Asymptotically optimal Bayesian tests for composite hypotheses

We consider two asymptotical ($\varepsilon\to 0$) problems of testing hypotheses $H_{0,\varepsilon}=\{P_{\varepsilon,\theta},\,\theta\in\Theta_0\}$ against $H_\varepsilon=\{P_{\varepsilon,\theta},\,\theta\in\Theta\diagdown\Theta_0\}$ with $\Theta_0\subset E^m$ being the subset of the parameter space $\Theta\subset E^n$, $0\le m<n$. Under sufficiently general assumptions about the families $P_{\varepsilon,\theta}$ and the densities $\pi_\varepsilon$ and $\pi_{\varepsilon,0}$ on $\Theta\diagdown\Theta_0$ and $\Theta_0$ we construct asymptotically optimal famalies of Bayesian tests and investigate the asymptotics of probabilities of errors.