  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 $G\mu$ 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 $G\mu \sim 5 \times 10^{-7}$ for Nambu-Goto string simulations
that include an integrated Sachs-Wolfe (ISW) contribution only and do not
include any recombination effects, before any parameters of the analysis are
optimised. The sensitivity of the method compares favourably with other
techniques applied to the same simulations.
