In this paper, we present initial investigations towards boosting posterior probability based speech recognition systems by estimating more informative posteriors taking into account acoustic context (e.g., the whole utterance), as well as possible prior information (such as phonetic and lexical knowledge). These posteriors are estimated based on HMM state posterior probability definition (typically used in standard HMMs training). This approach provides a new, principled, theoretical framework for hierarchical estimation/use of more informative posteriors integrating appropriate context and prior knowledge. In the present work, we used the resulting posteriors as local scores for decoding. On the OGI numbers database, this resulted in significant performance improvement, compared to using MLP estimated posteriors for decoding (hybrid HMM/ANN approach) for clean and more specially for noisy speech. The system is also shown to be much less sensitive to tuning factors (such as phone deletion penalty, language model scaling) compared to the standard HMM/ANN and HMM/GMM systems, thus practically it does not need to be tuned to achieve the best possible performance.