  We study approximation schemes for shift spaces over a finite alphabet using
(pseudo)metrics connected to Ornstein's $\bar{d}$ metric. This leads to a class
of shift spaces we call $\bar{d}$-approachable. A shift space
$\bar{d}$-approachable when its canonical sequence of Markov approximations
converges to it also in the $\bar{d}$ sense. We give a topological
characterisation of chain mixing $\bar{d}$-approachable shift spaces. As an
application we provide a new criterion for entropy density of ergodic measures.
Entropy-density of a shift space means that every invariant measure $\mu$ of
such a shift space is the weak$^*$ limit of a sequence $\mu_n$ of ergodic
measures with the corresponding sequence of entropies $h(\mu_n)$ converging to
$h(\mu)$. We prove ergodic measures are entropy-dense for every shift space
that can be approximated in the $\bar{d}$ pseudometric by a sequence of
transitive sofic shifts. This criterion can be applied to many examples that
were out of the reach of previously known techniques including hereditary
$\mathscr{B}$-free shifts and some minimal or proximal systems. The class of
symbolic dynamical systems covered by our results includes also shift spaces
where entropy density was established previously using the (almost)
specification property.
