  We develop a framework for spectrum sensing in cooperative
amplify-and-forward cognitive radio networks. We consider a stochastic model
where relays are assigned in cognitive radio networks to transmit the primary
user's signal to a cognitive Secondary Base Station (SBS). We develop the
Bayesian optimal decision rule under various scenarios of Channel State
Information (CSI) varying from perfect to imperfect CSI. In order to obtain the
optimal decision rule based on a Likelihood Ratio Test (LRT), the marginal
likelihood under each hypothesis relating to presence or absence of
transmission needs to be evaluated pointwise. However, in some cases the
evaluation of the LRT can not be performed analytically due to the
intractability of the multi-dimensional integrals involved. In other cases, the
distribution of the test statistic can not be obtained exactly. To circumvent
these difficulties we design two algorithms to approximate the marginal
likelihood, and obtain the decision rule. The first is based on Gaussian
Approximation where we quantify the accuracy of the approximation via a
multivariate version of the Berry-Esseen bound. The second algorithm is based
on Laplace approximation for the marginal likelihood, which results in a
non-convex optimisation problem which is solved efficiently via Bayesian
Expectation-Maximisation method. We also utilise a Laguerre series expansion to
approximate the distribution of the test statistic in cases where its
distribution can not be derived exactly. Performance is evaluated via analytic
bounds and compared to numerical simulations.
