## 1 Introduction
No regularization parameter selection method is currently suitable for all problems [26], which significantly impacts severely ill-conditioned acoustic inverse problems. To improve regularization methods, Bayesian inference becomes a powerful tool for enhancing sparsity and adaptive parameter estimation. Under the Bayesian framework, an iterative algorithm is derived based on Bayesian optimization criteria such as maximum posterior probability and minimum KL divergence. This can adaptively estimate regularization parameters and significantly improve calculation accuracy while ensuring improved computational efficiency. [PERSON] [27] conducted an in-depth theoretical study on the sparse Bayesian learning (SBL) algorithm and found that it has significant advantages. The SBL has demonstrated its ability to obtain genuinely sparse solutions, showing insensitivity to high correlations between columns in the sensing matrix. Moreover, the SBL exhibits flexibility in utilizing the structural information of the solution, displaying inherent parameter adaptability. Based on the advantages of the SBL, some scholars have redefined the inverse problem of sound source identification as a Bayesian inference problem. [PERSON] [28] proposed a high-resolution DOA estimation algorithm based on spatially alternating a multi-snapshot SBL. [PERSON] [29] also proposed an SBL algorithm for multi-snapshot high-resolution DOA estimation. The difference is that this algorithm uses derivative-based evidence maximization to estimate source power and noise variance. [PERSON] [30] proposed a fast sparse reconstruction method based on Bayesian compressed sensing (BCS), which is applied to the sound field reconstruction after sparsely decomposing the sound field usingsingular value decomposition. This method applies to spatially sparsely distributed and spatially extended sources. Considering that in the process of solving the inverse problem, the mismatch between the acoustic inversion model and the actual physical model will cause uncertainty in the sound source identification problem, [PERSON] [31] compared traditional beamforming and Bayesian focusing through sensitivity analysis. The results showed that Bayesian focusing is overall more resilient to sources of uncertainty than conventional beamforming. The above sound source identification methods are all implemented by dividing the search area into discrete grids. The crucial assumption is that potential sound sources are located at the search points. However, when the sources span a wide range, it is difficult to ensure that all the sources fall precisely on the grid positions, and then the received data cannot consist of a base of candidate discrete points. The difference between the actual sound source location and the nearest grid point will produce a basis mismatch, thereby reducing the accuracy of the sound source identification estimation. A direct way to alleviate basis mismatch is to refine the mesh, but at the cost of a sharp increase in computational effort and amplification of reconstruction errors due to a high correlation between adjacent basis points. Therefore, some scholars consider optimizing the mathematical model. According to different mathematical models, compressive beamforming can be divided into fixed on-grid [32], off-grid [13], and grid-free [33] methods. The off-grid and grid-free methods are introduced below. Instead of discretizing the focal region, the grid-free method establishes source sparsity metrics (such as atomic norm [34; 35; 36; 37] and total variational norm [38]) and directly processes the focus region as a continuum, thus theoretically avoiding the basis mismatch problem. However, grid-free methods require a priori signal-to-noise ratio (SNR) information. They are more suitable for linear arrays [34] or uniform rectangular [33] arrays, requiring extensive computation to extend to arbitrary arrays [39; 40].