains underexplored in large language models (LLMs). Prior work often conflates the cognitive decision to randomize with the mechanical generation of randomness, leading to incomplete evaluations. To address this, we propose a novel zero-sum game inspired by the Tian Ji Horse Race, where the Nash equilibrium corresponds to a maximal entropy strategy. The game's complexity masks this property from untrained humans and underdeveloped LLMs. We evaluate five LLMs across prompt styles -- framed, neutral, and hinted -- using competitive multi-tournament gameplay with system-provided random choices, isolating the decision to randomize. Results show that weaker models remain deterministic regardless of prompts, while stronger models exhibit increased randomization under explicit hints. When facing weaker models, strong LLMs adopt deterministic strategies to exploit biases, but converge toward equilibrium play when facing peers. Through win/loss outcomes and Bayes factor analysis, we demonstrate meaningful variation in LLMs' strategic reasoning capabilities, highlighting opportunities for improvement in abstract reasoning and adaptive learning. We make our implementation publicly available at https://github.com/ocelopus/llm-when-to-throw-coin to ensure full reproducibility.