now lets add them together but also add a multiplicative constant #c# to one of them ##Eq(d,d_1+c*d_2)## depending on #c# the auc of the addition chances There is an optimum value of c and if you use a value of c that is way to large, it can actually hurt your auc so assume: #Eq(c,1)#(unweighted addition) is a #c# that is way to big for toptagging so lets calculate the perfect c for a given distribution auc as function of c %show animation here ##Eq(mu_1B,0),Eq(mu_2B,0),Eq(mu_1S,1),Eq(mu_2S,c*alpha)## ##Eq(sigma_iB,sigma_iS),Eq(sigma_1,s_1),Eq(sigma_2,alpha*c*s_2)## ##Eq(mu_B,0),Eq(mu_S,1+c*alpha),Eq(sigma,sqrt(sigma_1**2+sigma_2**2))## fix the scale by demanding #Eq(mu_S,1)#, then maximum auc means minimum #sigma# (or #(sigma/s1)**2#) ##Eq((sigma/s1)**2,(1+(s_2/s_1)**2*alpha**2*c**2)/(1+alpha*c))## ##Eq(d/dc * (sigma/s1)**2,0)## ##Eq((1/(1+alpha*c)**3)*2*y*(c*alpha*(s_2/s_1)**2-1),0)## ##Eq(c,1/(alpha*(s_2/s_1)**2))## ##Eq(alpha,1.0),Eq(s_2,0.75),Eq(s_1,0.5)## compare to numerics: ##Eq(c,0.4444),Eq(c_n,0.4436),Eq(sigma_c_n,0.0024)##