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# 10 Gaussian Processes

Gaussian processes are continuous stochastic processes and thus may be interpreted as providing a probability distribution over functions. A probability distribution over continuous functions may be viewed, roughly, as an uncountably infinite collection of random variables, one for each valid input. The generality of the supported functions makes Gaussian priors popular choices for priors in general multivariate (non-linear) regression problems.

The defining feature of a Gaussian process is that the joint distribution of the function’s value at a finite number of input points is a multivariate normal distribution. This makes it tractable to both fit models from finite amounts of observed data and make predictions for finitely many new data points.

Unlike a simple multivariate normal distribution, which is parameterized by a mean vector and covariance matrix, a Gaussian process is parameterized by a mean function and covariance function. The mean and covariance functions apply to vectors of inputs and return a mean vector and covariance matrix which provide the mean and covariance of the outputs corresponding to those input points in the functions drawn from the process.

Gaussian processes can be encoded in Stan by implementing their mean and covariance functions and plugging the result into the Gaussian form of their sampling distribution, or by using the specialized covariance functions outlined below. This form of model is straightforward and may be used for simulation, model fitting, or posterior predictive inference. A more efficient Stan implementation for the GP with a normally distributed outcome marginalizes over the latent Gaussian process, and applies a Cholesky-factor reparameterization of the Gaussian to compute the likelihood and the posterior predictive distribution analytically.

After defining Gaussian processes, this chapter covers the basic implementations for simulation, hyperparameter estimation, and posterior predictive inference for univariate regressions, multivariate regressions, and multivariate logistic regressions. Gaussian processes are general, and by necessity this chapter only touches on some basic models. For more information, see Rasmussen and Williams (2006).