Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differentiable manifolds. Using our results, we provide a disturbingly simple example in which the restricted density and the disintegration density drastically disagree. Motivated by applications in approximate Bayesian inference and Bayesian inverse problems, we further study the modes of disintegrations. We show that the recently introduced notion of a "conditional mode" does not coincide in general with the modes of the conditional measure obtained through disintegration, but rather the modes of the restricted measure. We also discuss the implications of the discrepancy between the two measures in practice, advocating for the utility of both approaches depending on the modelling context.

We analyze the statistical problem of recovering an atomic signal, modeled as a discrete uniform distribution $μ$, from a binned Poisson convolution model. This question is motivated, among others, by super-resolution laser microscopy applications, where precise estimation of $μ$ provides insights into spatial formations of cellular protein assemblies. Our main results quantify the local minimax risk of estimating $μ$ for a broad class of smooth convolution kernels. This local perspective enables us to sharply quantify optimal estimation rates as a function of the clustering structure of the underlying signal. Moreover, our results are expressed under a multiscale loss function, which reveals that different parts of the underlying signal can be recovered at different rates depending on their local geometry. Overall, these results paint an optimistic perspective on the Poisson deconvolution problem, showing that accurate recovery is achievable under a much broader class of signals than suggested by existing global minimax analyses. Beyond Poisson deconvolution, our results also allow us to establish the local minimax rate of parameter estimation in Gaussian mixture models with uniform weights. We apply our methods to experimental super-resolution microscopy data to identify the location and configuration of individual DNA origamis. In addition, we complement our findings with numerical experiments on runtime and statistical recovery that showcase the practical performance of our estimators and their trade-offs.

Bayesian neural networks and deep ensemble methods have been proposed for uncertainty quantification; however, they are computationally intensive and require large storage. By utilizing a single deterministic model, we can solve the above issue. We propose an effective method based on feature space density to quantify uncertainty for distributional shifts and out-of-distribution (OOD) detection. Specifically, we leverage the information potential field derived from kernel density estimation to approximate the feature space density of the training set. By comparing this density with the feature space representation of test samples, we can effectively determine whether a distributional shift has occurred. Experiments were conducted on a 2D synthetic dataset (Two Moons and Three Spirals) as well as an OOD detection task (CIFAR-10 vs. SVHN). The results demonstrate that our method outperforms baseline models.

--Modern cyber attacks unfold through multiple stages, requiring defenders to dynamically prioritize mitigations under uncertainty. While game-theoretic models capture attacker-defender interactions, existing approaches often rely on static assumptions and lack integration with real-time threat intelligence, limiting their adaptability. This paper presents Cy-GATE, a game-theoretic framework modeling attacker-defender interactions, using large language models (LLMs) with retrieval-augmented generation (RAG) to enhance tactic selection and patch prioritization. Applied to a two-agent scenario, CyGATE frames cyber conflicts as a partially observable stochastic game (POSG) across Cyber Kill Chain stages. Both agents use belief states to navigate uncertainty, with the attacker adapting tactics and the defender re-prioritizing patches based on evolving risks and observed adversary behavior . The framework's flexible architecture enables extension to multi-agent scenarios involving coordinated attackers, collaborative defenders, or complex enterprise environments with multiple stakeholders. The evolving cybersecurity landscape presents increasingly sophisticated threats that necessitate adaptive, proactive defense strategies. Patch management, a cornerstone of cyber defense, requires intelligent prioritization of vulnerabilities under resource constraints such as maintenance windows and operational cost [1] [2] . However, traditional scoring systems like common vulnerability scoring system (CVSS) [3] fail to capture the evolving nature of cyber threats, where attackers adapt their strategies based on defender actions. Game theory provides a structured framework for modeling attacker-defender interactions [4], with chained or multistage games particularly suited to representing complex attack progressions along the Cyber Kill Chain (CKC) [5][6][7]. These models allow defenders to reason about long-term risks and preempt cascading compromises. Despite these advancements, existing models remain constrained by fixed strategies, static payoff structures, and minimal integration of threat intelligence, failing to dynamically prioritize vulnerabilities based on evolving exploitation trends [8]. Traditional game-theoretical approaches typically use predefined rules to analyze strategies, hence are limited in dynamic cyber environments where adversaries continuously adapt, operate under uncertainty, and employ unpredictable tactics [9].

In analyses with severe data-limitations, augmenting the target dataset with information from ancillary datasets in the application domain, called source datasets, can lead to significantly improved statistical procedures. However, existing methods for this transfer learning struggle to deal with situations where the source datasets are also limited and not guaranteed to be well-aligned with the target dataset. A typical strategy is to use the empirical loss minimizer on the source data as a prior mean for the target parameters, which places the estimation of source parameters outside of the Bayesian formalism. Our key conceptual contribution is to use a risk minimizer conditional on source parameters instead. This allows us to construct a single joint prior distribution for all parameters from the source datasets as well as the target dataset. As a consequence, we benefit from full Bayesian uncertainty quantification and can perform model averaging via Gibbs sampling over indicator variables governing the inclusion of each source dataset. We show how a particular instantiation of our prior leads to a Bayesian Lasso in a transformed coordinate system and discuss computational techniques to scale our approach to moderately sized datasets. We also demonstrate that recently proposed minimax-frequentist transfer learning techniques may be viewed as an approximate Maximum a Posteriori approach to our model. Finally, we demonstrate superior predictive performance relative to the frequentist baseline on a genetics application, especially when the source data are limited.

Despite the recent proliferation of machine learning methods like SINDy that promise automatic discovery of governing equations from time-series data, there remain significant challenges to discovering models from noisy datasets. One reason is that the linear regression underlying these methods assumes that all noise resides in the training target (the regressand), which is the time derivative, whereas the measurement noise is in the states (the regressors). Recent methods like modified-SINDy and DySMHO address this error-in-variable problem by leveraging information from the model's temporal evolution, but they are also imposing the equation as a hard constraint, which effectively assumes no error in the regressand. Without relaxation, this hard constraint prevents assimilation of data longer than Lyapunov time. Instead, the fulfilment of the model equation should be treated as a soft constraint to account for the small yet critical error introduced by numerical truncation. The uncertainties in both the regressor and the regressand invite the use of orthogonal distance regression (ODR). By incorporating ODR with the Bayesian framework for model selection, we introduce a novel method for model discovery, termed ODR-BINDy, and assess its performance against current SINDy variants using the Lorenz63, Rossler, and Van Der Pol systems as case studies. Our findings indicate that ODR-BINDy consistently outperforms all existing methods in recovering the correct model from sparse and noisy datasets. For instance, our ODR-BINDy method reliably recovers the Lorenz63 equation from data with noise contamination levels of up to 30%.

Biological systems commonly exhibit complex spatiotemporal patterns whose underlying generative mechanisms pose a significant analytical challenge. Traditional approaches to spatiodynamic inference rely on dimensionality reduction through summary statistics, which sacrifice complexity and interdependent structure intrinsic to these data in favor of parameter identifiability. This imposes a fundamental constraint on reliably extracting mechanistic insights from spatiotemporal data, highlighting the need for analytical frameworks that preserve the full richness of these dynamical systems. To address this, we developed a simulation-based inference framework that employs vision transformer-driven variational encoding to generate compact representations of the data, exploiting the inherent contextual dependencies. These representations are subsequently integrated into a likelihood-free Bayesian approach for parameter inference. The central idea is to construct a fine-grained, structured mesh of latent representations from simulated dynamics through systematic exploration of the parameter space. This encoded mesh of latent embeddings then serves as a reference map for retrieving parameter values that correspond to observed data. By integrating generative modeling with Bayesian principles, our approach provides a unified inference framework to identify both spatial and temporal patterns that manifest in multivariate dynamical systems.

Practitioners making decisions based on causal effects typically ignore structural uncertainty. We analyze when this uncertainty is consequential enough to warrant methodological solutions (Bayesian model averaging over competing causal structures). Focusing on bivariate relationships ($X \rightarrow Y$ vs. $X \leftarrow Y$), we establish that model averaging is beneficial when: (1) structural uncertainty is moderate to high, (2) causal effects differ substantially between structures, and (3) loss functions are sufficiently sensitive to the size of the causal effect. We prove optimality results of our suggested methodological solution under regularity conditions and demonstrate through simulations that modern causal discovery methods can provide, within limits, the necessary quantification. Our framework complements existing robust causal inference approaches by addressing a distinct source of uncertainty typically overlooked in practice.

Large language models (LLMs) have shown strong performance across natural language reasoning tasks, yet their reasoning processes remain brittle and difficult to interpret. Prompting techniques like Chain-of-Thought (CoT) enhance reliability by eliciting intermediate reasoning steps or aggregating multiple outputs. However, they lack mechanisms for enforcing logical structure and assessing internal coherence. We introduce Theorem-of-Thought (ToTh), a novel framework that models reasoning as collaboration among three parallel agents, each simulating a distinct mode of inference: abductive, deductive, and inductive. Each agent produces a reasoning trace, which is structured into a formal reasoning graph. To evaluate consistency, we apply Bayesian belief propagation guided by natural language inference (NLI), assigning confidence scores to each step. The most coherent graph is selected to derive the final answer. Experiments on symbolic (WebOfLies) and numerical (MultiArith) reasoning benchmarks show that ToTh consistently outperforms CoT, Self-Consistency, and CoT-Decoding across multiple LLMs, while producing interpretable and logically grounded reasoning chains. Our findings suggest a promising direction for building more robust and cognitively inspired LLM reasoning. The implementation is available at https://github.com/KurbanIntelligenceLab/theorem-of-thought.

The widespread availability of off-the-shelf machine learning models poses a challenge: which model, of the many available candidates, should be chosen for a given data analysis task? This question of model selection is traditionally answered by collecting and annotating a validation dataset -- a costly and time-intensive process. We propose a method for active model selection, using predictions from candidate models to prioritize the labeling of test data points that efficiently differentiate the best candidate. Our method, CODA, performs consensus-driven active model selection by modeling relationships between classifiers, categories, and data points within a probabilistic framework. The framework uses the consensus and disagreement between models in the candidate pool to guide the label acquisition process, and Bayesian inference to update beliefs about which model is best as more information is collected. We validate our approach by curating a collection of 26 benchmark tasks capturing a range of model selection scenarios. CODA outperforms existing methods for active model selection significantly, reducing the annotation effort required to discover the best model by upwards of 70% compared to the previous state-of-the-art. Code and data are available at https://github.com/justinkay/coda.