Listing A1: Plot the cumulant function of a log ratio exponential family induced by two 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)$ assume(v1>0)$assume(v2>0)$ theta1(mu,v):=mu/v$ theta2(mu,v):=-1/(2*v)$; F(theta1,theta2):=((-theta1**2)/(4*theta2))+(1/2)*log(-%pi/theta2)$ JF(alpha,theta1,theta2,theta1p,theta2p):=alpha*F(theta1,theta2)+(1-alpha)*F(theta1p,theta2p)-F(alpha*theta1+(1-alpha)*theta1p,alpha*theta2+(1-alpha)*theta2p);   m1:0;v1:1;m2:1;v2:2;   plot2d([JF(alpha,theta1(m1,v1),theta2(m1,v1),theta1(m2,v2),theta2(m2,v2)), -JF(alpha,theta1(m1,v1),theta2(m1,v1),theta1(m2,v2),theta2(m2,v2)),  [discrete,[[0.4215580558605244,-0.15],[0.4215580558605244,0.15]]], [discrete, [0.4215580558605244], [0.1155433222682347]], [discrete, [0.4215580558605244], [-0.1155433222682347]] ], [alpha,0,1], [xlabel,"alpha"], [ylabel,"F_{pq}(alpha)=-D_{B,alpha}[p:q]"], [style,  [lines,1,1],[lines,1,2],  [lines,2,0], [points, 3,3],[points, 3,3]],[legend, "skew Bhattacharyya D_{B,alpha}[p:q]","LREF log-normalizer F_{pq}(alpha)","","","" ], [color, blue, red,   black, black,black],[point_type,asterisk]);