Shannon Entropy Rate of Hidden Markov Processes

The paper defines the mixed-state presentation (MSP) and its maps explicitly (Eqs. (8)–(9)), shows the MSP is a place-dependent IFS, and under stated IFS conditions identifies the attractor with the mixed-state set R and the invariant measure with the Blackwell measure (Sec. V; Theorem 1 context) . It then states Blackwell’s integral entropy-rate formula (Eq. (16)) and replaces it by a time-average via an ergodic theorem for IFS orbits (its Eq. (17)) . The candidate solution follows the same structure: MSP–IFS identification and forward invariance, Blackwell’s integral, and time-averaging via Birkhoff. One minor correction: the paper’s printed Eq. (17) appears to miss an explicit time-sum, which the model correctly includes; this aligns with the earlier stated ergodic theorem for IFS orbits (Eq. (6)) . Aside from this presentational glitch and standard caveats about contractivity/ergodicity assumptions, both are correct and essentially the same proof strategy.