Local state (or phone) posterior probabilities are often investigated as local classifiers (e.g., hybrid HMM/ANN systems) or as transformed acoustic features (e.g., ``Tandem'') towards improved speech recognition systems. In this paper, we present initial results towards boosting these approaches by improving the local state, phone, or word posterior estimates, using all possible acoustic information (as available in the whole utterance), as well as possible prior information (such as topological constraints). Furthermore, this approach results in a family of new HMM based systems, where only (local and global) posterior probabilities are used, while also providing a new, principled, approach towards a hierarchical use/integration of these posteriors, from the frame level up to the sentence level. Initial results on several speech (as well as other multimodal) tasks resulted in significant improvements. In this paper, we present recognition results on Numbers'95 and on a reduced vocabulary version (1000 words) of the DARPA Conversational Telephone Speech-to-text (CTS) task.