Gaussian Process
A Gaussian Process (GP) is a Bayesian, non-parametric approach to modeling functions. Instead of fitting a fixed set of parameters, GPs assume that any collection of points is drawn from a multivariate Gaussian distribution. This allows not only for predictions but also for uncertainty quantification at every point in the input space.
Key use cases
- Bayesian optimization: widely applied in hyperparameter tuning for AI models.
- Time-series forecasting: when uncertainty and confidence intervals are critical.
- Robotics and control systems: for adaptive decision-making under uncertainty.
Why it matters
GPs are particularly valued in domains where confidence estimates are as important as the prediction itself—for instance, in medical diagnostics or scientific modeling.
Challenges
- Computationally expensive with large datasets.
- Requires careful choice of kernel (covariance function), which encodes prior assumptions about the data.
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Gaussian Processes (GPs) are non-parametric models, meaning they do not assume a fixed number of parameters but instead adapt their complexity to the data. This flexibility allows them to capture highly nonlinear relationships while providing uncertainty estimates for each prediction. The core idea is that any finite collection of function values follows a joint Gaussian distribution defined by a mean function and a covariance function (kernel).
The choice of kernel is critical since it encodes assumptions about smoothness, periodicity, or other structural properties of the data. Popular kernels include the Radial Basis Function (RBF), Matérn, and periodic kernels, each suited to different contexts such as smooth signals, rougher spatial data, or cyclical time series. Often, multiple kernels are combined to model complex phenomena more effectively.
Despite their interpretability and theoretical elegance, GPs face practical challenges. Standard implementations require operations on covariance matrices of size n × n, leading to cubic computational complexity in the number of data points. To address scalability issues, researchers have developed sparse approximations, inducing point methods, and variational inference approaches that extend GPs to larger datasets.
Beyond regression and prediction, GPs are also used for classification, time-series forecasting, and even reinforcement learning. For instance, Gaussian Process-based models can guide exploration in Bayesian optimization or act as priors in probabilistic deep learning. This versatility, combined with their principled handling of uncertainty, makes GPs a cornerstone of probabilistic machine learning.
Further reading: Rasmussen & Williams (2006); Bishop, Pattern Recognition and Machine Learning (2006).
