Description length of canonical and microcanonical models

The (non-)equivalence of canonical and microcanonical ensembles is a fundamental question in statistical physics, concerning whether the use of soft and hard constraints in the maximum-entropy construction leads to the same description of a system. Despite the fact that maximum-entropy models are also commonly used in statistical inference, pattern detection, and hypothesis testing, a complete understanding of the effects of ensemble non-equivalence on statistical modeling is still missing. Here, we study this problem from a rigorous model selection perspective by comparing canonical and microcanonical models via the Minimum Description Length (MDL) principle, which yields a trade-off between likelihood, measuring model accuracy, and complexity, measuring model flexibility and its potential to overfit data. We compute the Normalized Maximum Likelihood (NML) of both formulations and find that: (i) microcanonical models always achieve higher likelihood but are always more complex; (ii) the optimal model choice depends on the empirical values of the constraints -- the canonical model performs best when its fit to the observed data exceeds its uniform average fit across all realizations; (iii) in the thermodynamic limit, the difference in description length per node vanishes when ensemble equivalence holds but persists otherwise, showing that non-equivalence implies extensive differences between large canonical and microcanonical models. Finally, we compare the NML approach to Bayesian methods, showing that (iv) the choice of priors, practically irrelevant in equivalent models, becomes crucial when an extensive number of constraints is enforced, possibly leading to very different outcomes.