Bayesian machine learning models-especially Bayesian neural networks (BNNs)-offer powerful black-box approaches for prediction and uncertainty quantification. However, these models frequently exhibit inconsistent prediction quality across input regions, and conventional global metrics (e.g., the mean squared error (MSE)) are inadequate for capturing such local discrepancies. To overcome this limitation, we introduce a novel kernel-based framework for local calibration testing that assesses how well predicted distributions reflect both the function to be learned and inherent uncertainties. In our approach, spherical input-space kernels are used to define relevant subsets in the neighborhood of a point to be tested. This enables the online assessment of these localized regions using calibration metrics or statistical tests. By aggregating results across multiple kernel widths, our method yields both robust binary decisions and a continuous analysis over arbitrary inputs. Numerical experiments on single- and multi-dimensional regression tasks demonstrate the efficiency and scalability of our approach, underscoring its potential for real-time and large-scale applications.

We aim to approximate the distribution of the first-passage time of a particle moving according to a Gaussian process with increasing trend, i. e., the distribution of the first time a particle described, e.g., by a state-space model such as a constant-velocity or constant-acceleration model, arrives at a fixed location. Since the known approaches from the literature either consider processes from different families or lead to highly complex approximations, we seek a fast-to-compute method for the problem. Motivated by an engineering particle transport task for which we can assume that once a particle has arrived at this 10-cation it cannot move back, we derive an analytic approximation for the first-passage time probabilities and calculate its inverse cumulative distribution function analytically and the moments numerically. Furthermore, we propose a Gaussian approximation based on a linearization approach. The strengths and limitations of our methods are discussed and by comparison with Monte Carlo simulations, we show that in particular, the first one satisfies the requirements of engineering problems in terms of accuracy and computation time.