# On the Use of Structured Prior Models for Bayesian Compressive Sensing of Modulated Signals
## 2 Proposed Method
### Group Sparse Model for Modulated Signals
#### 2.3.3 Setting the Mixing Hyperparameters  
Setting the global and local parameters in the block sparse model (5) is quite important. In this paper, we propose the following hierarchical model for these parameters:  
\[\forall b\in\{1,\,\ldots,\,B\},\quad v_{b}\sim\mathcal{IG}(\alpha_{b},\beta_ {b})\] \[\forall q\in\{0,\,\ldots,\,Q\},\quad\sqrt{\gamma_{q}}|\phi_{q} \sim\mathcal{C}^{+}(0,\sqrt{\phi_{q}}),\quad\sqrt{\phi_{q}}\sim\mathcal{C}^{ +}(0,1)\]  
where \(\mathcal{IG}(\alpha_{b},\beta_{b})\) is the inverse Gamma prior with positive constant parameters \(\alpha_{b}\) and \(\beta_{b}\) and \(\mathcal{C}^{+}(0,e)\) denotes the half Cauchy distribution with location \(0\) and scale \(e\). The intuition for this choice is explained, as follows. First, the global variables \(v_{b}\) adapts the underlying sparsity in the active block: an appropriate small values of \(v_{b}\) may lead to an over-shrinking of the active coefficients especially those with small amplitudes. Thus, we preferred to set its prior in such a way to avoid vanishing values of \(v_{b}\) if active group \(b\) happensto be very sparse. This is motivated by our assumption of a known number of carrier harmonics, i.e., in each active block, there exists at least one non-zero coefficient. Note that, though, [48,50] recommend using the half Cauchy prior for both local and global parameters to improve shrinking, we found, in our experiments, that the half Cauchy prior for global parameters deteriorates the reconstruction of the small peaks as compared to the Inverse Gamma. We think that the Inverse Gamma is a suitable choice for two reasons: (1) its density vanishes in the origin that will artificially push away the values of \(v_{b}\) from zero, and (2) it is a conjugate density for the Gaussian distribution that will facilitate the calculation afterwards. Second, unlike for the global scales, the local variables \(\gamma_{q}\) are preferred to have vanishing values to appropriately model ultra-sparse signals. This is ensured while using the half Cauchy prior that, in addition to its Cauchy-like tails, has the merit when compared to the Inverse Gamma, of being non-zero at \(\gamma_{q}=0\). This allows for the prior of \(\mathbf{x}_{q}\) to have an infinitely tall spike in the origin resulting on good shrinking properties [51,52]. The hyperparameter \(\phi_{q}\) offers another degree of freedom in the mixing prior to \(\gamma_{q}\). Third, the half-Cauchy prior does not bring additional difficulty in the posterior inference as compared to Inverse Gamma using the trick of parameter expansion [48,53]. In fact, the above model can be equivalently written, as follows  
\[\forall b\in\{1,\,\ldots,\,B\}, v_{b}\sim\mathcal{IG}(\mathbf{a}_{b},\mathbf{\beta}_{b})\] \[\forall q\in\{0,\,\ldots,\,Q\}, \gamma_{q}|\eta_{q}\sim\mathcal{IG}\bigg{(}\frac{1}{2},\frac{1}{ \eta_{q}}\bigg{)},\eta_{q}|\phi_{q}\sim\mathcal{IG}\bigg{(}\frac{1}{2},\frac{ 1}{\phi_{q}}\bigg{)}\] \[\phi_{q}|\xi_{q}\sim\mathcal{IG}\bigg{(}\frac{1}{2},\frac{1}{\xi _{q}}\bigg{)},\xi_{q}\sim\mathcal{IG}\bigg{(}\frac{1}{2},1\bigg{)}\]