# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## Appendix A Hierarchical Univariate Models  
Hierarchical prior models proposed for Bayesian CS in the literature can be categorized into two families. First, continuous models propose a relaxation of the zeroness requirement of sparse signals by using a density with sharp peak at zero to promote values close to zero and heavy tails to allow few large peaks. These models have comparable shrinking behavior as \(\ell^{p}\) norm for \(p>0\). The most popular hierarchical model used in this first family is the scale mixtures of Gaussian distributions. Such a model writes as a Gaussian density with random variance following some mixing distribution. The hierarchical model of Fourier coefficients under such prior writes as  
\[x_{k}|\lambda_{k} \sim\mathcal{CN}(0,\lambda_{k}) \tag{1}\] \[\lambda_{k} \sim\mathcal{P}_{m} \tag{2}\]  
where \(\mathcal{CN}\) (\(\mathcal{CN}\) is used since the Fourier coefficients are complex; \(u\sim\mathcal{CN}(0,a)\) means that the real and the imaginary parts of \(u\) follow the Gaussian distribution of zero mean and variance \(a\).) is the complex normal distribution and \(\lambda_{k}\geqslant 0\) is the mixing parameter following some mixing distribution \(\mathcal{P}_{m}\). A wide large family of heavy tailed multivariate scale mixtures of Gaussians used for sparse recovery can be found with appropriate choices of \(\mathcal{P}_{m}\). Commonly used ones are the discrete mixture [59], the exponential mixing that implies a Laplacian prior for \(x_{k}\)[60], the inverse Gamma mixing resulting in a Studentprior for \(x_{k}\)[47] while the Horseshoe and the Horsehoe+ priors were derived using half Cauchy mixing distributions for the standard deviations [61; 62]. The second family are discrete spike-and-slab priors. The latter enforce explicitly some values of the signal to be exactly zero via the spike component while non-zero coefficients are modeled with a continuous model such as an Uniform distribution or a Gaussian distribution with a large variance [54; 63]. Under the two-component spike-and-slab, the prior hierarchical model for each Fourier coefficient reads  
\[x_{k}|z_{k},\lambda_{k}\sim(1-z_{k})\delta_{0}+z_{k}\mathcal{CN}(0,\lambda_{k}) \tag{11}\]  
\[z_{k}|\beta_{k}\sim\mathcal{B}(\beta_{k}) \tag{12}\]  
where \(\delta_{0}\) is the point mass distribution at \(0\) and \(z_{k}\) is a binary variable following Bernoulli distribution with probability \(\beta_{k}\leqslant 0\) (i.e., \(p(z_{k}=1)=\beta_{k}\)). The binary spike and slab model has been extensively used for Bayesian variable selection [59; 63; 64], factors regression [65] and compressive sensing [54; 55; 57]. The spike component which is the mass distribution concentrated at zero, shrinks small values towards zero, while the slab component is a Gaussian density allowing plausible values for non-zero components controlled by the variance \(\lambda_{k}\). The selection variable \(z_{k}\) is an indicator of the sparsity level in the spectrum. The variable \(\beta_{k}\) controls the prior probability of a non-zero element \(x_{k}\); a large value of \(\beta_{k}\) tends to produce a nonzero value with a large probability whereas a small value of \(\beta_{k}\) tends to generate a zero value. Although the binary spike-and-slab offers a good representation for strongly-sparse signals (many coefficients of the signal are exactly zero), they may suffer from several computational difficulties compared to continuous models [64].