Cosmic strings are a well-motivated extension to the standard cosmological model and could induce a subdominant component in the anisotropies of the cosmic microwave background (CMB), in addition to the standard inflationary component. The detection of strings, while observationally challenging, would provide a direct probe of physics at very high energy scales. We develop a new framework for cosmic string inference, constructing a Bayesian analysis in wavelet space where the string-induced CMB component has distinct statistical properties to the standard inflationary component. Our wavelet-Bayesian framework provides a principled approach to compute the posterior distribution of the string tension $Gμ$ and the Bayesian evidence ratio comparing the string model to the standard inflationary model. Furthermore, we present a technique to recover an estimate of any string-induced CMB map embedded in observational data. Using Planck-like simulations we demonstrate the application of our framework and evaluate its performance. The method is sensitive to $Gμ ∼ 5 \times 10^{-7}$ for Nambu-Goto string simulations that include an integrated Sachs-Wolfe (ISW) contribution only, before any parameters of the analysis are optimised. ; Comment: 17 pages, 13 figures