  As Gaussian processes are used to answer increasingly complex questions,
analytic solutions become scarcer and scarcer. Monte Carlo methods act as a
convenient bridge for connecting intractable mathematical expressions with
actionable estimates via sampling. Conventional approaches for simulating
Gaussian process posteriors view samples as draws from marginal distributions
of process values at finite sets of input locations. This distribution-centric
characterization leads to generative strategies that scale cubically in the
size of the desired random vector. These methods are prohibitively expensive in
cases where we would, ideally, like to draw high-dimensional vectors or even
continuous sample paths. In this work, we investigate a different line of
reasoning: rather than focusing on distributions, we articulate Gaussian
conditionals at the level of random variables. We show how this pathwise
interpretation of conditioning gives rise to a general family of approximations
that lend themselves to efficiently sampling Gaussian process posteriors.
Starting from first principles, we derive these methods and analyze the
approximation errors they introduce. We, then, ground these results by
exploring the practical implications of pathwise conditioning in various
applied settings, such as global optimization and reinforcement learning.
