  We derive effective actions for SU(2) Polyakov loops using inverse Monte
Carlo techniques. In a first approach, we determine the effective couplings by
requiring that the effective ensemble reproduces the single-site distribution
of the Polyakov loops. The latter is flat below the critical temperature
implying that the (untraced) Polyakov loop is distributed uniformly over its
target space, the SU(2) group manifold. This allows for an analytic
determination of the Binder cumulant and the distribution of the mean-field,
which turns out to be approximately Gaussian. In a second approach, we employ
novel lattice Schwinger-Dyson equations which reflect the SU(2) x SU(2)
invariance of the functional Haar measure. Expanding the effective action in
terms of SU(2) group characters makes the numerics sufficiently stable so that
we are able to extract a total number of 14 couplings. The resulting action is
short-ranged and reproduces the Yang-Mills correlators very well.
