Maximum entropy reconstruction of discrete-time Markov chains under partial information

This paper addresses the reconstruction of discrete-time Markov chains from incomplete information using the maximum entropy rate principle. Three settings are considered: prescribed graph topology alone, topology with a fully specified stationary distribution, and topology with partially specified stationary probabilities. We investigate the relationship between entropy-maximizing transition mechanisms and Luce's choice axiom, establishing conditions under which optimal transition matrices admit a Luce-score representation. When the stationary distribution is fully prescribed, we develop a constructive feasibility test and a system-theoretic iterative algorithm based on a Lyapunov-like function. Unlike conventional matrix-scaling approaches, the proposed algorithm operates directly on Luce scores. We illustrate the results for selected graph topologies, including complete graphs without self-loops and odd cycles. We also provide a counterexample to a previously stated uniqueness claim, showing that distinct, nonproportional Luce-score vectors can generate the same entropy-maximizing transition matrix. For partially prescribed stationary distributions, we extend the score-based approach to jointly reconstruct the unknown stationary probabilities and transition mechanism through an iterative procedure involving a parameterized Lyapunov-like function. Overall, the proposed framework integrates graph compatibility, entropy-rate maximization, and probabilistic choice representations, offering constructive tools for reconstructing Markov chains under different levels of stationary information and structural constraints.