Hyperparameter Marginal Likelihood Optimization

Gaussian Processes (GPs) provide a flexible, non-parametric way to model functions, placing a prior directly over the function space. The covariance function, or kernel, encodes assumptions about the function's properties, like smoothness or periodicity. GP regression allows predictions of $f_*$ at new points $x_*$ along with uncertainty estimates $\mathbb{E}[f_* | \mathbf{y}]$ and $\text{Var}[f_* | \mathbf{y}]$.

However, the kernel itself, like the popular Radial Basis Function (RBF) kernel, $k(x, x') = \sigma_f^2 \exp\left(-\frac{\|x - x'\|^2}{2l^2}\right)$, has parameters: the signal variance $\sigma_f^2$ and the lengthscale $l$. Similarly, the Gaussian likelihood often assumed in GP regression, $p(y|f) = \mathcal{N}(y|f, \sigma_n^2)$, introduces a noise variance parameter $\sigma_n^2$. These parameters, collectively denoted as $\boldsymbol{\theta} = \{ \sigma_f^2, l, \sigma_n^2 \}$, are called hyperparameters. How do we choose optimal values for them?

Unlike typical supervised learning where we might use cross-validation to tune hyperparameters, the Bayesian nature of GPs offers a more integrated approach: optimizing the marginal likelihood (also known as the evidence).

The Marginal Likelihood: Integrating Out the Function

The core idea is to find the hyperparameter values $\boldsymbol{\theta}$ that maximize the probability of observing the actual data $\mathbf{y}$ given the inputs $\mathbf{X}$ and the hyperparameters $\boldsymbol{\theta}$. This probability is obtained by integrating out (or marginalizing) the unknown function values $\mathbf{f}$ at the training points:

$$p(\mathbf{y} | \mathbf{X}, \boldsymbol{\theta}) = \int p(\mathbf{y} | \mathbf{f}, \sigma_n^2) p(\mathbf{f} | \mathbf{X}, \boldsymbol{\theta}_{\text{kernel}}) d\mathbf{f}$$

Here, $\boldsymbol{\theta}_{\text{kernel}}$ represents the kernel hyperparameters (like $l$ and $\sigma_f^2$), and we explicitly show the noise variance $\sigma_n^2$ from the likelihood term.

Because both the prior $p(\mathbf{f} | \mathbf{X}, \boldsymbol{\theta}_{\text{kernel}})$ and the likelihood $p(\mathbf{y} | \mathbf{f}, \sigma_n^2)$ are Gaussian (assuming a Gaussian likelihood for regression), this integral can be solved analytically. The result is itself a Gaussian distribution for $\mathbf{y}$:

$$\mathbf{y} | \mathbf{X}, \boldsymbol{\theta} \sim \mathcal{N}(\mathbf{0}, K_y) = \mathcal{N}(\mathbf{0}, K_f + \sigma_n^2 I)$$

where $K_f$ is the $N \times N$ covariance matrix evaluated at the training inputs $\mathbf{X}$ using the kernel parameters, $K_{ij} = k(x_i, x_j | \boldsymbol{\theta}_{\text{kernel}})$, and $I$ is the identity matrix. $K_y = K_f + \sigma_n^2 I$ is the covariance matrix of the observed data points $\mathbf{y}$, incorporating both function variability (from $K_f$) and observation noise (from $\sigma_n^2 I$).

Log Marginal Likelihood for Optimization

For numerical stability and convenience, we typically work with the log marginal likelihood:

$$\log p(\mathbf{y} | \mathbf{X}, \boldsymbol{\theta}) = -\frac{1}{2} \mathbf{y}^T K_y^{-1} \mathbf{y} - \frac{1}{2} \log |K_y| - \frac{N}{2} \log(2\pi)$$

Let's break down the important terms:

1. Data Fit Term: $-\frac{1}{2} \mathbf{y}^T K_y^{-1} \mathbf{y}$ This term measures how well the model explains the observed data $\mathbf{y}$. It rewards hyperparameter settings where the observed outputs $\mathbf{y}$ are probable under the Gaussian distribution defined by the covariance $K_y$. If the predictions based on $K_y$ are far from the actual $\mathbf{y}$, this term becomes a large negative number (a penalty).

2. Complexity Penalty Term: $-\frac{1}{2} \log |K_y|$ This term involves the logarithm of the determinant of the covariance matrix $K_y$. The determinant $|K_y|$ measures the "volume" spanned by the distribution. Kernels that lead to very high correlations between points (e.g., very large lengthscales) or very high variance (large $\sigma_f^2$ or $\sigma_n^2$) result in a larger determinant, and thus a larger penalty from this term. This term acts as an automatic "Occam's Razor," penalizing overly complex models (e.g., those that are too flexible or assume too much noise).

3. Normalization Constant: $-\frac{N}{2} \log(2\pi)$ This term is constant with respect to the hyperparameters $\boldsymbol{\theta}$ and can be ignored during optimization.

Maximizing the log marginal likelihood forces a trade-off: finding hyperparameters that allow the model to fit the data well (term 1) without becoming unnecessarily complex (term 2).

Optimization Procedure

The goal is to find the hyperparameters $\boldsymbol{\theta}^*$ that maximize the log marginal likelihood:

$$\boldsymbol{\theta}^* = \arg \max_{\boldsymbol{\theta}} \log p(\mathbf{y} | \mathbf{X}, \boldsymbol{\theta})$$

Since the log marginal likelihood function is generally differentiable with respect to the hyperparameters $\boldsymbol{\theta}$ (including kernel parameters like $l, \sigma_f^2$ and the noise variance $\sigma_n^2$), we can use standard gradient-based optimization algorithms. Common choices include:

• Conjugate Gradients
• L-BFGS-B (Limited-memory Broyden–Fletcher–Goldfarb–Shanno with Bounds, useful for constrained hyperparameters like variances and lengthscales which must be positive).

To use these methods, we need the partial derivatives of the log marginal likelihood with respect to each hyperparameter $\theta_j \in \boldsymbol{\theta}$. The derivative has a relatively clean analytical form:

$$\frac{\partial}{\partial \theta_j} \log p(\mathbf{y} | \mathbf{X}, \boldsymbol{\theta}) = \frac{1}{2} \mathbf{y}^T K_y^{-1} \frac{\partial K_y}{\partial \theta_j} K_y^{-1} \mathbf{y} - \frac{1}{2} \text{Tr}\left( K_y^{-1} \frac{\partial K_y}{\partial \theta_j} \right)$$

Here, $\frac{\partial K_y}{\partial \theta_j}$ is the matrix containing the element-wise partial derivatives of the kernel function (plus the noise term if $\theta_j = \sigma_n^2$) with respect to the hyperparameter $\theta_j$. These derivatives are usually straightforward to compute for standard kernels. For instance, if $K_y = K_f + \sigma_n^2 I$ and $\theta_j$ is a kernel parameter like lengthscale $l$, then $\frac{\partial K_y}{\partial \theta_j} = \frac{\partial K_f}{\partial l}$. If $\theta_j = \sigma_n^2$, then $\frac{\partial K_y}{\partial \theta_j} = I$.

Modern GP libraries (like GPy, GPflow, scikit-learn's GaussianProcessRegressor, GPytorch) automatically compute these gradients and perform the optimization for you. You typically just need to define the kernel structure and provide the data.

# Example using a scikit-learn like interface
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C, WhiteKernel

# Define the kernel with initial hyperparameters (these will be optimized)
# Kernel: k(x, x') = C(constant_value) * RBF(length_scale) + WhiteKernel(noise_level)
# We often optimize the log-transformed parameters for stability
kernel = C(1.0) * RBF(length_scale=1.0) + WhiteKernel(noise_level=1.0)

# Create the GP model
# n_restarts_optimizer helps find a better optimum by restarting from different points
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=10, random_state=42)

# Fit the model: This step performs the log marginal likelihood maximization
# X_train, y_train are your training data
# gp.fit(X_train, y_train)

# After fitting, the optimized hyperparameters are stored
# print(f"Optimized kernel: {gp.kernel_}")
# print(f"Log-Marginal-Likelihood: {gp.log_marginal_likelihood(gp.kernel_.theta)}")

# Example Output (Illustrative):
# Optimized kernel: 1.5**2 * RBF(length_scale=0.8) + WhiteKernel(noise_level=0.1)
# Log-Marginal-Likelihood: -15.23

Challenges and Approaches

While powerful, optimizing the marginal likelihood isn't without challenges:

1. Computational Cost: The main bottleneck is computing $K_y^{-1}$ and $\log |K_y|$, both of which typically scale as $O(N^3)$ where $N$ is the number of training points. This makes direct optimization difficult for large datasets ( motivating the scalable approximations discussed later). The gradient calculation also involves matrix multiplications that add to the cost, but it remains dominated by the $O(N^3)$ term per iteration.
2. Non-Convexity: The log marginal likelihood surface can be non-convex, meaning optimization algorithms might converge to local optima rather than the global maximum. This is why practical implementations often include multiple restarts from different initial hyperparameter values (n_restarts_optimizer in the example above).
3. Initialization: The choice of initial hyperparameter values can significantly impact the optimization result, especially on complex surfaces. Reasonable initial guesses based on domain knowledge or data properties (like the range of inputs and outputs) can be helpful.

Let's visualize how the log marginal likelihood (LML) might vary for a simple 1D regression problem with an RBF kernel, considering only the lengthscale ($l$) and signal variance ($\sigma_f^2$) hyperparameters (keeping noise $\sigma_n^2$ fixed for simplicity).

A contour plot showing the Log Marginal Likelihood (LML) values for different combinations of RBF kernel lengthscale and signal variance. The optimization algorithm searches for the peak (highest LML value) on this surface. The potentially complex shape highlights why multiple restarts might be needed to find the global optimum.

By finding the hyperparameters that maximize the probability of the observed data, marginal likelihood optimization provides a principled and often effective method for tuning GP models, directly embedding model selection within the Bayesian inference framework. This contrasts with methods like cross-validation, offering a more data-efficient approach, particularly when data is scarce. Once the optimal hyperparameters $\boldsymbol{\theta}^*$ are found, they are plugged back into the GP predictive equations for making predictions on new data.