The Annals of Mathematical Statistics

On the Identifiability Problem for Functions of Finite Markov Chains

Edgar J. Gilbert

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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 $A$.

Article information

Source
Ann. Math. Statist., Volume 30, Number 3 (1959), 688-697.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706199

Digital Object Identifier
doi:10.1214/aoms/1177706199

Mathematical Reviews number (MathSciNet)
MR107304

Zentralblatt MATH identifier
0089.34503

JSTOR
links.jstor.org

Citation

Gilbert, Edgar J. On the Identifiability Problem for Functions of Finite Markov Chains. Ann. Math. Statist. 30 (1959), no. 3, 688--697. doi:10.1214/aoms/1177706199. https://projecteuclid.org/euclid.aoms/1177706199


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