Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback
carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups
are unimodular. And even in noncommutative cases unimodular Lie groups share many of the useful features of Euclidean space. The rotation and Euclidean motion groups, which are perhaps the most relevant Lie groups to problems in geometric mechanics, are unimodular, as are the unitary groups that play important roles in quantum computing.
The extension of core information theoretic inequalities defined in the setting of Euclidean space
to this broad class of Lie groups is potentially relevant to a number of problems
relating to information gathering in mobile robotics, satellite attitude control, tomographic image reconstruction, biomolecular
structure determination, and quantum information theory. In this paper, several definitions are extended from
the Euclidean setting to that of Lie groups (including entropy and the Fisher information matrix), and inequalities analogous to those in classical information theory are derived and stated in the form of fifteen small theorems.
In all such inequalities, addition of random variables is replaced with the group product, and the appropriate generalization of convolution of probability densities is employed. An example from the field of robotics demonstrates how several of these results can be applied to quantify the amount of information gained
by pooling different sensory inputs.
Mathematics Subject Classification: Primary: 22E15, 94A15; Secondary: 22E70.