Gaussian Processes

What are Gaussian Processes?

Gaussian Processes (GPs) are a non-parametric Bayesian modeling technique used for regression, classification, and optimization. GPs model the function space directly, rather than having a fixed set of parameters like in parametric models. GPs are particularly useful when dealing with small datasets, noisy data, or complex functions with unknown structure.

How do Gaussian Processes work?

Gaussian Processes work by defining a prior distribution over functions, which is specified by a mean function and a covariance function. The covariance function, also known as the kernel, determines the smoothness and structure of the functions in the process. Given a set of data points, Gaussian Processes compute the posterior distribution over functions, which can be used to make predictions and quantify uncertainty.

Example of Gaussian Process regression with Python and scikit-learn:

import numpy as np
import matplotlib.pyplot as plt
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C

# Generate sample data
x = np.random.uniform(-3, 3, size=(20, 1))
y = np.sin(x) + np.random.randn(20, 1) * 0.1

# Define the kernel function
kernel = C(1.0, (1e-3, 1e3)) * RBF(1.0, (1e-3, 1e3))

# Fit a Gaussian Process model
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=9), y)

# Make predictions
x_new = np.linspace(-3, 3, 100).reshape(100, 1)
y_pred, sigma = gp.predict(x_new, return_std=True)

# Visualize the results
plt.scatter(x, y, color='blue')
plt.plot(x_new, y_pred, color='red', linewidth=2)
plt.fill_between(x_new.ravel(), y_pred.ravel() - 2 * sigma, y_pred.ravel() + 2 * sigma, alpha=0.2, color='orange')
plt.title('Gaussian Process Regression')

In this example, we generate a noisy sine curve and use a Gaussian Process model with an RBF kernel to fit the data. The resulting plot shows the original data points, the GP regression line, and the uncertainty bounds.

Resources for learning Gaussian Processes: