We introduce an inversion method that uses dimension reduction for the retrieval of atmospheric methane (CH4) profiles. Uncertainty analysis is performed using the Markov chain Monte Carlo (MCMC) statistical estimation. These techniques are used to retrieve CH4 profiles from the ground-based spectral measurements by the Fourier Transform Spectrometer (FTS) instrument at Sodankyla (67.4 degrees N, 26.6 degrees E), Northern Finland. The Sodankyla FTS is part of the Total Carbon Column Observing Network (TCCON), a global network that observes solar spectra in near-infrared wavelengths. The high spectral resolution of the data provides approximately 3 degrees of freedom about the vertical structure of CH4 between around 0 and 40km. We reduce the dimension of the inverse problem by using principal component analysis. Smooth and realistic profiles are sought by estimating three parameters for the profile shape. We use Bayesian framework with adaptive MCMC to better characterize the full posterior distribution of the solution and uncertainties related to the retrieval. The retrieved profiles are validated against in situ AirCore soundings which provide an accurate reference up to 20-30km. The method is presented in a general form, so that it can easily be adapted for other applications, such as different trace gases or satellite-borne measurements where more accurate profile information and better analysis of the uncertainties would be highly valuable.