  Lattice field theory with the $\theta$ term suffers from the sign problem.
The sign problem appears as flattening of the free energy.
  As an alternative to the conventional method, the Fourier transform method
  (FTM), we apply the maximum entropy method (MEM) to Monte Carlo data obtained
using the CP$^3$ model with the $\theta$ term.
  For data without flattening, we obtain the most probable images of the
partition function ${\hat{\cal Z}}(\theta)$ with rather small errors. The
results are quantitatively close to the result obtained with the
  FTM. Motivated by this fact, we systematically investigate flattening in
terms of the MEM.
  Obtained images ${\hat{\cal Z}}(\theta)$ are consistent with the FTM for
small values of $\theta$, while the behavior of ${\hat{\cal Z}}(\theta)$
depends strongly on the default model for large values of $\theta$.
  This behavior of ${\hat{\cal Z}}(\theta)$ reflects the flattening phenomenon.
