Classical discriminant analysis assumes identically distributed training data, yet in many applications observations are collected over time and the class-conditional distributions drift. This population drift renders stationary classifiers unreliable. We propose a principled, model-based framework that embeds discriminant analysis within state-space models to obtain nonstationary linear discriminant analysis (NSLDA) and nonstationary quadratic discriminant analysis (NSQDA). For linear-Gaussian dynamics, we adapt Kalman smoothing to handle multiple samples per time step and develop two practical extensions: (i) an expectation-maximization (EM) approach that jointly estimates unknown system parameters, and (ii) a Gaussian mixture model (GMM)-Kalman method that simultaneously recovers unobserved time labels and parameters, a scenario common in practice. To address nonlinear or non-Gaussian drift, we employ particle smoothing to estimate time-varying class centroids, yielding fully nonstationary discriminant rules. Extensive simulations demonstrate consistent improvements over stationary linear discriminant analysis (LDA), quadratic discriminant analysis (QDA), and support vector machine (SVM) baselines, with robustness to noise, missing data, and class imbalance. This paper establishes a unified and data-efficient foundation for discriminant analysis under temporal distribution shift.

Somewhat surprisingly, learning in a Markov Decision Process is most often considered under the performance criteria of consistency or regret minimization (see e.g.

To address this issue, various approximate Bayesian inference methods have been proposed, such as Markov Chain Monte Carlo [MCMC, see e.g.

We also devise a reinforcement learning algorithm for SICMDPs that we call SI-CRL.

Modest statistical differences between the sampling performances of the D-Wave quantum annealer (QA) and the classical Markov Chain Monte Carlo (MCMC), when applied to Restricted Boltzmann Machines (RBMs), are explored to explain, and possibly address, the absence of significant and consistent improvements in RBM trainability when the D-Wave sampling was used in previous investigations. A novel hybrid sampling approach, combining the classical and the QA contributions, is investigated as a promising way to benefit from the modest differences between the two sampling methods. No improvements in the RBM training are achieved in this work, thereby suggesting that the differences between the QA-based and MCMC sampling, mainly found in the medium-to-low probability regions of the distribution, which are less important for the quality of the sample, are insufficient to benefit the training. Difficulties in achieving sufficiently high quality of embedding RBMs into the lattice of the newer generation of D-Wave hardware could be further complicating the task. On the other hand, the ability to generate samples of sufficient variety from lower-probability parts of the distribution has a potential to benefit other machine learning applications, such as the mitigation of catastrophic forgetting (CF) during incremental learning. The feasibility of using QA-generated patterns of desirable classes for CF mitigation by the generative replay is demonstrated in this work for the first time. While the efficiency of the CF mitigation using the D-Wave QA was comparable to that of the classical mitigation, both the speed of generating a large number of distinct desirable patterns and the potential for further improvement make this approach promising for a variety of challenging machine learning applications.

In this paper, we introduce a categorial generalization of RL, termed universal reinforcement learning (URL), building on powerful mathematical abstractions from the study of coinduction on non-well-founded sets and universal coalgebras, topos theory, and categorial models of asynchronous parallel distributed computation. In the first half of the paper, we review the basic RL framework, illustrate the use of categories and functors in RL, showing how they lead to interesting insights. In particular, we also introduce a standard model of asynchronous distributed minimization proposed by Bertsekas and Tsitsiklis, and describe the relationship between metric coinduction and their proof of the Asynchronous Convergence Theorem. The space of algorithms for MDPs or PSRs can be modeled as a functor category, where the co-domain category forms a topos, which admits all (co)limits, possesses a subobject classifier, and has exponential objects. In the second half of the paper, we move on to universal coalgebras. Dynamical system models, such as Markov decision processes (MDPs), partially observed MDPs (POMDPs), a predictive state representation (PSRs), and linear dynamical systems (LDSs) are all special types of coalgebras. We describe a broad family of universal coalgebras, extending the dynamic system models studied previously in RL. The core problem in finding fixed points in RL to determine the exact or approximate (action) value function is generalized in URL to determining the final coalgebra asynchronously in a parallel distributed manner.