Lower　bound　estimation　plays　an　important　role　for　establishing　the　minimax　risk.　A　key　step　in　lower　bound　estimation　is　deriving　a　lower　bound　of　the　affinity　between　two　probability　measures.　This　paper　provides　a　simple　method　to　estimate　the　affinity　between　mixture　probability　measures.　Then　we　apply　the　lower　bound　of　the　affinity　to　establish　the　minimax　lower　bound　for　a　family　of　sparse　covariance　matrices,　which　contains　Cai-Ren-Zhou＇s　theorem　in　[T.　Cai,　Z.　Ren　and　H.　Zhou,　Estimating　structured　high-dimensional　covariance　and　precision　matrices:　Optimal　rates　and　adaptive　estimation,　Electron.　J.　Stat.　10(1)　(2016)　1-59]　as　a　special　example.