# Sparse Plane Wave Approximation of Acoustic Modes to Address Basis Mismatch
## 3 Proposed Combined Two-Stage Method
### Stage 2: Estimation of PWAs and Directions Using Parametric SBL
#### 3.2.3 Hidden Random Variable Inference  
We intended to obtain the posterior \(P(\mathbf{w},\tau,\mathbf{\kappa}|\mathbf{\psi})\) to evaluate inferences regarding hidden variables \(\mathcal{H}=\{\mathbf{w},\tau,\mathbf{\kappa}\}\) with the probabilistic model. However, the closed-form expression of the posterior is generally intractable owing to the difficulty in calculating the multidimensional integral when computing the model evidence \(P(\mathbf{\psi})\). Therefore, we resorted to the VBEM method, which involves recursively approximating the posterior to infer \(\mathcal{H}\) at the expectation step and optimizing \(\mathbf{\vartheta}\) and \(\mathbf{\varphi}\) at the maximization step until convergence. We assumed a distribution \(Q(\mathcal{H})\) with the factorized form in the expectation calculation step, that is, \(Q(\mathcal{H})=Q_{1}(\mathbf{w})Q_{2}(\tau)Q_{3}(\mathbf{\kappa})\). The posterior \(P(\mathcal{H}|\mathbf{\psi})\) can then be  
Figure 1: Probabilistic model of parametric SBL represented as a Bayesian network (for conventional SBL, the parameters \(a,b\), \(c\), and \(d\), are often set to be very small values, such as \(10^{-6}\), in this study). approximated by minimizing the KL divergence between \(P(\mathcal{H}|\mathbf{\psi})\) and \(Q(\mathcal{H})\) for given parameters \(\mathbf{\vartheta}\) and \(\mathbf{\varphi}\), which is equivalent to solving the following equation  
\[\ln\hat{Q}_{j}(\mathcal{H}_{j})=\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_ {j})}[\ln P(\mathcal{H},\mathbf{\psi})], \tag{13}\]  
where \(\mathbb{E}_{Q(\mathcal{H}\setminus\mathcal{H}_{j})}[\cdot]\) represents the expectation of \(\prod_{i\
eq j}Q_{i}(\mathcal{H}_{i})\). Substituting the priors of \(\mathcal{H}\), the updates of each \(\mathcal{H}\) at the \(i\)th iteration can be summarized as follows [26]:  
\[\mathbf{\alpha}^{(i)} =(c+1)\oslash[d+\text{diag}(\mathbf{\mu}^{(i-1)}(\mathbf{\mu}^{(i-1)})^{ \text{H}}+\mathbf{\Sigma}^{(i-1)})], \tag{14}\] \[\tau^{(i)} =\frac{a+M}{b+||\mathbf{\psi}-\mathbf{H}(\mathbf{\theta},\mathbf{\varphi})\mathbf{ \mu}^{(i-1)}||_{2}^{2}+\text{Tr}(\mathbf{H}(\mathbf{\theta},\mathbf{\varphi})\mathbf{\Sigma}^ {(i-1)}(\mathbf{H}(\mathbf{\theta},\mathbf{\varphi}))^{\text{H}})},\] (15) \[\mathbf{\Sigma}^{(i)} =[\tau^{(i)}(\mathbf{H}(\mathbf{\vartheta},\mathbf{\varphi}))^{\text{H}}\mathbf{ H}(\mathbf{\vartheta},\mathbf{\varphi})+\text{diag}(\mathbf{\alpha}^{(i)})]^{-1},\] (16) \[\mathbf{w}^{(i)} =\mathbf{\mu}^{(i)}=\tau^{(i)}\mathbf{\Sigma}^{(i)}(\mathbf{H}(\mathbf{\vartheta},\mathbf{\varphi}))^{\text{H}}\mathbf{\psi}, \tag{17}\]  
where \(\oslash\) denotes the component-wise division, and \(\text{Tr}(\cdot)\) represents the trace operation. At the maximization step, we estimated the unknown parameters \(\mathbf{\vartheta}\) and \(\mathbf{\varphi}\) separately by maximizing the expected log-likelihood for one parameter while keeping the other fixed. This is equivalent to solving the equation below,  
\[\hat{\mathbf{x}}=\arg\min_{\mathbf{x}}\{\mathcal{L}(\mathbf{x})\},\] \[\mathcal{L}(\mathbf{x})=||\mathbf{\psi}-\mathbf{H}(\mathbf{x})\mathbf{\mu}^{(i)}||_{ 2}^{2}+\text{Tr}(\mathbf{H}(\mathbf{x})\mathbf{\Sigma}^{(i)}(\mathbf{H}(\mathbf{x}))^{\text{H}}), \tag{18}\]  
where \(\mathbf{x}\) represents \(\mathbf{\vartheta}\) or \(\mathbf{\varphi}\). We used [PERSON]'s method to solve this nonlinear problem.