In fields such as finance, climate science, and neuroscience, inferring causal relationships from time series data poses a formidable challenge. While contemporary techniques can handle nonlinear relationships between variables and flexible noise distributions, they rely on the simplifying assumption that data originates from the same underlying causal model. In this work, we relax this assumption and perform causal discovery from time series data originating from mixtures of different causal models. We infer both the underlying structural causal models and the posterior probability for each sample belonging to a specific mixture component. Our approach employs an end-to-end training process that maximizes an evidence-lower bound for data likelihood. Through extensive experimentation on both synthetic and real-world datasets, we demonstrate that our method surpasses state-of-the-art benchmarks in causal discovery tasks, particularly when the data emanates from diverse underlying causal graphs. Theoretically, we prove the identifiability of such a model under some mild assumptions.

A hierarchical Bayesian framework is introduced for developing rich mixture models for real-valued time series, partly motivated by important applications in financial time series analysis. At the top level, meaningful discrete states are identified as appropriately quantised values of some of the most recent samples. These observable states are described as a discrete context-tree model. At the bottom level, a different, arbitrary model for real-valued time series -- a base model -- is associated with each state. This defines a very general framework that can be used in conjunction with any existing model class to build flexible and interpretable mixture models. We call this the Bayesian Context Trees State Space Model, or the BCT-X framework. Efficient algorithms are introduced that allow for effective, exact Bayesian inference and learning in this setting; in particular, the maximum a posteriori probability (MAP) context-tree model can be identified. These algorithms can be updated sequentially, facilitating efficient online forecasting. The utility of the general framework is illustrated in two particular instances: When autoregressive (AR) models are used as base models, resulting in a nonlinear AR mixture model, and when conditional heteroscedastic (ARCH) models are used, resulting in a mixture model that offers a powerful and systematic way of modelling the well-known volatility asymmetries in financial data. In forecasting, the BCT-X methods are found to outperform state-of-the-art techniques on simulated and real-world data, both in terms of accuracy and computational requirements. In modelling, the BCT-X structure finds natural structure present in the data. In particular, the BCT-ARCH model reveals a novel, important feature of stock market index data, in the form of an enhanced leverage effect.

Gaussian processes are the model of choice in Bayesian optimization and active learning. Yet, they are highly dependent on cleverly chosen hyperparameters to reach their full potential, and little effort is devoted to finding good hyperparameters in the literature. We demonstrate the impact of selecting good hyperparameters for GPs and present two acquisition functions that explicitly prioritize hyperparameter learning. Statistical distance-based Active Learning (SAL) considers the average disagreement between samples from the posterior, as measured by a statistical distance. SAL outperforms the state-of-the-art in Bayesian active learning on several test functions. We then introduce Self-Correcting Bayesian Optimization (SCoreBO), which extends SAL to perform Bayesian optimization and active learning simultaneously. SCoreBO learns the model hyperparameters at improved rates compared to vanilla BO, while outperforming the latest Bayesian optimization methods on traditional benchmarks. Moreover, we demonstrate the importance of self-correction on atypical Bayesian optimization tasks.

StepMix is an open-source Python package for the pseudo-likelihood estimation (one-, two- and three-step approaches) of generalized finite mixture models (latent profile and latent class analysis) with external variables (covariates and distal outcomes). In many applications in social sciences, the main objective is not only to cluster individuals into latent classes, but also to use these classes to develop more complex statistical models. These models generally divide into a measurement model that relates the latent classes to observed indicators, and a structural model that relates covariates and outcome variables to the latent classes. The measurement and structural models can be estimated jointly using the so-called one-step approach or sequentially using stepwise methods, which present significant advantages for practitioners regarding the interpretability of the estimated latent classes. In addition to the one-step approach, StepMix implements the most important stepwise estimation methods from the literature, including the bias-adjusted three-step methods with Bolk-Croon-Hagenaars and maximum likelihood corrections and the more recent two-step approach. These pseudo-likelihood estimators are presented in this paper under a unified framework as specific expectation-maximization subroutines. To facilitate and promote their adoption among the data science community, StepMix follows the object-oriented design of the scikit-learn library and provides an additional R wrapper.

In recent years, methods for Bayesian inference have been widely used in many different problems in physics where detection and characterization are necessary. Data analysis in gravitational-wave astronomy is a prime example of such a case. Bayesian inference has been very successful because this technique provides a representation of the parameters as a posterior probability distribution, with uncertainties informed by the precision of the experimental measurements. During the last couple of decades, many specific advances have been proposed and employed in order to solve a large variety of different problems. In this work, we present a Markov Chain Monte Carlo (MCMC) algorithm that integrates many of those concepts into a single MCMC package. For this purpose, we have built {\tt Eryn}, a user-friendly and multipurpose toolbox for Bayesian inference, which can be utilized for solving parameter estimation and model selection problems, ranging from simple inference questions, to those with large-scale model variation requiring trans-dimensional MCMC methods, like the LISA global fit problem. In this paper, we describe this sampler package and illustrate its capabilities on a variety of use cases.

Detecting change-points in data is challenging because of the range of possible types of change and types of behaviour of data when there is no change. Statistically efficient methods for detecting a change will depend on both of these features, and it can be difficult for a practitioner to develop an appropriate detection method for their application of interest. We show how to automatically generate new offline detection methods based on training a neural network. Our approach is motivated by many existing tests for the presence of a change-point being representable by a simple neural network, and thus a neural network trained with sufficient data should have performance at least as good as these methods. We present theory that quantifies the error rate for such an approach, and how it depends on the amount of training data. Empirical results show that, even with limited training data, its performance is competitive with the standard CUSUM-based classifier for detecting a change in mean when the noise is independent and Gaussian, and can substantially outperform it in the presence of auto-correlated or heavy-tailed noise. Our method also shows strong results in detecting and localising changes in activity based on accelerometer data.

While hyperparameter tuning shows theoretical promise, its practical efficacy is not universally superior, as evidenced in Figure 4. The results thus offer a balanced perspective that marries theoretical rigor with empirical validation, fulfilling both academic and practical requirements. Implications for the Field of Machine Learning and Predictive Modeling Robustness and Generalization: The ensemble and stacking methods offer a mathematically substantiated pathway to improve the generalization capabilities of predictive models. Interpretability: The feature integration techniques not only improve model performance but also offer better interpretability by highlighting important features through mathematical formulations. Optimization: The proven convergence of Bayesian Optimization to the global optimum has far-reaching implications for hyperparameter tuning in models, as formalized by the EI equation. Unified Framework: This research provides a unified, mathematically rigorous framework for integrating Bayesian and non-Bayesian approaches, thereby setting a new benchmark for hybrid predictive systems. Future Research Directions Scalability: Investigating the scalability of the proposed methods, particularly in the context of the ensemble and Bayesian optimization equations, for larger datasets and more models. Real-world Applications: Extending this research to specific domains like healthcare, finance, and natural language processing to assess the practical utility of the proposed methods. Advanced Optimization Techniques: Exploring other optimization techniques that could further improve the efficiency and effectiveness of the proposed hybrid models, perhaps by introducing new mathematical formulations.

Model-based Reinforcement Learning (MBRL) aims to make agents more sample-efficient, adaptive, and explainable by learning an explicit model of the environment. While the capabilities of MBRL agents have significantly improved in recent years, how to best learn the model is still an unresolved question. The majority of MBRL algorithms aim at training the model to make accurate predictions about the environment and subsequently using the model to determine the most rewarding actions. However, recent research has shown that model predictive accuracy is often not correlated with action quality, tracing the root cause to the \emph{objective mismatch} between accurate dynamics model learning and policy optimization of rewards. A number of interrelated solution categories to the objective mismatch problem have emerged as MBRL continues to mature as a research area. In this work, we provide an in-depth survey of these solution categories and propose a taxonomy to foster future research.

Multi-agent reinforcement learning (MARL) has achieved re markable empirical successes in solving complicated games involving sequential and strategic d ecision-making across multiple agents ( Vinyals et al., 2019; Brown and Sandholm, 2018; Silver et al., 2016). These achievements have catalyzed many research efforts focusing on developing efficient MARL algorithms in a theoretically principled manner. Specifically, a multi-agent system is ty pically modeled as a general-sum Markov Game (MG) ( Littman, 1994), with the primary aim of efficiently discerning a certain equ ilibrium notion among multiple agents from data collected via online interactions. Some popular equilibrium notions include Nash equilibrium (NE), correlated equ ilibrium (CE), and coarse correlated equilibrium (CCE). However, multi-agent general-sum Markov Games (MGs) bring forth various challenges. In particular, empirical application suffers from the large st ate space. Such a challenge necessitates the use of the function approximation as an effective way to ex tract the essential features of RL problems and avoid dealing directly with the large state spa ce. Yet, adopting function approximation in a general-sum MG brings about additional complexities no t found in single-agent RL or a zero-sum MG.

At present, most surface-quality prediction methods can only perform single-task prediction which results in under-utilised datasets, repetitive work and increased experimental costs. To counter this, the authors propose a Bayesian hierarchical model to predict surface-roughness measurements for a turning machining process. The hierarchical model is compared to multiple independent Bayesian linear regression models to showcase the benefits of partial pooling in a machining setting with respect to prediction accuracy and uncertainty quantification.