  An important task in machine learning and statistics is the approximation of
a probability measure by an empirical measure supported on a discrete point
set. Stein Points are a class of algorithms for this task, which proceed by
sequentially minimising a Stein discrepancy between the empirical measure and
the target and, hence, require the solution of a non-convex optimisation
problem to obtain each new point. This paper removes the need to solve this
optimisation problem by, instead, selecting each new point based on a Markov
chain sample path. This significantly reduces the computational cost of Stein
Points and leads to a suite of algorithms that are straightforward to
implement. The new algorithms are illustrated on a set of challenging Bayesian
inference problems, and rigorous theoretical guarantees of consistency are
established.
