construct probabilistic couplings

Traditional Markov decision processes (MDPs) model only the marginal distribution of actions, making it difficult to capture the joint dependencies among counterfactual one-step outcomes under multiple actions. To address this limitation, this work proposes a formal framework termed Joint MDP (JMDP), which introduces a multi-action generation interface to explicitly model the coupled dynamics among actions under shared exogenous randomness. Under a one-step coupling assumption, the authors derive Bellman operators for higher-order moments of returns and develop corresponding dynamic programming and incremental learning algorithms with convergence guarantees. This study establishes the first theoretical foundation for modeling joint distributions in reinforcement learning, enabling accurate estimation of higher-order return moments.

Existing optimal transport approaches to causal inference suffer from non-uniqueness of transport maps and conceptual conflation among multivariate monotonicity notions, leaving the selection of counterfactual transport maps under fixed marginals theoretically unjustified. Method: We establish necessary and sufficient conditions for the equivalence of cyclically monotone, quantile-preserving, and triangularly monotone maps; formulate counterfactual inference as a transport map selection problem under fixed marginals; and systematically characterize map identifiability under causal graphs and structural equation models. Contribution/Results: Our work unifies statistical optimal transport and causal inference within a single theoretical framework. It precisely delineates the applicability domains of each map class and reveals how causal assumptions fundamentally constrain admissible map structures. By grounding counterfactual reasoning in testable transport-theoretic principles, the study provides a rigorous, empirically verifiable foundation for marginal-preserving causal mapping—resolving long-standing ambiguities in transport-based counterfactual estimation.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

Gaussian processes (GPs) struggle to rigorously incorporate uncountably infinite-dimensional functional prior information—such as boundary conditions or global physical constraints satisfied by PDE solutions. Method: This paper proposes a unified modeling framework grounded in reproducing kernel Hilbert spaces (RKHS), establishing for the first time a rigorous equivalence between the GP conditional expectation and orthogonal projection in RKHS. This enables direct embedding of functional constraints (e.g., Dirichlet or Neumann boundary conditions) into the GP prior, bypassing conventional pseudo-point approximations. Contribution/Results: We provide theoretical guarantees on existence, uniqueness, and convergence of the constrained GP posterior. Computationally, we design a practical numerical approximation algorithm. Experiments on PDE inverse problems demonstrate substantial improvements in uncertainty quantification accuracy and posterior consistency. The framework delivers a rigorous, general, and computationally tractable paradigm for integrating domain knowledge into Bayesian modeling.

This work addresses the intractability of exact Bayesian inference for latent states in continuous-time Markov jump processes—such as chemical reaction networks and stochastic Lotka–Volterra systems. We propose an analytically tractable expectation propagation (EP) framework grounded in entropy matching: by embedding entropy matching into the EP formalism, we derive closed-form approximations to the latent-state posterior distribution and integrate them with an approximate EM algorithm for efficient parameter estimation. Our approach overcomes the high computational complexity and poor scalability inherent in conventional methods for discrete-state continuous-time models. Evaluated on multiple systems biology benchmarks, the method achieves high-accuracy latent-state inference and parameter estimation while significantly improving computational scalability and practical applicability.

This study addresses the problem of generating counterfactual outcomes under hypothetical interventions from observational data. To this end, it proposes a flow matching framework that integrates doubly robust training with coupling learning to enable end-to-end counterfactual generation, while introducing Gaussian smoothed interpolation and stochastic samplers to enhance inference efficiency. Theoretically, the work establishes a coupling-sensitive KL error bound, demonstrating its dependence on displacement moments rather than global regularity, and provides finite-step convergence guarantees. Empirically, experiments on synthetic and image benchmarks validate these theoretical findings, revealing that stochastic samplers significantly outperform deterministic ODE samplers under limited computational budgets.

This study addresses the limitation of Gaussian process causal models in handling counterfactual inference with discrete variables by proposing a unified probabilistic framework that integrates GP predictors with explicit exogenous noise mechanisms. Methodologically, exact conditional noise abduction procedures are derived for binary, nominal, and ordinal variables, employing uniform thresholds, Gumbel-max competition, and latent Gaussian cut-point models to achieve discrete noise propagation, which theoretically recovers both observational and interventional distributions. Experimental results demonstrate that incorrect noise coupling choices yield counterfactual errors that fail to converge even with increasing data volumes, underscoring the critical role of structural soundness.

This work establishes novel maximal inequalities for empirical processes under graph-dependent observations, revealing that their convergence rates are jointly governed by the complexity of the function class, the underlying graph structure, and the decay of dependence. By integrating graph coloring, block partitioning, and graph-adapted dependence coefficients, the authors develop a coupling strategy that effectively disentangles the complexity of the indexing class from the graph-induced dependence, thereby overcoming the classical √n convergence barrier. The theoretical framework accommodates graphs with polynomial or exponential growth as well as directed binary networks, yielding Glivenko–Cantelli-type results and explicit quantifications of effective sample size. These advances enable uniform laws of large numbers in applications such as network autoregressive models, nonlinear local propagation dynamics, and settings involving treatment interference.

Traditional predictive models struggle to capture the structural uncertainty inherent in counterfactual worlds and often yield invalid outputs under partially identifiable causal models. This work proposes formalizing world models as positive semi-definite coupling kernels \( K(T,T') \) defined over admissible possible worlds, where diagonal entries correspond to standard posteriors and off-diagonal entries—introduced here for the first time—encode cross-world counterfactual couplings. By integrating structural causal models, ontological axioms, and targeted infeasibility learning, the method efficiently bounds counterfactual responses in polynomial time. Empirical results demonstrate that 28% of models avoid invalid predictions; ontological constraints tighten uncertainty bounds by up to one-third; and targeted learning substantially improves convergence efficiency.

研究通过无训练方法将冻结的时间序列基础模型边际结合成多变量预测样本路径，以改善依赖关系诊断。