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Causality plays an important role in understanding intelligent behavior, and there is a wealth of literature on mathematical models for causality, most of which is focused on causal graphs. Causal graphs are a powerful tool for a wide range of applications, in particular when the relevant variables are known and at the same level of abstraction. However, the given variables can also be unstructured data, like pixels of an image. Meanwhile, the causal variables, such as the positions of objects in the image, can be arbitrary deterministic functions of the given variables. Moreover, the causal variables may form a hierarchy of abstractions, in which the macro-level variables are deterministic functions of the micro-level variables. Causal graphs are limited when it comes to modeling this kind of situation. In the presence of deterministic relationships there is generally no causal graph that satisfies both the Markov condition and the faithfulness condition. We introduce factored space models as an alternative to causal graphs which naturally represent both probabilistic and deterministic relationships at all levels of abstraction. Moreover, we introduce structural independence and establish that it is equivalent to statistical independence in every distribution that factorizes over the factored space. This theorem generalizes the classical soundness and completeness theorem for d-separation.

Generative diffusion models apply the concept of Langevin dynamics in physics to machine leaning, attracting a lot of interests from engineering, statistics and physics, but a complete picture about inherent mechanisms is still lacking. In this paper, we provide a transparent physics analysis of diffusion models, formulating the fluctuation theorem, entropy production, equilibrium measure, and Franz-Parisi potential to understand the dynamic process and intrinsic phase transitions. Our analysis is rooted in a path integral representation of both forward and backward dynamics, and in treating the reverse diffusion generative process as a statistical inference, where the time-dependent state variables serve as quenched disorder akin to that in spin glass theory. Our study thus links stochastic thermodynamics, statistical inference and geometry based analysis together to yield a coherent picture about how the generative diffusion models work.

Many partial differential equations (PDEs) such as Navier--Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDE as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. We derive the dual weak form for the linear, one-dimensional, transient convection-diffusion equation. A Galerkin discretization is used to obtain the discrete equations, with the trial and test functions chosen as linear combination of either RePU activation functions (shallow neural network) or B-spline basis functions; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin implementation is used with tensor-product B-splines as approximating functions. Numerical results are presented for the steady-state and transient convection-diffusion equation, and transient heat conduction. The proposed method delivers sound accuracy for ODEs and PDEs and rates of convergence are established in the $L^2$ norm and $H^1$ seminorm for the steady-state convection-diffusion problem.

The motivation for sparse learners is to compress the inputs (features) by selecting only the ones needed for good generalization. Linear models with LASSO-type regularization achieve this by setting the weights of irrelevant features to zero, effectively identifying and ignoring them. In artificial neural networks, this selective focus can be achieved by pruning the input layer. Given a cost function enhanced with a sparsity-promoting penalty, our proposal selects a regularization term $\lambda$ (without the use of cross-validation or a validation set) that creates a local minimum in the cost function at the origin where no features are selected. This local minimum acts as a baseline, meaning that if there is no strong enough signal to justify a feature inclusion, the local minimum remains at zero with a high prescribed probability. The method is flexible, applying to complex models ranging from shallow to deep artificial neural networks and supporting various cost functions and sparsity-promoting penalties. We empirically show a remarkable phase transition in the probability of retrieving the relevant features, as well as good generalization thanks to the choice of $\lambda$, the non-convex penalty and the optimization scheme developed. This approach can be seen as a form of compressed sensing for complex models, allowing us to distill high-dimensional data into a compact, interpretable subset of meaningful features.

A time-varying bivariate copula joint model, which models the repeatedly measured longitudinal outcome at each time point and the survival data jointly by both the random effects and time-varying bivariate copulas, is proposed in this paper. A regular joint model normally supposes there exist subject-specific latent random effects or classes shared by the longitudinal and time-to-event processes and the two processes are conditionally independent given these latent variables. Under this assumption, the joint likelihood of the two processes is straightforward to derive and their association, as well as heterogeneity among the population, are naturally introduced by the unobservable latent variables. However, because of the unobservable nature of these latent variables, the conditional independence assumption is difficult to verify. Therefore, besides the random effects, a time-varying bivariate copula is introduced to account for the extra time-dependent association between the two processes. The proposed model includes a regular joint model as a special case under some copulas. Simulation studies indicates the parameter estimators in the proposed model are robust against copula misspecification and it has superior performance in predicting survival probabilities compared to the regular joint model. A real data application on the Primary biliary cirrhosis (PBC) data is performed.

While generalized linear mixed models are a fundamental tool in applied statistics, many specifications, such as those involving categorical factors with many levels or interaction terms, can be computationally challenging to estimate due to the need to compute or approximate high-dimensional integrals. Variational inference is a popular way to perform such computations, especially in the Bayesian context. However, naive use of such methods can provide unreliable uncertainty quantification. We show that this is indeed the case for mixed models, proving that standard mean-field variational inference dramatically underestimates posterior uncertainty in high-dimensions. We then show how appropriately relaxing the mean-field assumption leads to methods whose uncertainty quantification does not deteriorate in high-dimensions, and whose total computational cost scales linearly with the number of parameters and observations. Our theoretical and numerical results focus on mixed models with Gaussian or binomial likelihoods, and rely on connections to random graph theory to obtain sharp high-dimensional asymptotic analysis. We also provide generic results, which are of independent interest, relating the accuracy of variational inference to the convergence rate of the corresponding coordinate ascent algorithm that is used to find it. Our proposed methodology is implemented in the R package, see //github.com/mgoplerud/vglmer . Numerical results with simulated and real data examples illustrate the favourable computation cost versus accuracy trade-off of our approach compared to various alternatives.

We present an efficient algorithm for the application of sequences of planar rotations to a matrix. Applying such sequences efficiently is important in many numerical linear algebra algorithms for eigenvalues. Our algorithm is novel in three main ways. First, we introduce a new kernel that is optimized for register reuse in a novel way. Second, we introduce a blocking and packing scheme that improves the cache efficiency of the algorithm. Finally, we thoroughly analyze the memory operations of the algorithm which leads to important theoretical insights and makes it easier to select good parameters. Numerical experiments show that our algorithm outperforms the state-of-the-art and achieves a flop rate close to the theoretical peak on modern hardware.

We study a circular-circular multiplicative regression model, characterized by an angular error distribution assumed to be wrapped Cauchy. We propose a specification procedure for this model, focusing on adapting a recently proposed goodness-of-fit test for circular distributions. We derive its limiting properties and study the power performance of the test through extensive simulations, including the adaptation of some other well-known goodness-of-fit tests for this type of data. To emphasize the practical relevance of our methodology, we apply it to several small real-world datasets and wind direction measurements in the Black Forest region of southwestern Germany, demonstrating the power and versatility of the presented approach.

A complex multi-state redundant system undergoing preventive maintenance and experiencing multiple events is being considered in a continuous time frame. The online unit is susceptible to various types of failures, both internal and external in nature, with multiple degradation levels present, both internally and externally. Random inspections are continuously monitoring these degradation levels, and if they reach a critical state, the unit is directed to a repair facility for preventive maintenance. The repair facility is managed by a single repairperson, who follows a multiple vacation policy dependent on the operational status of the units. The repairperson is responsible for two primary tasks: corrective repairs and preventive maintenance. The time durations within the system follow phase-type distributions, and the model is constructed using Markovian Arrival Processes with marked arrivals. A variety of performance measures, including transient and stationary distributions, are calculated using matrix-analytic methods. This approach enables the expression of key results and overall system behaviour in a matrix-algorithmic format. In order to optimize the model, costs and rewards are integrated into the analysis. A numerical example is presented to showcase the model's flexibility and effectiveness in real-world applications.

Hidden Markov models (HMMs) are a versatile statistical framework commonly used in ecology to characterize behavioural patterns from animal movement data. In HMMs, the observed data depend on a finite number of underlying hidden states, generally interpreted as the animal's unobserved behaviour. The number of states is a crucial parameter, controlling the trade-off between ecological interpretability of behaviours (fewer states) and the goodness of fit of the model (more states). Selecting the number of states, commonly referred to as order selection, is notoriously challenging. Common model selection metrics, such as AIC and BIC, often perform poorly in determining the number of states, particularly when models are misspecified. Building on existing methods for HMMs and mixture models, we propose a double penalized likelihood maximum estimate (DPMLE) for the simultaneous estimation of the number of states and parameters of non-stationary HMMs. The DPMLE differs from traditional information criteria by using two penalty functions on the stationary probabilities and state-dependent parameters. For non-stationary HMMs, forward and backward probabilities are used to approximate stationary probabilities. Using a simulation study that includes scenarios with additional complexity in the data, we compare the performance of our method with that of AIC and BIC. We also illustrate how the DPMLE differs from AIC and BIC using narwhal (Monodon monoceros) movement data. The proposed method outperformed AIC and BIC in identifying the correct number of states under model misspecification. Furthermore, its capacity to handle non-stationary dynamics allowed for more realistic modeling of complex movement data, offering deeper insights into narwhal behaviour. Our method is a powerful tool for order selection in non-stationary HMMs, with potential applications extending beyond the field of ecology.