  Due to the principle of minimal information gain, the measurement of points
in an affine space $V$ determines a Legendrian submanifold of $V \times V^*
\times \mathbb R$. Such Legendrian submanifolds are equipped with additional
geometric structures that come from the central moments of the underlying
probability distributions and are invariant under the action of the group of
affine transformations on $V$. We investigate the action of this group of
affine transformations on Legendrian submanifolds of $V \times V^* \times
\mathbb R$ by giving a detailed overview of the structure of the algebra of
scalar differential invariants. We show how the central moments can be used to
construct the scalar differential invariants. In the end, we view the results
in the context of equilibrium thermodynamics of gases, where we notice that the
heat capacity is one of the differential invariants.
