Modern deep neural networks are powerful predictive tools yet often lack valid inference for causal parameters, such as treatment effects or entire survival curves. While frameworks like Double Machine Learning (DML) and Targeted Maximum Likelihood Estimation (TMLE) can debias machine-learning fits, existing neural implementations either rely on "targeted losses" that do not guarantee solving the efficient influence function equation or computationally expensive post-hoc "fluctuations" for multi-parameter settings. We propose Targeted Deep Architectures (TDA), a new framework that embeds TMLE directly into the network's parameter space with no restrictions on the backbone architecture. Specifically, TDA partitions model parameters - freezing all but a small "targeting" subset - and iteratively updates them along a targeting gradient, derived from projecting the influence functions onto the span of the gradients of the loss with respect to weights. This procedure yields plug-in estimates that remove first-order bias and produce asymptotically valid confidence intervals. Crucially, TDA easily extends to multi-dimensional causal estimands (e.g., entire survival curves) by merging separate targeting gradients into a single universal targeting update. Theoretically, TDA inherits classical TMLE properties, including double robustness and semiparametric efficiency. Empirically, on the benchmark IHDP dataset (average treatment effects) and simulated survival data with informative censoring, TDA reduces bias and improves coverage relative to both standard neural-network estimators and prior post-hoc approaches. In doing so, TDA establishes a direct, scalable pathway toward rigorous causal inference within modern deep architectures for complex multi-parameter targets.

Trajectory planning involving multi-agent interactions has been a long-standing challenge in the field of robotics, primarily burdened by the inherent yet intricate interactions among agents. While game-theoretic methods are widely acknowledged for their effectiveness in managing multi-agent interactions, significant impediments persist when it comes to accommodating the intentional uncertainties of agents. In the context of intentional uncertainties, the heavy computational burdens associated with existing game-theoretic methods are induced, leading to inefficiencies and poor scalability. In this paper, we propose a novel game-theoretic interactive trajectory planning method to effectively address the intentional uncertainties of agents, and it demonstrates both high efficiency and enhanced scalability. As the underpinning basis, we model the interactions between agents under intentional uncertainties as a general Bayesian game, and we show that its agent-form equivalence can be represented as a potential game under certain minor assumptions. The existence and attainability of the optimal interactive trajectories are illustrated, as the corresponding Bayesian Nash equilibrium can be attained by optimizing a unified optimization problem. Additionally, we present a distributed algorithm based on the dual consensus alternating direction method of multipliers (ADMM) tailored to the parallel solving of the problem, thereby significantly improving the scalability. The attendant outcomes from simulations and experiments demonstrate that the proposed method is effective across a range of scenarios characterized by general forms of intentional uncertainties. Its scalability surpasses that of existing centralized and decentralized baselines, allowing for real-time interactive trajectory planning in uncertain game settings.

Standard Bayesian approaches for linear time-invariant (LTI) system identification are hindered by parameter non-identifiability; the resulting complex, multi-modal posteriors make inference inefficient and impractical. We solve this problem by embedding canonical forms of LTI systems within the Bayesian framework. We rigorously establish that inference in these minimal parameterizations fully captures all invariant system dynamics (e.g., transfer functions, eigenvalues, predictive distributions of system outputs) while resolving identifiability. This approach unlocks the use of meaningful, structure-aware priors (e.g., enforcing stability via eigenvalues) and ensures conditions for a Bernstein--von Mises theorem -- a link between Bayesian and frequentist large-sample asymptotics that is broken in standard forms. Extensive simulations with modern MCMC methods highlight advantages over standard parameterizations: canonical forms achieve higher computational efficiency, generate interpretable and well-behaved posteriors, and provide robust uncertainty estimates, particularly from limited data.

Equivariant models leverage prior knowledge on symmetries to improve predictive performance, but misspecified architectural constraints can harm it instead. While work has explored learning or relaxing constraints, selecting among pretrained models with varying symmetry biases remains challenging. We examine this model selection task from an uncertainty-aware perspective, comparing frequentist (via Conformal Prediction), Bayesian (via the marginal likelihood), and calibration-based measures to naive error-based evaluation. We find that uncertainty metrics generally align with predictive performance, but Bayesian model evidence does so inconsistently. We attribute this to a mismatch in Bayesian and geometric notions of model complexity for the employed last-layer Laplace approximation, and discuss possible remedies. Our findings point towards the potential of uncertainty in guiding symmetry-aware model selection.

Tensor Network (TN) Kernel Machines speed up model learning by representing parameters as low-rank TNs, reducing computation and memory use. However, most TN-based Kernel methods are deterministic and ignore parameter uncertainty. Further, they require manual tuning of model complexity hyperparameters like tensor rank and feature dimensions, often through trial-and-error or computationally costly methods like cross-validation. We propose Bayesian Tensor Network Kernel Machines, a fully probabilistic framework that uses sparsity-inducing hierarchical priors on TN factors to automatically infer model complexity. This enables automatic inference of tensor rank and feature dimensions, while also identifying the most relevant features for prediction, thereby enhancing model interpretability. All the model parameters and hyperparameters are treated as latent variables with corresponding priors. Given the Bayesian approach and latent variable dependencies, we apply a mean-field variational inference to approximate their posteriors. We show that applying a mean-field approximation to TN factors yields a Bayesian ALS algorithm with the same computational complexity as its deterministic counterpart, enabling uncertainty quantification at no extra computational cost. Experiments on synthetic and real-world datasets demonstrate the superior performance of our model in prediction accuracy, uncertainty quantification, interpretability, and scalability.

--Stochastic Petri Nets (SPNs) are an increasingly popular tool of choice for modeling discrete-event dynamics in areas such as epidemiology and systems biology, yet their parameter estimation remains challenging in general and in particular when transition rates depend on external covariates and explicit likelihoods are unavailable. We introduce a neural-surrogate (neural-network-based approximation of the posterior distribution) framework that predicts the coefficients of known covariate-dependent rate functions directly from noisy, partially observed token trajectories. Our model employs a lightweight 1D Convolutional Residual Network trained end-to-end on Gillespie-simulated SPN realizations, learning to invert system dynamics under realistic conditions of event dropout. During inference, Monte Carlo dropout provides calibrated uncertainty bounds together with point estimates. On synthetic SPNs with 20% missing events, our surrogate recovers rate-function coefficients with an RMSE = 0.108 and substantially runs faster than traditional Bayesian approaches. These results demonstrate that data-driven, likelihood-free surrogates can enable accurate, robust, and real-time parameter recovery in complex, partially observed discrete-event systems.

Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can be commonly expressed by polynomial size arithmetic circuits computing multilinear polynomials. This raises the question, can all functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $\textsf{FP}\neq\#\textsf{P}$ (an assumption implied by $\textsf{P} \neq \textsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small circuits for that function's multilinear representation.

Protecting cyberspace requires not only advanced tools but also a shift in how we reason about threats, trust, and autonomy. Traditional cybersecurity methods rely on manual responses and brittle heuristics. To build proactive and intelligent defense systems, we need integrated theoretical frameworks and software tools. Game theory provides a rigorous foundation for modeling adversarial behavior, designing strategic defenses, and enabling trust in autonomous systems. Meanwhile, software tools process cyber data, visualize attack surfaces, verify compliance, and suggest mitigations. Yet a disconnect remains between theory and practical implementation. The rise of Large Language Models (LLMs) and agentic AI offers a new path to bridge this gap. LLM-powered agents can operationalize abstract strategies into real-world decisions. Conversely, game theory can inform the reasoning and coordination of these agents across complex workflows. LLMs also challenge classical game-theoretic assumptions, such as perfect rationality or static payoffs, prompting new models aligned with cognitive and computational realities. This co-evolution promises richer theoretical foundations and novel solution concepts. Agentic AI also reshapes software design: systems must now be modular, adaptive, and trust-aware from the outset. This chapter explores the intersection of game theory, agentic AI, and cybersecurity. We review key game-theoretic frameworks (e.g., static, dynamic, Bayesian, and signaling games) and solution concepts. We then examine how LLM agents can enhance cyber defense and introduce LLM-driven games that embed reasoning into AI agents. Finally, we explore multi-agent workflows and coordination games, outlining how this convergence fosters secure, intelligent, and adaptive cyber systems.

In the study of complex dynamical systems, understanding and accurately modeling the underlying physical processes is crucial for predicting system behavior and designing effective interventions. Yet real-world systems exhibit pronounced input (or system) variability and are observed through noisy, limited data conditions that confound traditional discovery methods that assume fixed-coefficient deterministic models. In this work, we theorize that accounting for system variability together with measurement noise is the key to consistently discover the governing equations underlying dynamical systems. As such, we introduce a stochastic inverse physics-discovery (SIP) framework that treats the unknown coefficients as random variables and infers their posterior distribution by minimizing the Kullback-Leibler divergence between the push-forward of the posterior samples and the empirical data distribution. Benchmarks on four canonical problems -- the Lotka-Volterra predator-prey system (multi- and single-trajectory), the historical Hudson Bay lynx-hare data, the chaotic Lorenz attractor, and fluid infiltration in porous media using low- and high-viscosity liquids -- show that SIP consistently identifies the correct equations and lowers coefficient root-mean-square error by an average of 82\% relative to the Sparse Identification of Nonlinear Dynamics (SINDy) approach and its Bayesian variant. The resulting posterior distributions yield 95\% credible intervals that closely track the observed trajectories, providing interpretable models with quantified uncertainty. SIP thus provides a robust, data-efficient approach for consistent physics discovery in noisy, variable, and data-limited settings.

There is no much doubt that biotic interactions shape community assembly and ultimately the spatial co-variations between species. There is a hope that the signal of these biotic interactions can be observed and retrieved by investigating the spatial associations between species while accounting for the direct effects of the environment. By definition, biotic interactions can be both symmetric and asymmetric. Yet, most models that attempt to retrieve species associations from co-occurrence or co-abundance data internally assume symmetric relationships between species. Here, we propose and validate a machine-learning framework able to retrieve bidirectional associations by analyzing species community and environmental data. Our framework (1) models pairwise species associations as directed influences from a source to a target species, parameterized with two species-specific latent embeddings: the effect of the source species on the community, and the response of the target species to the community; and (2) jointly fits these associations within a multi-species conditional generative model with different modes of interactions between environmental drivers and biotic associations. Using both simulated and empirical data, we demonstrate the ability of our framework to recover known asymmetric and symmetric associations and highlight the properties of the learned association networks. By comparing our approach to other existing models such as joint species distribution models and probabilistic graphical models, we show its superior capacity at retrieving symmetric and asymmetric interactions. The framework is intuitive, modular and broadly applicable across various taxonomic groups.