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Plot Gpr Prior Posterior

========================================================================== Illustration of prior and posterior Gaussian process for different kernels

This example illustrates the prior and posterior of a

class:

~sklearn.gaussian_process.GaussianProcessRegressor with different kernels. Mean, standard deviation, and 5 samples are shown for both prior and posterior distributions.

Here, we only give some illustration. To know more about kernels’ formulation, refer to the :ref:User Guide <gp_kernels>.

Imports for Visualizing GP Prior and Posterior with Different Kernels

Kernel choice defines the function space prior: Each kernel in the Gaussian process framework encodes assumptions about the functions we expect to see – RBF produces infinitely smooth functions, Matern controls smoothness via the nu parameter, ExpSineSquared generates periodic functions, RationalQuadratic mixes multiple length scales, and DotProduct produces polynomial functions. Before observing any data, sample_y draws random functions from the prior distribution defined by these kernels, visually demonstrating each kernel’s inductive bias. After fitting on training data, the same sampling method draws from the posterior distribution, showing how observations constrain the function space.

From prior to posterior via Bayesian conditioning: The GaussianProcessRegressor.predict method returns the posterior mean and standard deviation at each test point, computed analytically via the kernel matrix inverse (or its Cholesky factorization). The prior distribution has constant mean and uncertainty everywhere, while the posterior pinches down to near-zero uncertainty at training points and expands between them. The plot_gpr_samples helper visualizes this transition by drawing function samples and shading the +/-1 standard deviation band, making the Bayesian updating process intuitive.

# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause

# %%
# Helper function
# ---------------
#
# Before presenting each individual kernel available for Gaussian processes,
# we will define a helper function allowing us plotting samples drawn from
# the Gaussian process.
#
# This function will take a
# :class:`~sklearn.gaussian_process.GaussianProcessRegressor` model and will
# drawn sample from the Gaussian process. If the model was not fit, the samples
# are drawn from the prior distribution while after model fitting, the samples are
# drawn from the posterior distribution.
import matplotlib.pyplot as plt
import numpy as np

Helper for Plotting GP Prior and Posterior Samples

Visualizing the GP’s function-space distribution: The plot_gpr_samples function draws n_samples random functions from either the GP’s prior (before fitting) or posterior (after fitting) distribution using sample_y, then overlays the posterior mean and +/-1 standard deviation band. Before fitting, all sampled functions reflect only the kernel’s prior assumptions about smoothness and scale. After fitting, the sampled functions are constrained to pass near the training points (exactly through them in the noise-free case), with variation between samples indicating regions of higher posterior uncertainty where the data provides less information about the true function.

def plot_gpr_samples(gpr_model, n_samples, ax):
    """Plot samples drawn from the Gaussian process model.

    If the Gaussian process model is not trained then the drawn samples are
    drawn from the prior distribution. Otherwise, the samples are drawn from
    the posterior distribution. Be aware that a sample here corresponds to a
    function.

    Parameters
    ----------
    gpr_model : `GaussianProcessRegressor`
        A :class:`~sklearn.gaussian_process.GaussianProcessRegressor` model.
    n_samples : int
        The number of samples to draw from the Gaussian process distribution.
    ax : matplotlib axis
        The matplotlib axis where to plot the samples.
    """
    x = np.linspace(0, 5, 100)
    X = x.reshape(-1, 1)

    y_mean, y_std = gpr_model.predict(X, return_std=True)
    y_samples = gpr_model.sample_y(X, n_samples)

    for idx, single_prior in enumerate(y_samples.T):
        ax.plot(
            x,
            single_prior,
            linestyle="--",
            alpha=0.7,
            label=f"Sampled function #{idx + 1}",
        )
    ax.plot(x, y_mean, color="black", label="Mean")
    ax.fill_between(
        x,
        y_mean - y_std,
        y_mean + y_std,
        alpha=0.1,
        color="black",
        label=r"$\pm$ 1 std. dev.",
    )
    ax.set_xlabel("x")
    ax.set_ylabel("y")
    ax.set_ylim([-3, 3])


# %%
# Dataset and Gaussian process generation
# ---------------------------------------
# We will create a training dataset that we will use in the different sections.
rng = np.random.RandomState(4)
X_train = rng.uniform(0, 5, 10).reshape(-1, 1)
y_train = np.sin((X_train[:, 0] - 2.5) ** 2)
n_samples = 5

# %%
# Kernel cookbook
# ---------------
#
# In this section, we illustrate some samples drawn from the prior and posterior
# distributions of the Gaussian process with different kernels.
#
# Radial Basis Function kernel
# ............................
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF

kernel = 1.0 * RBF(length_scale=1.0, length_scale_bounds=(1e-1, 10.0))
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Radial Basis Function kernel", fontsize=18)
plt.tight_layout()

# %%
print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

# %%
# Rational Quadratic kernel
# .........................
from sklearn.gaussian_process.kernels import RationalQuadratic

kernel = 1.0 * RationalQuadratic(length_scale=1.0, alpha=0.1, alpha_bounds=(1e-5, 1e15))
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Rational Quadratic kernel", fontsize=18)
plt.tight_layout()

# %%
print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

# %%
# Exp-Sine-Squared kernel
# .......................
from sklearn.gaussian_process.kernels import ExpSineSquared

kernel = 1.0 * ExpSineSquared(
    length_scale=1.0,
    periodicity=3.0,
    length_scale_bounds=(0.1, 10.0),
    periodicity_bounds=(1.0, 10.0),
)
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Exp-Sine-Squared kernel", fontsize=18)
plt.tight_layout()

# %%
print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

# %%
# Dot-product kernel
# ..................
from sklearn.gaussian_process.kernels import ConstantKernel, DotProduct

kernel = ConstantKernel(0.1, (0.01, 10.0)) * (
    DotProduct(sigma_0=1.0, sigma_0_bounds=(0.1, 10.0)) ** 2
)
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0, normalize_y=True)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Dot-product kernel", fontsize=18)
plt.tight_layout()

# %%
print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)

# %%
# Matérn kernel
# ..............
from sklearn.gaussian_process.kernels import Matern

kernel = 1.0 * Matern(length_scale=1.0, length_scale_bounds=(1e-1, 10.0), nu=1.5)
gpr = GaussianProcessRegressor(kernel=kernel, random_state=0)

fig, axs = plt.subplots(nrows=2, sharex=True, sharey=True, figsize=(10, 8))

# plot prior
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[0])
axs[0].set_title("Samples from prior distribution")

# plot posterior
gpr.fit(X_train, y_train)
plot_gpr_samples(gpr, n_samples=n_samples, ax=axs[1])
axs[1].scatter(X_train[:, 0], y_train, color="red", zorder=10, label="Observations")
axs[1].legend(bbox_to_anchor=(1.05, 1.5), loc="upper left")
axs[1].set_title("Samples from posterior distribution")

fig.suptitle("Matérn kernel", fontsize=18)
plt.tight_layout()

# %%
print(f"Kernel parameters before fit:\n{kernel})")
print(
    f"Kernel parameters after fit: \n{gpr.kernel_} \n"
    f"Log-likelihood: {gpr.log_marginal_likelihood(gpr.kernel_.theta):.3f}"
)