0 ) ▁times ▁out ▁of ▁ 1 0 0 , ▁then ▁the ▁dis cre p ancy ▁is ▁unlikely ▁to ▁be ▁explained ▁by ▁chance ▁alone ▁and ▁the ▁null - h yp oth esis ▁is ▁rejected . ▁Statist ical ▁models ▁provide ▁exact ▁out comes ▁of ▁the ▁paramet ric ▁and ▁estimates ▁of ▁the ▁sample ▁statistics . ▁Hence , ▁the ▁bur den ▁of ▁proof ▁rest s ▁in ▁the ▁sample ▁statistics ▁that ▁provide ▁estimates ▁of ▁a ▁statistical ▁model . ▁Statist ical ▁models ▁contain ▁the ▁mathematical ▁proof ▁of ▁the ▁paramet ric ▁values ▁and ▁their ▁probability ▁distributions . ▁ ▁Bay es ▁theorem ▁▁▁▁ ▁Bay esian ▁statistics ▁are ▁based ▁on ▁a ▁different ▁philosoph ical ▁approach ▁for ▁proof ▁of ▁inference . ▁The ▁mathematical ▁formula ▁for ▁Bay es ' s ▁theorem ▁is : ▁ ▁The ▁formula ▁is ▁read ▁as ▁the ▁probability ▁of ▁the ▁parameter ▁( or ▁hypothesis ▁= h , ▁as ▁used ▁in ▁the ▁notation ▁on ▁ax i oms ) ▁“ g iven ” ▁the ▁data ▁( or ▁empir ical ▁observation ), ▁where ▁the ▁horizontal ▁bar ▁refers ▁to ▁" g iven ". ▁The ▁right ▁hand ▁side ▁of ▁the ▁formula ▁calcul ates ▁the ▁prior ▁probability ▁of ▁a ▁statistical ▁model ▁( Pr ▁[ Parameter ]) ▁with ▁the ▁lik elihood ▁( Pr ▁[ Data ▁| ▁Parameter ]) ▁to ▁produce ▁a ▁posterior ▁probability ▁distribution ▁of ▁the ▁parameter ▁( Pr ▁[ Parameter ▁| ▁Data ]). ▁The ▁posterior ▁probability ▁is ▁the ▁lik elihood ▁that ▁the ▁parameter ▁is ▁correct ▁given ▁the ▁observed ▁data ▁or ▁samples ▁statistics . ▁Hyp oth eses ▁can ▁be ▁compared ▁using ▁Bay esian ▁inference ▁by ▁means ▁of