Ising-Markov chains equivalence as an inverse problem of statistical physics

We develop a consistent framework for introducing an effective temperature for symbolic stochastic systems with known statistical properties but unknown underlying interaction energies. Focusing on binary finite-memory Markov chains, we establish an exact correspondence with one-dimensional Ising models with finite-range interactions. Within this setting, we demonstrate that information temperature can be defined through three independent approaches: (i) matching joint probability distributions, (ii) matching conditional probabilities, and (iii) solving an entropy-based inverse problem of statistical physics. For the present one-parameter model, the three constructions yield the same information temperature. The introduction of temperature by the inverse problem method provides a constructive realization of the Hammersley–Clifford (Markov-Gibbs) correspondence in the special case of stationary binary finite-memory Markov chains and complements it with an entropy-based inverse construction.