  In this paper, we consider a general observation model for restless
multi-armed bandit problems. The operation of the player needs to be based on
certain feedback mechanism that is error-prone due to resource constraints or
environmental or intrinsic noises. By establishing a general probabilistic
model for dynamics of feedback/observation, we formulate the problem as a
restless bandit with a countable belief state space starting from an arbitrary
initial belief (a priori information). We apply the achievable region method
with partial conservation law (PCL) to the infinite-state problem and analyze
its indexability and priority index (Whittle index). Finally, we propose an
approximation process to transform the problem into which the AG algorithm of
Ni\~no-Mora and Bertsimas for finite-state problems can be applied to.
Numerical experiments show that our algorithm has an excellent performance.
