In this paper we study the exponential decay of posterior probability of a set of sources and conditioning by rare sources for both uniform and general prior distributions of sources. The decay rate is determined by $L$-divergence and rare sources from a convex, closed set asymptotically conditionally concentrate on an $L$-projection. $L$-projection on a linear family of sources belongs to $\varLambda$-family of distributions. The results parallel those of Large Deviations for Empirical Measures (Sanov's Theorem and Conditional Limit Theorem).