In wideband spectrum sensing, compressive sensing approaches have been used at the receiver side to decrease the sampling rate, if the wideband signal can be represented as sparse in a given domain. While most studies consider the reconstruction of primary user's signal accurately, it is indeed more important to analyze the presence or absence of the signal correctly. Furthermore, these studies do not consider the achievable lower bounds of reconstruction error and how well the selected method performs correspondingly. Motivated by these issues, we investigate in detail the primary user detection performance of Bayesian compressive sensing (BCS) approach in this paper. Accordingly, we (i) determine the BCS signal reconstruction performance in terms of mean-square error (MSE), compression ratio and signal-to-noise ratio (SNR), and compare it with the conventionally used basis pursuit approach, (ii) determine how well BCS performs compared with the Bayesian Cramer-Rao lower bound (BCRLB) of the signal reconstruction error, and (iii) assess the probability of detection performance of BCS for various SNR and compression ratio values. The results of this study are important for determining the achievable performance of BCS based spectrum sensing.