On the Identifiability Problem for Functions of Finite Markov Chains

Abstract

Let M = \| m_{ij} \| be a 4 \times 4 irreducible aperiodic Markov matrix such that h_1 eq h_2, h_3 eq h_4, where h_i = m_{i1} + m_{i2}. Let x_1, x_2, \cdots be a stationary Markov process with transition matrix M, and let y_n = 0 when x_n = 1 or 2, y_n = 1 when x_n = 3 \text{or} 4. For any finite sequence s = (ε_1, ε_2, \cdots, ε_n) of 0's and 1's, let p(s) = \mathrm{Pr}{y_1 = ε_1, \cdots, y_n = ε_n}. If \begin{equation*}\tag{1}p^2(00) eq p(0)p(000) \text{and} p^2(01) eq p(1)p(010),\end{equation*} the joint distribution of y_1, y_2, \cdots is uniquely determined by the eight probabilities p(0), p(00), p(000), p(010), p(0000), p(0010), p(0100), p(0110), so that two matrices M determine the same joint distribution of y_1, y_2, \cdots whenever the eight probabilities listed agree, provided (1) is satisfied. The method consists in showing that the function p satisfies the recurrence relation \begin{equation*}\tag{2}p(s, ε, δ, 0) = p(s, ε, 0)a(ε, δ) + p(s, ε)b(ε, δ)\end{equation*} for all s and ε= 0 or 1, δ= 0 or 1, where a(ε, δ), b(ε, δ) are (easily computed) functions of M, and noting that, if (1) is satisfied, a(ε, δ) and b(ε, δ) are determined by the eight probabilities listed. The class of doubly stochastic matrices yielding the same joint distribution for y_1, y_2, \cdots is described somewhat more explicitly, and the case of a larger number of states is considered briefly.