Beyond Gaussian Processes: Pearson Type VII Processes for Robust Bayesian Regression

We propose a Bayesian framework for regression modeling that uses Pearson Type VII Processes (P7Ps) as an adaptable generalization of Gaussian Processes (GPs). The P7P is a scale mixture of Normal and Gamma distributions. It has flexible heavy-tailed properties that enhance its robustness against outliers and render it suitable for modelling heavytailed data. The P7P framework outperforms GP functions by providing explicit control over tail behavior and variance modulation via the shape and scale parameters. These factors dictate the weight of the tails and the overall dispersion of the process, facilitating a seamless transition between Gaussian and heavy-tailed regimes, thus improving modelling flexibility and robustness. We derive the predictive distribution for new data points and use the Laplace approximation for efficient posterior inference under the Pearson Type VII (P7) prior, which ensures scalability while maintaining analytical tractability. The Pearson Type VII Process for regression (P7PR) outperforms Gaussian Process for regression (GPR) in terms of resilience, adaptability and predictive accuracy, especially in challenging scenarios with heavy-tailed noise and outliers, as evidenced using comparative analysis based on Mean Square Error (MSE), Mean Absolute Error (MAE) and predictive log-likelihood (PLL), using both simulated and real-world datasets. These results validate the benefits of heavy-tailed priors in modelling non-Gaussian distributions and illustrate the resilience of P7PR as a Bayesian substitute for conventional GPR. Although the proposed P7PR framework exhibits robustness and strong predictive accuracy, its computational cost increases remarkably as the size of the datasets, the complexity of the model, and the posterior space expand. Consequently, in order to preserve scalability without compromising performance, practical implementations may necessitate moderate model sizes or approximate inference methods.