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 $h_1 \neq h_2, h_3 \neq h_4$, 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 $s = (\epsilon_1, \epsilon_2, \cdots, \epsilon_n)$ of 0's and 1's, let $p(s) = \mathrm{Pr}\{y_1 = \epsilon_1, \cdots, y_n = \epsilon_n\}$. If \begin{equation*}\tag{1}p^2(00) \neq p(0)p(000) \text{and} p^2(01) \neq 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, \epsilon, \delta, 0) = p(s, \epsilon, 0)a(\epsilon, \delta) + p(s, \epsilon)b(\epsilon, \delta)\end{equation*} for all $s$ and $\epsilon = 0$ or 1, $\delta = 0$ or 1, where $a(\epsilon, \delta), b(\epsilon, \delta)$ are (easily computed) functions of $M$, and noting that, if (1) is satisfied, $a(\epsilon, \delta)$ and $b(\epsilon, \delta)$ 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.