  Let $A$ be a transition probability kernel on a finite state space $\Delta^o
=\{1, \ldots , d\}$ such that $A(x,y)>0$ for all $x,y \in \Delta^o$. Consider a
reinforced chain given as a sequence $\{X_n, \; n \in \mathbb{N}_0\}$ of
$\Delta^o$-valued random variables, defined recursively according to, $$L^n =
\frac{1}{n}\sum_{i=0}^{n-1} \delta_{X_i}, \;\; P(X_{n+1} \in \cdot \mid X_0,
\ldots, X_n) = L^n A(\cdot).$$ We establish a large deviation principle for
$\{L^n\}$. The rate function takes a strikingly different form than the
Donsker-Varadhan rate function associated with the empirical measure of the
Markov chain with transition kernel $A$ and is described in terms of a novel
deterministic infinite horizon discounted cost control problem with an
associated linear controlled dynamics and a nonlinear running cost involving
the relative entropy function. Proofs are based on an analysis of time-reversal
of controlled dynamics in representations for log-transforms of exponential
moments, and on weak convergence methods.
