We introduce a novel class of graphical models, termed profile graphical models, that represent, within a single graph, how an external factor influences the dependence structure of a multivariate set of variables. This class is quite general and includes multiple graphs and chain graphs as special cases. Profile graphical models capture the conditional distributions of a multivariate random vector given different levels of a risk factor, and learn how the conditional independence structure among variables may vary across these risk profiles; we formally define this family of models and establish their corresponding Markov properties. We derive key structural and probabilistic properties that underpin a more powerful inferential framework than existing approaches, underscoring that our contribution extends beyond a novel graphical representation.Furthermore, we show that the resulting profile undirected graphical models are independence-compatible with two-block LWF chain graph models.We then develop a Bayesian approach for Gaussian undirected profile graphical models based on continuous spike-and-slab priors to learn shared sparsity structures across different levels of the risk factor. We also design a fast EM algorithm for efficient inference. Inferential properties are explored through simulation studies, including the comparison with competing methods. The practical utility of this class of models is demonstrated through the analysis of protein network data from various subtypes of acute myeloid leukemia. Our results show a more parsimonious network and greater patient heterogeneity than its competitors, highlighting its enhanced ability to capture subject-specific differences.

We present AutoStan, a framework in which a command-line interface (CLI) coding agent autonomously builds and iteratively improves Bayesian models written in Stan. The agent operates in a loop, writing a Stan model file, executing MCMC sampling, then deciding whether to keep or revert each change based on two complementary feedback signals: the negative log predictive density (NLPD) on held-out data and the sampler's own diagnostics (divergences, R-hat, effective sample size). We evaluate AutoStan on five datasets with diverse modeling structures. On a synthetic regression dataset with outliers, the agent progresses from naive linear regression to a model with Student-t robustness, nonlinear heteroscedastic structure, and an explicit contamination mixture, matching or outperforming TabPFN, a state-of-the-art black-box method, while remaining fully interpretable. Across four additional experiments, the same mechanism discovers hierarchical partial pooling, varying-slope models with correlated random effects, and a Poisson attack/defense model for soccer. No search algorithm, critic module, or domain-specific instructions are needed. This is, to our knowledge, the first demonstration that a CLI coding agent can autonomously write and iteratively improve Stan code for diverse Bayesian modeling problems.

Conditional independence is central to modern statistics, but beyond special parametric families it rarely admits an exact covariance characterization. We introduce the binary expansion group intersection network (BEGIN), a distribution-free graphical representation for multivariate binary data and bit-encoded multinomial variables. For arbitrary binary random vectors and bit representations of multinomial variables, we prove that conditional independence is equivalent to a sparse linear representation of conditional expectations, to a block factorization of the corresponding interaction covariance matrix, and to block diagonality of an associated generalized Schur complement. The resulting graph is indexed by the intersection of multiplicative groups of binary interactions, yielding an analogue of Gaussian graphical modeling beyond the Gaussian setting. This viewpoint treats data bits as atoms and local BEGIN molecules as building blocks for large Markov random fields. We also show how dyadic bit representations allow BEGIN to approximate conditional independence for general random vectors under mild regularity conditions. A key technical device is the Hadamard prism, a linear map that links interaction covariances to group structure.

Many causal discovery algorithms, including the celebrated FCI algorithm, output a Partial Ancestral Graph (PAG). PAGs serve as an abstract graphical representation of the underlying causal structure, modeled by directed acyclic graphs with latent and selection variables. This paper develops a characterization of the set of extended-type conditional independence relations that are invariant across all causal models represented by a PAG. This theory allows us to formulate a general measure-theoretic version of Pearl's causal calculus and a sound and complete identification algorithm for PAGs under selection bias. Our results also apply when PAGs are learned by certain algorithms that integrate observational data with experimental data and incorporate background knowledge.

We propose SAHMM-VAE, a source-wise adaptive Hidden Markov prior variational autoencoder for unsupervised blind source separation. Instead of treating the latent prior as a single generic regularizer, the proposed framework assigns each latent dimension its own adaptive regime-switching prior, so that different latent dimensions are pulled toward different source-specific temporal organizations during training. Under this formulation, source separation is not implemented as an external post-processing step; it is embedded directly into variational learning itself. The encoder, decoder, posterior parameters, and source-wise prior parameters are optimized jointly, where the encoder progressively learns an inference map that behaves like an approximate inverse of the mixing transformation, while the decoder plays the role of the generative mixing model. Through this coupled optimization, the gradual alignment between posterior source trajectories and heterogeneous HMM priors becomes the mechanism through which different latent dimensions separate into different source components. To instantiate this idea, we develop three branches within one common framework: a Gaussian-emission HMM prior, a Markov-switching autoregressive HMM prior, and an HMM state-flow prior with state-wise autoregressive flow transformations. Experiments show that the proposed framework achieves unsupervised source recovery while also learning meaningful source-wise switching structures. More broadly, the method extends our structured-prior VAE line from smooth, mixture-based, and flow-based latent priors to adaptive switching priors, and provides a useful basis for future work on interpretable and potentially identifiable latent source modeling.

Motifs often recur in musical works in altered forms, preserving aspects of their identity while undergoing local variation. This paper investigates how such motivic transformations occur within their musical context in symbolic music. To support this analysis, we develop a probabilistic framework for modeling motivic transformations and apply it to Beethoven's piano sonatas by integrating multiple datasets that provide melodic, rhythmic, harmonic, and motivic information within a unified analytical representation. Motif transformations are represented as multilabel variables by comparing each motif instance to a designated reference occurrence within its local context, ensuring consistent labeling across transformation families. We introduce a multilabel Conditional Random Field to model how motif-level musical features influence the occurrence of transformations and how different transformation families tend to co-occur. Our goal is to provide an interpretable, distributional analysis of motivic transformation patterns, enabling the study of their structural relationships and stylistic variation. By linking computational modeling with music-theoretical interpretation, the proposed framework supports quantitative investigation of musical structure and complexity in symbolic corpora and may facilitate the analysis of broader compositional patterns and writing practices.

Adapting large-scale foundation models to new domains with limited supervision remains a fundamental challenge due to latent distribution mismatch, unstable optimization dynamics, and miscalibrated uncertainty propagation. This paper introduces an uncertainty-aware probabilistic latent transport framework that formulates domain adaptation as a stochastic geometric alignment problem in representation space. A Bayesian transport operator is proposed to redistribute latent probability mass along Wasserstein-type geodesic trajectories, while a PAC-Bayesian regularization mechanism constrains posterior model complexity to mitigate catastrophic overfitting. The proposed formulation yields theoretical guarantees on convergence stability, loss landscape smoothness, and sample efficiency under distributional shift. Empirical analyses demonstrate substantial reduction in latent manifold discrepancy, accelerated transport energy decay, and improved covariance calibration compared with deterministic fine-tuning and adversarial domain adaptation baselines. Furthermore, bounded posterior uncertainty evolution indicates enhanced probabilistic reliability during cross-domain transfer. By establishing a principled connection between stochastic optimal transport geometry and statistical generalization theory, the proposed framework provides new insights into robust adaptation of modern foundation architectures operating in heterogeneous environments. These findings suggest that uncertainty-aware probabilistic alignment constitutes a promising paradigm for reliable transfer learning in next-generation deep representation systems.

Detection of occult hemorrhage (i.e., internal bleeding) in patients in intensive care units (ICUs) can pose significant challenges for critical care workers. Because blood loss may not always be clinically apparent, clinicians rely on monitoring vital signs for specific trends indicative of a hemorrhage event. The inherent difficulties of diagnosing such an event can lead to late intervention by clinicians which has catastrophic consequences. Therefore, a methodology for early detection of hemorrhage has wide utility. We develop a Bayesian regime switching model (RSM) that analyzes trends in patients' vitals and labs to provide a probabilistic assessment of the underlying physiological state that a patient is in at any given time. This article is motivated by a comprehensive dataset we curated from Mayo Clinic of 33,924 real ICU patient encounters. Longitudinal response measurements are modeled as a vector autoregressive process conditional on all latent states up to the current time point, and the latent states follow a Markov process. We present a novel Bayesian sampling routine to learn the posterior probability distribution of the latent physiological states, as well as develop an approach to account for pre-ICU-admission physiological changes. A simulation and real case study illustrate the effectiveness of our approach.

Causal representation learning seeks to uncover causal relationships among high-level latent variables from low-level, entangled, and noisy observations. Existing approaches often either rely on deep neural networks, which lack interpretability and formal guarantees, or impose restrictive assumptions like linearity, continuous-only observations, and strong structural priors. These limitations particularly challenge applications with a large number of discrete latent variables and mixed-type observations. To address these challenges, we propose discrete causal representation learning (DCRL), a generative framework that models a directed acyclic graph among discrete latent variables, along with a sparse bipartite graph linking latent and observed layers. This design accommodates continuous, count, and binary responses through flexible measurement models while maintaining interpretability. Under mild conditions, we prove that both the bipartite measurement graph and the latent causal graph are identifiable from the observed data distribution alone. We further propose a three-stage estimate-resample-discovery pipeline: penalized estimation of the generative model parameters, resampling of latent configurations from the fitted model, and score-based causal discovery on the resampled latents. We establish the consistency of this procedure, ensuring reliable recovery of the latent causal structure. Empirical studies on educational assessment and synthetic image data demonstrate that DCRL recovers sparse and interpretable latent causal structures.

Online reinforcement learning in infinite-horizon Markov decision processes (MDPs) remains less theoretically and algorithmically developed than its episodic counterpart, with many algorithms suffering from high ``burn-in'' costs and failing to adapt to benign instance-specific complexity. In this work, we address these shortcomings for two infinite-horizon objectives: the classical average-reward regret and the $γ$-regret. We develop a single tractable UCB-style algorithm applicable to both settings, which achieves the first optimal variance-dependent regret guarantees. Our regret bounds in both settings take the form $\tilde{O}( \sqrt{SA\,\text{Var}} + \text{lower-order terms})$, where $S,A$ are the state and action space sizes, and $\text{Var}$ captures cumulative transition variance. This implies minimax-optimal average-reward and $γ$-regret bounds in the worst case but also adapts to easier problem instances, for example yielding nearly constant regret in deterministic MDPs. Furthermore, our algorithm enjoys significantly improved lower-order terms for the average-reward setting. With prior knowledge of the optimal bias span $\Vert h^\star\Vert_\text{sp}$, our algorithm obtains lower-order terms scaling as $\Vert h^\star\Vert_\text{sp} S^2 A$, which we prove is optimal in both $\Vert h^\star\Vert_\text{sp}$ and $A$. Without prior knowledge, we prove that no algorithm can have lower-order terms smaller than $\Vert h^\star \Vert_\text{sp}^2 S A$, and we provide a prior-free algorithm whose lower-order terms scale as $\Vert h^\star\Vert_\text{sp}^2 S^3 A$, nearly matching this lower bound. Taken together, these results completely characterize the optimal dependence on $\Vert h^\star\Vert_\text{sp}$ in both leading and lower-order terms, and reveal a fundamental gap in what is achievable with and without prior knowledge.