Gaussian Process Regression: The Bayesian Approach to Curve Fitting

The marginal log-likelihood equation you've provided is a key component in the process of hyperparameter optimization for Gaussian Processes (GPs). Let's break down each term and understand how it contributes to finding optimal values for the hyperparameters, such as (\sigma_f) and (\ell).

Marginal Log-Likelihood

The marginal log-likelihood ( \log p(\mathbf{y} | \mathbf{x}, \boldsymbol{\theta}) ) is given by:

[ \log p(\mathbf{y} | \mathbf{x}, \boldsymbol{\theta}) = -\frac{1}{2}\mathbf{y}^\top K^{-1}\mathbf{y} - \frac{1}{2}\log|K| - \frac{n}{2}\log 2\pi ]

Where:

• ( \mathbf{y} ) is the vector of observations.
• ( \mathbf{x} ) is the vector (or matrix) of input features.
• ( K ) is the covariance matrix, which depends on the hyperparameters (\boldsymbol{\theta}).
• ( n ) is the number of data points.

Terms

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