  We study the notion of the scaled entropy of a filtration of $\sigma$-fields
(= decreasing sequence of $\sigma$-fields) introduced by the first author
({V4}). We suggest a method for computing this entropy for the sequence of
$\sigma$-fields of pasts of a Markov process determined by a random walk over
the trajectories of a Bernoulli action of a commutative or nilpotent countable
group (Theorems~5,~6). Since the scaled entropy is a metric invariant of the
filtration, it follows that the sequences of $\sigma$-fields of pasts of random
walks over the trajectories of Bernoulli actions of lattices (groups ${\Bbb
Z}^d$) are metrically nonisomorphic for different dimensions $d$, and for the
same $d$ but different values of the entropy of the Bernoulli scheme. We give a
brief survey of the metric theory of filtrations, in particular, formulate the
standardness criterion and describe its connections with the scaled entropy and
the notion of a tower of measures.
