Let \(\theta(t)\) be a nonobservable Markov chain with discrete time and with a finite number of states \(\{a_1,\dots, a_n\}\). Let the conditional distribution of the observations \(X_t\in \mathbb{R}^d\) be \({\mathbf P}\{X_t\leq x\mid \theta(t)\}= F_{\theta(t)}(x)\), where \(F_{a_1}(x),\dots, F_{a_n}(x)\) are known distribution functions in \(\mathbb{R}^d\). The problem of filtering consists in the reconstruction of \(\theta(t)\) from the observations \(X_k\), \(k\leq t\). In the case of a Markov chain with rate transitions, an asymptotic formula for the mean-square error of the optimal filter is obtained.