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An Intuitive Tutorial to Gaussian Process Regression
Jie Wang
TL;DR
Regression requires smooth functions rather than noisy independent Gaussian samples, while multiple functions may fit observed data equally well. The tutorial explains how correlated Gaussian variables, kernels, and GPR provide a principled function model with input-dependent predictive uncertainty, and illustrates standard implementation and packages.
Problem
Independent Gaussian samples produce noisy, nonsmooth functions, while regression may admit multiple functions fitting the same observations; a smoother probabilistic model is therefore needed.
Method
The tutorial builds GPR from multivariate normal distributions, covariance kernels, joint and conditional probabilities, a standard regression example, and Python packages for Gaussian processes.
Results
RBF covariance produces smooth sampled functions where an identity covariance produces uncorrelated, nonsmooth behavior; an optimized example reports σf = 0.0067 and l = 0.0967.
Takeaways & Limitations
GPR represents a prior distribution over infinitely many functions and provides predictive uncertainty determined by inputs before hyperparameter optimization.
Takeaways & Limitations
Standard GPR has O(N3) computational complexity and quadratic memory growth, making it impractical for large datasets without sparse alternatives.
Abstract
from arXiv · showhide
This tutorial aims to provide an intuitive introduction to Gaussian process regression (GPR). GPR models have been widely used in machine learning applications due to their representation flexibility and inherent capability to quantify uncertainty over predictions. The tutorial starts with explaining the basic concepts that a Gaussian process is built on, including multivariate normal distribution, kernels, non-parametric models, and joint and conditional probability. It then provides a concise description of GPR and an implementation of a standard GPR algorithm. In addition, the tutorial reviews packages for implementing state-of-the-art Gaussian process algorithms. This tutorial is accessible to a broad audience, including those new to machine learning, ensuring a clear understanding of GPR fundamentals.
MATHEMATICAL BASICS
The mathematical foundations of GPR progress from Gaussian and multivariate normal distributions to conditional probability and kernels. These concepts explain how correlated, smooth functions can be generated for regression.
- Gaussian Distribution: Gaussian distributions describe random variables through a mean and variance, with their probability density visualized as a bell curve.The tutorial illustrates 1000 samples from a univariate normal distribution and their probability density function.
- From Gaussian Vectors to Functions: Independent Gaussian vectors connected into lines are too noisy for regression, motivating correlations that make nearby inputs produce similar outputs.Correlating the variables forms a joint Gaussian distribution with smoother function-like samples.
- Multivariate Normal Distribution: Multivariate normal distributions jointly model correlated variables using a mean vector and covariance matrix.The covariance matrix stores pairwise covariances among jointly modeled variables.
- Joint and Conditional Probability: Conditional probability is more useful than joint probability for regression because it gives the distribution of one variable when another is fixed.Slicing a bivariate Gaussian at a constant x2 produces the conditional Gaussian distribution P(x1|x2).
- Kernels: Kernels encode prior knowledge about function smoothness by comparing input similarity and defining covariance between inputs.The dot product measures similarity, while a kernel provides the equivalent comparison through feature space without explicitly mapping the data.
- Kernelized Priors: An identity covariance produces noisy prior samples, whereas an RBF covariance produces smooth lines that resemble functions.Increasing the MVN dimension extends this construction toward kernelized prior functions over many inputs.
GAUSSIAN PROCESSES
Gaussian processes represent distributions over possible functions and update those distributions with observations to support regression predictions and uncertainty estimates. GPR uses joint and conditional Gaussian distributions, with kernels encoding relationships between inputs and noisy observations incorporated through covariance.
- GAUSSIAN PROCESSES: A Gaussian process defines a probability distribution over functions that fit observed points, rather than selecting only one fitted function.Observations update the prior into a posterior over functions, which can then be updated again as new data arrive.
- GAUSSIAN PROCESSES: The posterior mean provides the regression function, while variances indicate prediction confidence across possible functions.Finite samples from the function distribution are jointly Gaussian distributed.
- GAUSSIAN PROCESSES: The kernel matrix K defines function smoothness by encoding similarity between inputs: similar inputs are expected to produce similar outputs.The kernel is positive definite, and the mean is set to zero when data are normalized and no observations are available.
- GAUSSIAN PROCESSES: Given observed data, GPR estimates a mean function and predicts function values at new inputs.The regression process uses observed red points, an estimated mean function, and predictions at new points X∗.
- GAUSSIAN PROCESSES: GPR obtains predictions by conditioning the joint Gaussian distribution of observed and test function values.With noisy observations y = f(x) + ϵ, the observation covariance becomes K + σ2_nI before deriving predictive equations.
- GAUSSIAN PROCESSES: Before hyperparameter optimization, predictive uncertainty depends only on training and test inputs X and X∗, not observed outputs y.This dependence is identified as a distinctive property of Gaussian distributions.
ILLUSTRATIVE EXAMPLE
The tutorial implements standard GPR, demonstrates its posterior predictions and uncertainty, and explains how kernel hyperparameters affect smoothness. It also reviews Python packages for practical Gaussian-process implementations, including options for computationally intensive models.
- ILLUSTRATIVE EXAMPLE: The standard GPR implementation follows Rasmussen (2006), taking inputs, targets, a covariance function, noise level, and test inputs.Its outputs are the predictive mean, predictive variance, and log marginal likelihood.
- ILLUSTRATIVE EXAMPLE: Figure 10 plots observed data, 20 posterior mean-function samples, the GPR mean function, and three-times prediction variances over [-5, 5].The observed training points were generated uniformly between -5 and 5, and functions were evaluated at evenly spaced points.
- ILLUSTRATIVE EXAMPLE: Kernel selection strongly affects generalization, and practical kernels can be chosen for properties such as smoothness, sparsity, drastic changes, and differentiability.The tutorial notes established choices such as RBF alongside custom kernels tailored to model requirements.
- ILLUSTRATIVE EXAMPLE: In the RBF kernel, σf controls vertical function scale, while l controls how quickly correlation decreases with distance.Larger l produces smoother functions, whereas smaller l produces more fluctuations or wiggles.
- ILLUSTRATIVE EXAMPLE: After hyperparameter optimization, the predictive variance depends on inputs X and X∗ and outputs y, with σf = 0.0067 and l = 0.0967 in the reported example.The optimization was conducted using GPy, and the resulting regression is shown in Figure 12.
- ILLUSTRATIVE EXAMPLE: GPy, GPflow, and GPyTorch provide alternative Python implementations, with GPyTorch offering GPU acceleration and automatic gradients through PyTorch.GPy is described as mature and stable for tasks that are not computationally intensive, while standard GPR is expensive beyond a few dozen dimensions.
CONCLUSION
GPR represents uncertainty over functions while incorporating prior knowledge through kernels, and the tutorial presents the standard approach as a practical foundation. Its scalability is limited for large datasets, motivating sparse Gaussian-process methods.
- CONCLUSION: A Gaussian process provides prediction values together with uncertainty estimates by modeling a distribution over functions.Kernel functions incorporate prior knowledge about the nature of those functions.
- CONCLUSION: Standard, or vanilla, GPR has O(N^3) computational complexity and memory consumption that grows quadratically with data size.These constraints make standard GPR impractical for large datasets.
- CONCLUSION: Sparse Gaussian processes are employed to alleviate the computational complexity of standard GPR for large datasets.