Gaussian Process Posterior Mean

Problem

For a zero-mean Gaussian process with squared exponential kernel exp(-(a - b)^2 / (2 * length_scale^2)) and additive noise variance noise, the posterior mean at x_star is k_star^T (K + noise * I)^-1 y. Build the kernel matrix, solve the system with a Cholesky factorization, and return the predictive mean. Return 0.0 for empty or mismatched data or invalid hyperparameters.

Hint

The predictive mean is the kernel vector against the training targets, smoothed by the regularized kernel inverse.

Test cases

case 1

in: [[0],[2],1,1,1]

expected: 0.6065306597126333

case 2

in: [[0,1],[0,1],0.5,1,0.1]

expected: 0.5171292397015598

case 3

in: [[0,2],[1,3],1,1,0.5]

expected: 1.4835628288095244

case 4

in: [[],[],1,1,1]

expected: 0

case 5

in: [[0,1],[1],1,1,1]

expected: 0

Starter codePython

import math


def gp_posterior_mean(x_train, y_train, x_star, length_scale, noise):
    # Your code here
    pass

5 test cases · run your code against them in the editor