Undirected graphical models are compact representations of joint probability distributions over random variables. To solve inference tasks of interest, graphical models of arbitrary topology can be trained using empirical risk minimization.

We demonstrate that BootDQNprior+'s lagged target parameters, which are essential to its performance, arise from applying approximate inference to the BBO posterior.

Below, we proceed to describe our contributions in greater detail. Our framework builds upon the seminal work of Jordan et al. [1998], which introduced the celebrated

Gaussian Process models are a rich distribution over functions with inductive biases controlled by a kernel function. Learning occurs through optimisation of the kernel hyperparameters using the marginal likelihood as the objective.

Graphical models, capable of representing data with complex dependencies, are widely used across domains.

Hierarchical Bayesian methods enable information sharing across regression problems on multiple groups of data.

Bayesian optimization has become widely popular across various experimental sciences due to its favorable attributes: it can handle noisy data, perform well with relatively small datasets, and provide adaptive suggestions for sequential experimentation. While still in its infancy, Bayesian optimization has recently gained traction in bioprocess engineering. However, experimentation with biological systems is highly complex and the resulting experimental uncertainty requires specific extensions to classical Bayesian optimization. Moreover, current literature often targets readers with a strong statistical background, limiting its accessibility for practitioners. In light of these developments, this review has two aims: first, to provide an intuitive and practical introduction to Bayesian optimization; and second, to outline promising application areas and open algorithmic challenges, thereby highlighting opportunities for future research in machine learning.

We present BKP, a user-friendly and extensible R package that implements the Beta Kernel Process (BKP) -- a fully nonparametric and computationally efficient framework for modeling spatially varying binomial probabilities. The BKP model combines localized kernel-weighted likelihoods with conjugate beta priors, resulting in closed-form posterior inference without requiring latent variable augmentation or intensive MCMC sampling. The package supports binary and aggregated binomial responses, allows flexible choices of kernel functions and prior specification, and provides loss-based kernel hyperparameter tuning procedures. In addition, BKP extends naturally to the Dirichlet Kernel Process (DKP) for modeling spatially varying multinomial or compositional data. To our knowledge, this is the first publicly available R package for implementing BKP-based methods. We illustrate the use of BKP through several synthetic and real-world datasets, highlighting its interpretability, accuracy, and scalability. The package aims to facilitate practical application and future methodological development of kernel-based beta modeling in statistics and machine learning.