Gaussian processes

What are Gaussian Processes? Gaussian Processes Explained

Gaussian processes (GPs) are probabilistic models that define a distribution over functions. Unlike parametric models that represent functions with a fixed set of parameters, GPs provide a flexible and non-parametric approach to modeling functions, making them suitable for tasks such as regression and classification.

Here are the key concepts and characteristics of Gaussian processes:

Distribution over Functions: In a Gaussian process, a function is defined as a collection of random variables, one for each input point. Instead of specifying a specific functional form, GPs model the distribution of functions directly. The GP defines a prior distribution over functions, and this prior can be updated based on observed data to obtain a posterior distribution.

Prior and Posterior: The prior distribution of a GP represents the belief about functions before any data is observed. It is typically assumed to be a Gaussian distribution. The posterior distribution is obtained by conditioning the prior on observed data, incorporating the information provided by the data. The posterior distribution is also a Gaussian distribution, allowing for efficient computation and inference.

Mean and Covariance: A GP is fully specified by its mean function and covariance function (also called kernel function). The mean function represents the expected value of the function at each input point. The covariance function characterizes the relationship between the function values at different input points. It quantifies the similarity or correlation between function values and plays a crucial role in capturing the behavior and smoothness of the functions modeled by GPs.

Kernel Functions: Kernel functions define the covariance structure of a GP and determine its flexibility and ability to model different types of functions. Common kernel functions include the squared exponential kernel, Matérn kernel, and linear kernel. Each kernel has different properties, such as controlling the smoothness of the functions or incorporating periodicity.

Inference and Prediction: Inference in Gaussian processes involves updating the prior distribution based on observed data to obtain the posterior distribution. This is done using Bayes’ theorem. Given the posterior distribution, predictions can be made for new, unseen input points by computing the conditional distribution of the function values at those points. Predictions from a GP typically include both the point estimate and the uncertainty (variance) associated with the predictions.

Hyperparameters: Gaussian processes often involve hyperparameters that control the behavior of the model. Hyperparameters include parameters associated with the mean and covariance functions, such as length scales or noise variances. These hyperparameters are typically learned from the data by maximizing the likelihood or using other optimization techniques.

Gaussian processes have several advantages, including their flexibility, ability to model complex and non-linear functions, and the provision of uncertainty estimates for predictions. They can handle different types of data, including continuous, discrete, and mixed variables. Gaussian processes find applications in various fields, including regression, classification, optimization, time series analysis, spatial modeling, and reinforcement learning.

However, Gaussian processes can be computationally demanding for large datasets, as their complexity scales cubically with the number of data points. Various approximation methods and techniques, such as sparse GPs or kernel approximations, are used to mitigate this computational burden while maintaining the key characteristics of Gaussian processes.