  Information based thermodynamic logic is revisited. It consists of two parts:
Part A applies the modern theory of probability in which an arbitrary convex
function \phi is employed as an analytic "device" to express information as
statistical dependency contained in the topological sub-\sigma-algebra
structure. Via thermo-doubling, Fenchel-Young equality (FYE) that consists of
\phi(x) and its conjugate \psi(y) establishes the notion of equilibrium between
x and y through duality symmetry and the principle of maximum entropy/minimum
free energy. Part B deals with a given set of repetitive measurements, where an
inherent convex function emerges via the mathematics of large deviations.
Logarithm-based Shannon entropy with \phi(x)=-\log x figures prominently for
i.i.d. sample statistics. Information can be a measure of the agreement between
a statistical observation and its theoretical models. Maximum likelihood
principle arises here and FYE provides a thermodynamic energetic narrative of
recurrent data.
