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In this paper, we develop a sparse variational Bayesian (VB) extension of the space-alternating generalized expectation-maximization (SAGE) algorithm for the high resolution estimation of the parameters of relevant multipath components in the response of frequency and spatially selective wireless channels. The application context of the algorithm considered in this contribution is parameter estimation from channel sounding measurements for radio channel modeling purpose. The new sparse VB-SAGE algorithm extends the classical SAGE algorithm in two respects: i) by monotonically minimizing the variational free energy, distributions of the multipath component parameters can be obtained instead of parameter point estimates and ii) the estimation of the number of relevant multipath components and the estimation of the component parameters are implemented jointly. The sparsity is achieved by defining parametric sparsity priors for the weights of the multipath components. We revisit the Gaussian sparsity priors within the sparse VB-SAGE framework and extend the results by considering Laplace priors. The structure of the VB-SAGE algorithm allows for an analytical stability analysis of the update expression for the sparsity parameters. This analysis leads to fast, computationally simple, yet powerful, adaptive selection criteria applied to the single multipath component considered at each iteration. The selection criteria are adjusted on a per-component-SNR basis to better account for model mismatches, e.g., diffuse scattering, calibration and discretization errors, allowing for a robust extraction of the relevant multipath components. The performance of the sparse VB-SAGE algorithm and its advantages over conventional channel estimation methods are demonstrated in synthetic single-input-multiple-output (SIMO) time-invariant channels. The algorithm is also applied to real measurement data in a multiple-input-multiple-output (MIMO) time-invariant context.