Bearing　fault　diagnosis　is　one　of　the　most　important　topics　in　the　condition-based　maintenance　and　is　also　a　challenging　problem　because　of　the　heavy　noise　interference.　Sparse　representation　methods　have　been　proven　effective　to　solve　such　a　challenging　problem,　especially　through　L1　norm　regularization　(Laplacian).　In　this　paper,　we　analyze　the　sparse　signal　model　deeply　in　the　Bayesian　aspect　and　conclude　that　the　distribution　of　the　wavelet　coefficients　(transforming　by　TQWT)　is　well　modeled　by　a　hyper-Laplacian.　However,　such　a　prior　makes　the　optimization　problem　non-convex.　We　adopt　an　effective　algorithm　called　the　generalized　shrinkage　algorithm　(GISA)　to　solve　this　non-convex　problem.　Furthermore,　we　use　the　k-sparsity　strategy　to　replace　the　regularization　parameter　for　adaptivity.　Finally,　a　simulation　study　and　a　bearing　fault　experiment　demonstrate　the　performance　of　the　proposed　GISA　with　k-sparsity,　and　the　comparison　study　shows　that　the　hyper-Laplacian　distribution　can　estimate　the　bearing　fault　information　more　accurately　than　the　Laplacian　distribution.