Listing A2: Calculate symbolically the exact Chernoff information between two univariate normal distributions.

varalpha(v1,v2,alpha):=(v1*v2)/((1-alpha)*v1+alpha*v2)$ mualpha(mu1,v1,mu2,v2,alpha):=(alpha*mu1*v2+(1-alpha)*mu2*v1)/((1-alpha)*v1+alpha*v2)$   /* Kullback--Leibler divergence */ KLD(mu1,v1,mu2,v2):=(1/2)*((((mu2-mu1)**2)/v2)+(v1/v2)-log(v1/v2)-1)$   assume(alpha>0)$assume(alpha<1)$ assume(v1>0)$assume(v2>0)$ theta1(mu,v):=mu/v$ theta2(mu,v):=-1/(2*v)$; F(theta1,theta2):=-theta1**2/(4*theta2)+0.5*log(-1/theta2)$   eq: (theta1(mu1,v1)-theta1(mu2,v2))*mualpha(mu1,v1,mu2,v2,alpha)+(theta2(mu1,v1)-theta2(mu2,v2))*(mualpha(mu1,v1,mu2,v2,alpha)**2+varalpha(v1,v2,alpha))-F(theta1(mu1,v1),theta2(mu1,v1))+F(theta1(mu2,v2),theta2(mu2,v2)); solalpha: solve(eq,alpha)$ alphastar:rhs(solalpha[1]);   ChernoffInformation: KLD(mualpha(mu1,v1,mu2,v2,alphastar),varalpha(v1,v2,alphastar),mu1,v1)$ print("Chernoff information=")$ratsimp(ChernoffInformation);