Deep　belief　network　(DBN)　is　an　effective　learning　model　based　on　deep　learning.　It　can　hierarchically　transform　the　input　data　via　stacked　feature　detectors.　As　a　predictive　model,　DBN　has　shown　a　promising　prospect　in　model　predictive　control　(MPC).　However,　its　successful　application　relies　seriously　on　the　suitable　structure　size　(the　numbers　of　hidden　layers　and　neurons),　which　is　challenging　to　determine.　In　this　work,　we　present　a　theoretical　bound　on　its　minimum　structure　size　in　order　to　accurately　approximate　a　desired　control　law　of　MPC,　called　DBN-MPC.　First,　according　to　the　Markov　assumption,　a　controlled　system　is　equivalent　to　a　quadratic　program,　which　only　depends　on　the　current　system　state.　Second,　a　universal　theorem　is　proposed　to　give　a　bound　on　the　minimum　structure　size　of　DBN　from　the　perspective　of　piecewise　affine　function　analysis.　Third,　a　partial　least　square　regression　is　used　to　fine-tune　DBN　to　overcome　the　problems　of　local-minimum　and　time-consuming　training　process.　Finally,　we　demonstrate　the　effectiveness　of　the　proposed　method　through　two　classical　experiments:　1)　tracking　control　of　a　benchmark　dynamical　system　and　2)　temperature　control　of　a　practical　second-order　continuous　stirred　tank　reactor　(CSTR)　system.　The　experimental　results　generally　give　an　answer　to　the　question　that　how　deep　is　deep　enough　for　DBN　to　approximate　an　MPC　law.