Consider a polynomial scheme of \(n\) independent trials with \(r\) outcomes having probabilities \(p_1, \ldots p_r\) (\(\sum p_i = 1\)), and say that the hypothesis \(H({\mathbf p}_t,n)\) is true when for the vector of probabilities \({\mathbf p}_t = (p_1, \ldots, p_r)\). Given a set of alternative hypotheses \(H({\mathbf p}_j,n)\), \({\mathbf p}_j = (p_{j,1}, \ldots, p_{j,r})\), \(1 \leq j \leq m\), and a rule of choice, based on an observation of outcomes of \(n\) trials, let \(\alpha_{t,n}\) be the probability of rejecting \(H({\mathbf p}_t,n)\). The author [Sov. Math., Dokl. 14, 345-349 (1973); translation from Dokl. Akad. Nauk SSSR 209, 54-57 (1973; Zbl 0307.62040)] has already established the existence of an optimal sequence of (univalent) Bayesian rules such that \[ (1)\qquad \lim_{n \to \infty} n^{-1} \ln \max_{1 \leq t \leq m} \alpha_{t,n} = -\rho, \] \(\rho\) being the minimum of the Chernov distances between \({\mathbf p}_i\), \({\mathbf p}_j\), \(1 \leq i,j \leq n\). This paper proposes a multivalent Bayesian decision rule where the choice of true \({\mathbf p}_t\), on an observation, is proposed in \(k \leq m - 1\) most plausible variants \({\mathbf p}_{i,1}, \ldots , {\mathbf p}_{i,k}\). Explicit and asymptotic (as \(n \to \infty\)) upper bounds for error probabilities of that rule are obtained. Extending (1), they contain at most \(C_{m-1}^k\) generalized Chernov distances.