  We study the sign problem in lattice field theory with a $\theta$ term. We
apply the maximum entropy method (MEM) to flattening phenomenon of the free
energy density $f(\theta)$, which originates from the sign problem. In our
previous paper, we applied the MEM by employing the Gaussian topological charge
distribution $P(Q)$ as mock data. In the present paper, we consider models in
which `true' flattening of $f(\theta)$ occurs. These may be regarded as good
examples for studying whether the MEM could correctly detect non trivial phase
structure.
