Cumulative distribution function of the posterior predictive distribution of a Gaussian process Say I have calculated a posterior predictive distribution using Gaussian process regression. How can I assess the probability that an input X is less than or equal to some arbitrary value x? Is the posterior predictive distribution univariate or multivariate normal? 
By posterior predictive distribution I mean the mean, variance and the covariance assessed for a set of new points. I want to know how I compute the probability of a value less than or equal to x from the mean, variance and covariance. Either it is from the mean and variance using a normal cdf or from the mean and covariance using a multivariate normal cdf. I am hoping someone will know which of the two options. 
 A: If we're talking about standard GP regression where you're predicting a scalar output and assuming Gaussian noise, the posterior predictive distribution for a matrix of $n$ test points $X^* \in \mathbb R^{n \times d}$ is an $n$ dimensional multivariate Gaussian with mean and covariance given by
$$\boldsymbol{\mu} = K(X^*,X) \left(K(X,X) + \sigma^2 I\right)^{-1} \boldsymbol{y}$$
$$\Sigma = K(X^*,X^*) - K(X^*,X) \left(K(X,X) + \sigma^2 I\right)^{-1}K(X,X^*)$$
$\boldsymbol{y}$ is the vector of training outputs, $X$ is the matrix of training input data, $\sigma^2$ is the noise variance.
$K(X,Y)$ is the kernel matrix with $(i,j)^{th}$ entry $k(x_i,y_j)$, where $k$ is your GP kernel.
As with any multivariate Gaussian, you can get the marginal posterior predictive distributions (univariate Gaussians) for single datapoints by picking out the corresponding element of $\boldsymbol{\mu}$ and the corresponding diagonal element of $\Sigma$.
The posterior probability that a single prediction $f_i$ is less than some value then comes directly from the univariate normal cdf with mean $\mu_i$ and variance $\Sigma_{ii}$.
The best reference for GP-related questions is 
C. E. Rasmussen & C. K. I. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006. It's currently available in full, free of charge here.
