  In this paper we are concerned with understanding the nature of program
metrics for calculi with higher-order types, seen as natural generalizations of
program equivalences. Some of the metrics we are interested in are well-known,
such as those based on the interpretation of terms in metric spaces and those
obtained by generalizing observational equivalence. We also introduce a new
one, called the interactive metric, built by applying the well-known
Int-Construction to the category of metric complete partial orders. Our aim is
then to understand how these metrics relate to each other, i.e., whether and in
which cases one such metric refines another, in analogy with corresponding
well-studied problems about program equivalences. The results we obtain are
twofold. We first show that the metrics of semantic origin, i.e., the
denotational and interactive ones, lie \emph{in between} the observational and
equational metrics and that in some cases, these inclusions are strict. Then,
we give a result about the relationship between the denotational and
interactive metrics, revealing that the former is less discriminating than the
latter. All our results are given for a linear lambda-calculus, and some of
them can be generalized to calculi with graded comonads, in the style of Fuzz.
