The concept of reconfiguring wireless propagation environments using intelligent reflecting surfaces (IRS)s has recently emerged, where an IRS comprises of a large number of passive reflecting elements that can smartly reflect the impinging electromagnetic waves for performance enhancement. Previous works have shown promising gains assuming the availability of perfect channel state information (CSI) at the base station (BS) and the IRS, which is impractical due to the passive nature of the reflecting elements. This paper makes one of the preliminary contributions of studying an IRS-assisted multi-user multiple-input single-output (MISO) communication system under imperfect CSI. Different from the few recent works that develop least-squares (LS) estimates of the IRS-assisted channel vectors, we exploit the prior knowledge of the large-scale fading statistics at the BS to derive the Bayesian minimum mean squared error (MMSE) channel estimates under a protocol in which the IRS applies a set of optimal phase shifts vectors over multiple channel estimation sub-phases. The resulting mean squared error (MSE) is both analytically and numerically shown to be lower than that achieved by the LS estimates. Joint designs for the precoding and power allocation at the BS and reflect beamforming at the IRS are proposed to maximize the minimum user signal-to-interference-plus-noise ratio (SINR) subject to a transmit power constraint. Performance evaluation results illustrate the efficiency of the proposed system and study its susceptibility to channel estimation errors.