theorem logRatio_antisymmetric {x : Fin J → ℝ}
{j k : Fin J} (hj : 0 < x j) (hk : 0 < x k) :
logRatio x j k = -logRatio x k j := by
simp only [logRatio]
rw [Real.log_div (ne_of_gt hj) (ne_of_gt hk),
Real.log_div (ne_of_gt hk) (ne_of_gt hj)]
ringCES Estimation Theory: Connecting Theory to Data