▁one ▁of ▁the ▁arguments ▁approach ▁negative ▁infinity , ▁and ▁the ▁upper ▁bound ▁is ▁met ▁when ▁all ▁the ▁arguments ▁are ▁equal . ▁ ▁Writing ▁ ▁the ▁partial ▁derivatives ▁are : ▁ ▁This ▁can ▁be ▁calculated ▁via ▁log arith mic ▁different iation . ▁ ▁Express ing ▁the ▁partial ▁derivatives ▁as ▁a ▁vector ▁with ▁the ▁gradient ▁yields ▁the ▁soft max ▁function , ▁the ▁mult ivari able ▁analog ▁of ▁the ▁log istic ▁function . ▁ ▁The ▁convex ▁conjug ate ▁of ▁Log Sum Exp ▁is ▁the ▁negative ▁entropy . ▁ ▁log - sum - exp ▁trick ▁for ▁log - domain ▁calculations ▁The ▁L SE ▁function ▁is ▁often ▁encountered ▁when ▁the ▁usual ▁arithmetic ▁comput ations ▁are ▁performed ▁on ▁a ▁log arith mic ▁scale , ▁as ▁in ▁log ▁probability . ▁ ▁Similar ▁to ▁multiplication ▁operations ▁in ▁linear - scale ▁becoming ▁simple ▁add itions ▁in ▁log - scale , ▁an ▁addition ▁operation ▁in ▁linear - scale ▁becomes ▁the ▁L SE ▁in ▁log - scale . ▁ ▁A ▁common ▁purpose ▁of ▁using ▁log - domain ▁comput ations ▁is ▁to ▁increase ▁accuracy ▁and ▁avoid ▁under flow ▁and ▁overflow ▁problems ▁when ▁very ▁small ▁or ▁very ▁large ▁numbers ▁are ▁represented ▁directly ▁( i . e . ▁in ▁a ▁linear ▁domain ) ▁using ▁limited - prec ision ▁floating ▁point ▁numbers . ▁ ▁Unfortunately , ▁the ▁use ▁of ▁L SE ▁directly ▁in ▁this ▁case ▁can ▁again ▁cause ▁overflow / under flow ▁problems . ▁Therefore , ▁the ▁following ▁equivalent ▁must ▁be ▁used ▁instead ▁( es pecially ▁when ▁the ▁accuracy ▁of ▁the ▁above ▁' max ' ▁approximation ▁is ▁not