## 2 Pattern-Coupled Bayesian Model for Sound Field Reconstruction
### Pattern-Coupled Hierarchical Bayesian Model  
Given the measured sound pressure field \(\widetilde{\mathbf{P}}\), Bayesian inference is employed in this work to estimate the coefficients \(\mathbf{W}\) through the posterior distribution \(p\left(\mathbf{W}\middle|\widetilde{\mathbf{P}}\right)\). [PERSON] rule enables the computation of this posterior by combing the likelihood \(p\left(\widetilde{\mathbf{P}}\middle|\mathbf{W}\right)\), the prior \(p(\mathbf{W})\), and the evidence \(p\left(\widetilde{\mathbf{P}}\right)\). In this context, the evidence \(p\left(\widetilde{\mathbf{P}}\right)\) is a constant term with respect to the coefficients \(\mathbf{W}\) and can be treated as a normalization factor. Consequently, the posterior \(p\left(\mathbf{W}\middle|\widetilde{\mathbf{P}}\right)\) is determined by the product of the likelihood \(p\left(\widetilde{\mathbf{P}}\middle|\mathbf{W}\right)\) and the prior \(p(\mathbf{W})\), allowing for an efficient estimation of \(\mathbf{W}\). Assuming the noise \(\mathbf{E}\) follows a Gaussian distribution with a mean of 0 and a variance of \(\sigma_{e}^{2}\), where \(\sigma_{e}\) is estimated from the measured background noise. According to the principle of maximum entropy, the likelihood \(p\left(\widetilde{\mathbf{P}}\middle|\mathbf{W}\right)\) is modeled as an independent and identically distributed (iid) Gaussian distribution:  
\[p\left(\widetilde{\mathbf{P}}\middle|\mathbf{W}\right)=\prod_{m=1}^{M}\prod_{n= 1}^{N}\mathcal{N}\left(\widetilde{p}_{m,n}\middle|\left(\mathbf{D}^{\prime} \star\mathbf{W}\right)_{m,n^{\prime}}\sigma_{e}^{2}\right), \tag{8}\]  
where \(\left(\mathbf{D}^{\prime}\star\mathbf{W}\right)_{m,n}\) denotes the element of \(\mathbf{D}^{\prime}\star\mathbf{W}\) at the \(m\)th row and \(n\)th column. Within the SBL framework, the coefficients \(\mathbf{W}\) are assigned a Gaussian prior distribution. Eachcoefficient \(w_{l,n}\) is modeled with a mean of zero and an independent variance \(\sigma_{l,n}^{2}\), expressed in terms of a set of non-negative hyperparameters \(\gamma_{l,n}\), where \(\sigma_{l,n}^{2}=\gamma_{l,n}^{-1}\). For clarity, the hyperparameters are organized into a matrix \(\mathbf{\gamma}\in\mathbb{R}^{L\times N}\). These hyperparameters \(\mathbf{\gamma}\) control the sparsity of \(\mathbf{W}\), enabling the SBL framework to enforce structured sparsity effectively. Following the pattern-coupled structured SBL method [32], the hyperparameters \(\gamma_{l,n}\) are replaced by \(\widetilde{\gamma}_{l,n}\):  
\[p(\mathbf{W}|\mathbf{\gamma})=\prod_{l=1}^{L}\prod_{n=1}^{N}\mathcal{N}\Big{(}w_{l,n} \Big{|}0,\widehat{\gamma}_{l,n}^{-1}\Big{)}. \tag{9}\]  
The reason for this replacement is that \(\widetilde{\gamma}_{l,n}\) can better capture spatial and temporal dependencies, enabling the model to reflect complex coupling structures. Unlike conventional SBL, \(\widetilde{\gamma}_{l,n}\) is not entirely independent but instead depends on spatial and temporal relationships, which can be represented using various coupling structures, as discussed in the next section. For the hyperparameters \(\mathbf{\gamma}\), independent Gamma distributions are used as priors, expressed as:  
\[p(\mathbf{\gamma})=\prod_{l=1}^{L}\prod_{n=1}^{N}\text{Gamma}(\gamma_{l,n}|\alpha, \beta_{n})\,, \tag{10}\]  
where \(\alpha\) is a fixed parameter, and \(\beta_{n}\) is a function of the temporal index \(n\), as detailed in [32; 34]. Using [PERSON] theorem, the joint posterior distribution of \(\mathbf{W}\) and \(\mathbf{\gamma}\) given the measured data \(\widetilde{\mathbf{P}}\) is derived as:  
\[p\Big{(}\mathbf{W},\mathbf{\gamma}\Big{|}\widetilde{\mathbf{P}}\Big{)}=\frac{p\Big{(} \widetilde{\mathbf{P}}\Big{|}\mathbf{W}\Big{)}p(\mathbf{W}|\mathbf{\gamma})\,p(\mathbf{\gamma})}{ p\Big{(}\widetilde{\mathbf{P}}\Big{)}}, \tag{11}\]  
allowing the inference of both \(\mathbf{W}\) and \(\mathbf{\gamma}\).