  Terms of Church's $\lambda$-calculus can be considered equivalent along many
different definitions, but context equivalence is certainly the most direct and
universally accepted one. If the underlying calculus becomes probabilistic,
however, equivalence is too discriminating: terms which have totally unrelated
behaviours are treated the same as terms which behave very similarly. We study
the problem of evaluating the distance between affine $\lambda$-terms. The most
natural definition for it, namely a natural generalisation of context
equivalence, is shown to be characterised by a notion of trace distance, and to
be bounded from above by a coinductively defined distance based on the
Kantorovich metric on distributions. A different, again fully-abstract,
tuple-based notion of trace distance is shown to be able to handle nontrivial
examples.
