The principle of maximum caliber was proposed by Edwin T. Jaynes in 1980,^[1] in an article titled The Minimum Entropy Production Principle in the context of deriving a principle for non-equilibrium statistical mechanics.

The principle of maximum caliber can be considered as a generalization of the principle of maximum entropy defined over the paths space, the caliber ${\displaystyle S}$  is of the form

${\displaystyle S[\rho [x()]]=\int D_{x}\rho [x()]\ln {\rho [x()] \over \pi [x()]}}$ 

where for n-constraints

${\displaystyle \int D_{x}\rho [x()]A_{n}[x()]=\langle A_{n}[x()]\rangle =a_{n}}$ 

it is shown that the probability functional is

${\displaystyle \rho [x()]=\exp \left\{-\sum _{i=0}^{n}\alpha _{n}A_{n}[x()]\right\}.}$ 

In the same way, for n dynamical constraints defined in the interval ${\displaystyle t\in [0,T]}$  of the form

${\displaystyle \int D_{x}\rho [x()]L_{n}(x(t),{\dot {x}}(t),t)=\langle L_{n}(x(t),{\dot {x}}(t),t)\rangle =\ell (t)}$ 

it is shown that the probability functional is

${\displaystyle \rho [x()]=\exp \left\{-\sum _{i=0}^{n}\int _{0}^{T}dt\,\alpha _{n}(t)L_{n}(x(t),{\dot {x}}(t),t)\right\}.}$ 

Following Jaynes' hypothesis, there exist publications in which the principle of maximum caliber appears to emerge as a result of the construction of a framework which describes a statistical representation of systems with many degrees of freedom.^[2][3][4]

• Maximal entropy random walk