A rule for assigning a probability distribution when information is incomplete: select the distribution that maximizes (Shannon) entropy subject to the known constraints, thereby remaining maximally noncommittal about unknown details.

Principle

When only partial constraints (typically expectation values or normalization) are known, prefer the distribution with largest entropy consistent with those constraints to avoid introducing unjustified structure.

Demonstration

Given only the mean energy of a system, maximizing entropy under that constraint yields the Gibbs (exponential-family) distribution; given only a known average of a coin bias-related statistic, MaxEnt produces the least-informative distribution that matches that statistic.

Misapplication

Using MaxEnt with incorrectly specified constraints, with constraints derived from noisy or biased summaries, or replacing Bayesian updating when relevant prior information exists can produce misleading inferences. Using the wrong entropy functional (not suited to the domain) also misapplies the principle.

Consequence

Correct application yields exponential-family distributions, objective least-biased estimates given the constraints, and a canonical representation of ignorance that often aligns with observed statistical regularities.

Reversal

Minimizing entropy under the same constraints concentrates probability mass and imposes unjustified assumptions, producing overly confident or degenerate models.

Boundary

Applies when constraints are well-defined and represent all known information; it is not a substitute for causal models, does not prescribe the choice of constraint set, and depends on the chosen entropy measure and domain (discrete, continuous, or functional).

Semantic Tension

Competes with subjective Bayesian choice of priors and with methods that incorporate additional structural assumptions (e.g., parametric modeling or hierarchical priors); tension arises over what counts as 'known constraints' and whether MaxEnt or a chosen prior better encodes prior knowledge.

Synthesis

MaxEnt formalizes the idea of making the least-committal probabilistic inference consistent with specified constraints: maximize entropy to transform those constraints into a canonical distribution (typically exponential-family) that encodes only the justified information.