▁that ▁an ▁R BF ▁network ▁with ▁enough ▁hidden ▁neur ons ▁can ▁approximate ▁any ▁continuous ▁function ▁on ▁a ▁closed , ▁bounded ▁set ▁with ▁arbitrary ▁precision . ▁ ▁The ▁parameters ▁, ▁, ▁and ▁ ▁are ▁determined ▁in ▁a ▁manner ▁that ▁optim izes ▁the ▁fit ▁between ▁ ▁and ▁the ▁data . ▁ ▁Normal ized ▁ ▁Normal ized ▁architecture ▁In ▁addition ▁to ▁the ▁above ▁un normal ized ▁architecture , ▁R BF ▁networks ▁can ▁be ▁normal ized . ▁In ▁this ▁case ▁the ▁mapping ▁is ▁ ▁where ▁ ▁is ▁known ▁as ▁a ▁" normal ized ▁radial ▁basis ▁function ". ▁ ▁The oret ical ▁motiv ation ▁for ▁normal ization ▁There ▁is ▁theoretical ▁just ification ▁for ▁this ▁architecture ▁in ▁the ▁case ▁of ▁st ochastic ▁data ▁flow . ▁Assume ▁a ▁st ochastic ▁kernel ▁approximation ▁for ▁the ▁joint ▁probability ▁density ▁ ▁where ▁the ▁weights ▁▁ ▁and ▁ ▁are ▁exempl ars ▁from ▁the ▁data ▁and ▁we ▁require ▁the ▁k ern els ▁to ▁be ▁normal ized ▁ ▁and ▁. ▁ ▁The ▁probability ▁dens ities ▁in ▁the ▁input ▁and ▁output ▁spaces ▁are ▁ ▁and ▁ ▁The ▁expectation ▁of ▁y ▁given ▁an ▁input ▁ ▁is ▁ ▁where ▁ ▁is ▁the ▁conditional ▁probability ▁of ▁y ▁given ▁. ▁The ▁conditional ▁probability ▁is ▁related ▁to ▁the ▁joint ▁probability ▁through ▁Bay es ▁theorem ▁ ▁which ▁yields ▁ ▁. ▁ ▁This ▁becomes ▁ ▁when ▁the ▁integr ations ▁are ▁performed . ▁ ▁Local ▁linear ▁models ▁It ▁is ▁sometimes ▁convenient ▁to ▁expand ▁the ▁architecture ▁to ▁include ▁local ▁linear ▁models . ▁In ▁that ▁case ▁the ▁architect ures ▁become , ▁to ▁first ▁order , ▁ ▁and