2.5. The two-state Markov process

The general concepts introduced above are easily applied to a two-state, first-order Markov process that can be written as

with transition rates p and q. The self-transition rate for A→A is 1−p, and the rate of B→B is 1−q. The complete transition matrix T reads

and has eigenvalues λ₀ = 1 and λ₁ = 1 − (p + q). The eigenvalue λ₀ = 1 is assured by the Perron-Frobenius theorem as T is a stochastic matrix, i.e., ∑jTij=1 for all i. The normalized positive eigenvector to λ₀ is the equilibrium or stationary distribution p_st of the process,

We set pA=qp+q and pB=pp+q. With the auxiliary functions φ, ψ : [0, 1] → ℝ defined as φ(x) = −xlogx and ψ(x) = φ(x)+φ(1−x), the analytical quantities H_pq, h_pq and a_pq for the 2-state first-order Markov process acquire a very simple form.

The Shannon entropy of the 2-state Markov process is

Due to the Markov property, the entropy rate is h_pq = H(X_n+1 ∣ X_n) and evaluates to

The Markov property reduces the full expression for information storage I(Xn+1;Xn(k)) to a_pq = I(X_n+1; X_n):

The total entropy is conserved between active information storage and the entropy rate:

To validate the proposed approach, these analytical results will later be compared with numerical results of a hidden Markov process classified as first-order Markovian by the PAIF method.