## 1 Introduction  
Image registration is a fundamental technique in medical imaging that plays a critical role in clinical applications such as image-guided radiation therapy, image-guided surgery, and minimally invasive treatments [1, 2]. The primary goal of image registration is to determine an optimal transformation that aligns images from different datasets, which is essential when critical information is distributed across multiple images, such as those captured at different time points or from distinct modalities [3, 4]. Accurate registration is crucial for integrating relevant information, enhancing clinical decision-making, and ultimately improving patient outcomes [5, 6]. Typically, image registration involves defining one image as the \"fixed\" image while another, the \"moving\" image, undergoes transformation with the fixed image as a reference [6, 7]. Over the years, registration techniques have evolved significantly, from classical intensity-based methods to advanced machine learning-driven approaches [4, 8]. Recently, deep learning-based methods have shown particular promise, achieving efficient and accurate registration through complex transformation models, often using convolutional neural networks (CNNs) to capture intricate patterns in an unsupervised manner [9]. Within this context, deformable registration has emerged as a key approach to aligning anatomical structures that undergo substantial flexibility and variation. Unlike rigid transformations, deformable registration allows for nonlinear adjustments that adapt to local tissue variations, making it especially valuable for imaging complex organs in dynamic scenarios, such as the brain or lungs [8, 10]. Techniques such as B-spline free-form deformations (FFDs) [8] and diffeomorphic methods, including large displacement diffeomorphic metric mapping (LDDMM), are extensively used for such tasks, ensuring the preservation of anatomical topology during alignment [11]. However, deformable models must manage flexibility carefully to prevent unnatural deformations in sensitive anatomical regions; this balance is typically achieved through regularization terms that guide the transformations. A significant recent advancement is the use of 3D point clouds--data representations that capture anatomical structures' geometric shapes and spatial configurations instead of conventional pixel-based methods in deformable registration. Point clouds, composed of discrete points in 3D space, enable a highly detailed and precise representation of structures, making them particularly suitable for applications requiring anatomical fidelity, such as brain MRI analysis [12]. This shift allows deformable registration approaches to capture complex structural details, yielding a more accurate alignment of 3D anatomical structures and facilitating in-depth analyses of brain regions. Despite these advancements, several challenges in deformable registration persist. High computational costs, the limited interpretability of deformation fields, and a propensity for anatomically implausible transformations in critical brain regions remain problematic [13, 14]. Additionally, the effectiveness of deep learning-based methods is often limited by the availability of large training datasets, which can be scarce in medical imaging contexts [15]. Moreover, local pixel-based loss functions, commonly employed in these models, may overlook the global context of medical images, leading to the potential misalignment or distortion of critical structures. To address these challenges, we propose a novel probabilistic framework utilizing Conformal Bayesian Optimization (CBO) for the deformable registration of 3D point clouds representing brain structures. CBO builds upon Bayesian optimization (BO), introducing conformal prediction techniques that enable calibrated uncertainty estimates throughout the optimization process [16, 17].