  Let X_1,...., X_n be a collection of iid discrete random variables, and
Y_1,..., Y_m a set of noisy observations of such variables. Assume each
observation Y_a to be a random function of some a random subset of the X_i's,
and consider the conditional distribution of X_i given the observations, namely
\mu_i(x_i)\equiv\prob\{X_i=x_i|Y\} (a posteriori probability).
  We establish a general relation between the distribution of \mu_i, and the
fixed points of the associated density evolution operator. Such relation holds
asymptotically in the large system limit, provided the average number of
variables an observation depends on is bounded. We discuss the relevance of our
result to a number of applications, ranging from sparse graph codes, to
multi-user detection, to group testing.
