The formalism of Wiener filtering is developed here for the purpose of
reconstructing the large-scale structure of the universe from noisy,
sparse, and incomplete data. The method is based on a linear minimum v
ariance solution, given data and an assumed prior model which specifie
s the covariance matrix of the held to be reconstructed While earlier
applications of the Wiener filer have focused on estimation, namely su
ppressing the noise in the measured quantities, we extend the method h
ere to perform both prediction and dynamical reconstruction. The Wiene
r filter is used to predict the values of unmeasured quantities, such
as the density held in unsampled regions of space, or to deconvolve bl
urred data The method is developed, within the context of linear gravi
tational instability theory, to perform dynamical reconstruction of on
e held which is dynamically related to some other observed held. This
is the case, for example, in the reconstruction of the real space gala
xy distribution from its redshift distribution or the prediction of th
e radial velocity held from the observed density field. When the field
to be reconstructed is a Gaussian random held, such as the primordial
perturbation field predicted by the canonical model of cosmology, the
Wiener filter can be pushed to its fullest potential. In such a case
the Wiener estimator coincides with the Bayesian estimator designed to
maximize the posterior probability. The Wiener filter can be also der
ived by assuming a quadratic regularization function, in analogy with
the ''maximum entropy'' method. The mean field obtained by the minimal
variance solution can be supplemented with constrained realizations o
f the Gaussian held to create random recitations of the residual from
the mean.