  This paper is concerned with collective variables, or reaction coordinates,
that map a discrete-in-time Markov process $X_n$ in $\mathbb{R}^d$ to a (much)
smaller dimension $k \ll d$. We define the effective dynamics under a given
collective variable map $\xi$ as the best Markovian representation of $X_n$
under $\xi$. The novelty of the paper is that it gives strict criteria for
selecting optimal collective variables via the properties of the effective
dynamics. In particular, we show that the transition density of the effective
dynamics of the optimal collective variable solves a relative entropy
minimization problem from certain family of densities to the transition density
of $X_n$. We also show that many transfer operator-based data-driven numerical
approaches essentially learn quantities of the effective dynamics. Furthermore,
we obtain various error estimates for the effective dynamics in approximating
dominant timescales / eigenvalues and transition rates of the original process
$X_n$ and how optimal collective variables minimize these errors. Our results
contribute to the development of theoretical tools for the understanding of
complex dynamical systems, e.g. molecular kinetics, on large timescales. These
results shed light on the relations among existing data-driven numerical
approaches for identifying good collective variables, and they also motivate
the development of new methods.
