Contents
How is the posterior derived in the Gaussian process?
The Computing the posterior section derives the posterior from the prior and the likelihood. And it describes how to make predictions using the posterior. The Parameter learning section finds optimal concrete values for model parameters. The posterior from the previous section is a symbolic expression that mentions model parameters.
How to sample functions from the Gaussian process?
To sample functions from the Gaussian process we need to define the mean and covariance functions. The covariance function k(xa,xb) k ( x a, x b) models the joint variability of the Gaussian process random variables.
How is a Gaussian process a multivariate distribution?
1 A Gaussian process is a distribution over functions fully specified by a mean and covariance function. 2 Every finite set of the Gaussian process distribution is a multivariate Gaussian. 3 The posterior predictions of a Gaussian process are weighted averages of the observed data where the weighting is based on the coveriance and mean functions.
How are points close together in a Gaussian process?
Observe that points close together in the input domain of x x are strongly correlated ( y1 y 1 is close to y2 y 2 ), while points further away from eachother are almost independent.
How to use the Gaussian process in regression?
In a regression task, we have a set of training data points in pairs (X₁, Y₁), (X₂, Y₂), …, (Xₙ, Yₙ), where Xᵢ, Yᵢ, are real values. We use the symbol ℝ to denote the set of all real values. And use (X, Y) to denote the training data, where X and Y are both vectors of length n.
What to do when things are not Gaussian?
Variational Gaussian Process — What To Do When Things Are Not Gaussian introduces variational inference to allow us to use a non-Gaussian likelihood in a Gaussian Process model. Sparse and Variational Gaussian Process — What To Do When Data is Large introduces inducing variables to allow us to scale a Gaussian Process model to large datasets.