This paper develops a new signal modeling framework using Bayesian model averaging formulation and the redundant or translation-invariant wavelet transform. The aim of this framework is to provide a paradigm general enough to effectively treat fundamental problems arising in wavelet-based signal processing, segmentation, and modeling. Unlike many other attempts to mitigate the translation-dependent nature of wavelet analysis and processing, this framework is based on a well-defined statistical model averaging paradigm and improves over standard translation-invariant schemes for wavelet denoising. In addition to deriving new and more powerful signal modeling and denoising schemes, we demonstrate that certain existing methods are special suboptimal solutions of our proposed model averaging criterion. Experimental results demonstrate the promise of this framework.