Joint activity detection and channel estimation is a crucial task in grant-free random access for massive machine-type communications. To enhance communication quality, intelligent reflecting surfaces (IRSs) have been proposed as a promising technology by controlling the propagation environment with passive reflecting elements. However, due to their passive nature and the non-Gaussian device-IRS-BS composite channels, IRSs introduce significant challenges for joint activity detection and channel estimation. To this end, this paper establishes an accurate statistical model for the composite channel, demonstrating that it follows a variance-gamma (VG) distribution. Based on the exact channel statistics, this paper employs a Bernoulli-VG prior and extends the standard approximate message passing algorithm to learn the Gamma-distributed channel variance within an expectation-maximization framework. Additionally, this paper introduces a novel approach to enforce consistency in device activity status across all base station antennas by transforming the activity detection task into the estimation of a dedicated active probability for each device. Extensive simulations validate the proposed VG channel model and demonstrate significant improvement due to imposing consistent activity probability across multiple antennas.

Activity detection is an important task in the next generation Internet-of-things (IoT) networks. Existing algorithms mostly require precise information about the network, such as large-scale fading, noise variance, and small-scale fading statistics. Acquiring such information would take a significant overhead and their estimated values might not be accurate. This problem is even more severe in cell-free networks as more parameters are acquired. Therefore, this paper sets out to investigate this problem without the above mentioned information. In order to handle so many unknown parameters, this paper employs a Bayesian approach, where they are endowed with prior distributions as regularizations. Together with the likelihood function, a maximum a posteriori (MAP) estimator is derived. Simulations demonstrate that the proposed method outperforms state-of-the-art methods especially under imprecise information.