On the Identifiability Problem for Functions of Finite Markov Chains

Abstract

A stationary sequence $\{Y_n:n = 1, 2, \cdots\}$ of random variables with $D$ values (states) is said to be a function of a finite Markov chain if there is an integer $N \geqq D$, an $N \times N$ irreducible aperiodic Markov matrix $M$, a stationary Markov chain $\{X_n\}$ with transition matrix $M$, and a function $f$ such that $Y_n = f(X_n)$. For any finite sequence $s$ of states of $\{Y_n\}$, let $p(s) = P\{(Y_1, \cdots, Y_n) = s\}$. For any state $\epsilon$, let $s\epsilon t$ be the sequence $s$ followed by $\epsilon$ followed by the sequence $t$. For every state $\epsilon$, let $n(\epsilon)$ be the largest integer $n$ such that there are finite sequences $s_1, \cdots, s_n, t_1, \cdots, t_n$ such that the matrix $\| p(s_i\epsilon t_j):1 \leqq i, j \leqq n \|$ is nonsingular. If $\{Y_n\}$ is a function of a finite Markov chain, then $\Sigma n(\epsilon) \leqq N$. There is a finite set $\{s_1, \cdots, s_N, t_1, \cdots, t_N\}$ of finite sequences such that $p(s)$ satisfies the recurrence relations \begin{equation*}\tag{1}p(s\epsilon t) = \sum_{f(i)=\epsilon} a_i(s)p(s_i\epsilon t),\end{equation*} where $a_i(s)$ either is zero for all $s$ or else is a ratio of determinants involving only $p(s\epsilon t_k)$ and $p(s_j\epsilon t_k)$ for $f(j) = f(k) = f(i)$. If $\{Y_n\}$ has $D$ states and is a function of a Markov chain having $N$ states, then the entire distribution of $\{Y_n\}$ is determined by the distribution of sequences of length $\leqq 2(N - D + 1)$. For each $N$ and $D$, a function of a Markov chain is exhibited which attains this bound. If there is a Markov chain $\{X_n\}$ with $N = \Sigma n(\epsilon)$ states such that $\{Y_n\}$ is a function of $\{X_n\}$, then $\{Y_n\}$ is said to be a regular function of a Markov chain. If $\{Y_n\}$ is a regular function of a Markov chain having transition matrix $M$, then $M = X^{-1} AX$, where $A$ is an $N \times N$ matrix with elements $a_{ij} = a_j(s_if(i))$--defined by (1) above. $X = \|x_{ij}\|$ is a nonsingular $N \times N$ matrix such that $x_{ij} = 0$ unless $f(i) = f(j)$, the first row of each nonzero submatrix along the diagonal consists of positive numbers, and $\Sigma_jx_{ij} = p(s_if(i))$. Any $N \times N$ Markov matrix giving the same distribution for $\{Y_n\}$ can be written in this form, with the same $A$ and with an $X$ having the above properties. Any matrix of this form which has all elements nonnegative is a Markov matrix giving the same distribution for $\{Y_n\}$. There are $\Sigma\{n(\epsilon)\}^2 - N$ "unidentifiable" parameters in the matrix $X$, and at most $N^2 - \Sigma\{n(\epsilon)\}^2$ "identifiable" parameters, determined by the distribution of $\{Y_n\}$, in the matrix $A$.