Markov Chain Monte Carlo methods have revolutionised mathematical computation
and enabled statistical inference within many previously intractable models. In
this context, Hamiltonian dynamics have been proposed as an efficient way of
building chains which can explore probability densities efficiently. The method
emerges from physics and geometry and these links have been extensively studied
by a series of authors through the last thirty years. However, there is
currently a gap between the intuitions and knowledge of users of the
methodology and our deep understanding of these theoretical foundations. The
aim of this review is to provide a comprehensive introduction to the geometric
tools used in Hamiltonian Monte Carlo at a level accessible to statisticians,
machine learners and other users of the methodology with only a basic
understanding of Monte Carlo methods. This will be complemented with some
discussion of the most recent advances in the field which we believe will
become increasingly relevant to applied scientists.