Bayesian　estimation　of　finite　Gamma　mixture　model　(GaMM)　has　attracted　considerable　attention　recently　due　to　its　capability　of　modeling　positive　data.　With　conventional　variational　inference　(VI)　frameworks,　we　cannot　derive　an　analytically　tractable　solution　for　the　variational　posterior,　since　the　expectation　of　the　joint　distribution　of　all　the　random　variables　cannot　be　estimated　in　a　closed　form.　Therefore,　numerical　techniques　are　commonly　utilized　to　simulate　the　posterior　distribution.　However,　the　optimization　process　of　these　methods　can　be　prohibitively　slow　for　practical　applications.　In　order　to　obtain　closed-form　solutions,　some　lower-bound　approximations　are　then　introduced　into　the　evidence　lower　bound　(ELBO),　following　the　recently　proposed　extended　variational　inference　(EVI)　framework.　The　problem　in　numerical　simulation　can　be　overcome.　In　this　paper,　we　address　the　Bayesian　estimation　of　the　finite　Gamma　mixture　model　(GaMM)　under　the　EVI　framework　in　a　flexible　way.　Moreover,　the　optimal　mixture　component　number　can　be　automatically　determined　based　on　the　observed　data　and　the　over-fitting　problem　related　to　the　conventional　expectation–maximization　(EM)　is　overcome.　We　demonstrate　the　excellent　performance　of　the　proposed　method　with　synthesized　data　and　real　data　evaluations.　In　the　real　data　evaluation,　we　compare　the　proposed　method　on　object　detection　and　image　categorization　tasks　with　referred　methods　and　find　statistically　significant　improvement　on　accuracies　and　runtime.　©　2020　Elsevier　B.V.