  This paper concerns the Bayesian approach to inverse acoustic scattering
problems of inferring the position and shape of a sound-soft obstacle from
phaseless far-field data generated by point source waves. To improve the
convergence rate, we use the Gibbs sampling and preconditioned Crank-Nicolson
(pCN) algorithm with random proposal variance to implement the Markov chain
Monte Carlo (MCMC) method. This usually leads to heavy computational cost,
since the unknown obstacle is parameterized in high dimensions. To overcome
this challenge, we examine a surrogate model constructed by the generalized
polynomial chaos (gPC) method to reduce the computational cost. Numerical
examples are provided to illustrate the effectiveness of the proposed method.
