  We present local mappings that relate the marginal probabilities of a global
probability mass function represented by its primal normal factor graph to the
corresponding marginal probabilities in its dual normal factor graph. The
mapping is based on the Fourier transform of the local factors of the models.
Details of the mapping are provided for the Ising model, where it is proved
that the local extrema of the fixed points are attained at the phase transition
of the two-dimensional nearest-neighbor Ising model. The results are further
extended to the Potts model, to the clock model, and to Gaussian Markov random
fields. By employing the mapping, we can transform simultaneously all the
estimated marginal probabilities from the dual domain to the primal domain (and
vice versa), which is advantageous if estimating the marginals can be carried
out more efficiently in the dual domain. An example of particular significance
is the ferromagnetic Ising model in a positive external magnetic field. For
this model, there exists a rapidly mixing Markov chain (called the
subgraphs--world process) to generate configurations in the dual normal factor
graph of the model. Our numerical experiments illustrate that the proposed
procedure can provide more accurate estimates of marginal probabilities of a
global probability mass function in various settings.
