Determining the physical characteristics of a star is an inverse problem consisting of estimating the parameters of models for the stellar structure and evolution, and knowing certain observable quantities. We use a Bayesian approach to solve this problem for α Cen A, which allows us to incorporate prior information on the parameters to be estimated, in order to better constrain the problem. Our strategy is based on the use of a Markov chain Monte Carlo (MCMC) algorithm to estimate the posterior probability densities of the stellar parameters: mass, age, initial chemical composition, etc. We use the stellar evolutionary code astec to model the star. To constrain this model both seismic and non-seismic observations were considered. Several different strategies were tested to fit these values, using either two free parameters or five free parameters in astec. We are thus able to show evidence that MCMC methods become efficient with respect to more classical grid-based strategies when the number of parameters increases. The results of our MCMC algorithm allow us to derive estimates for the stellar parameters and robust uncertainties thanks to the statistical analysis of the posterior probability densities. We are also able to compute odds for the presence of a convective core in α Cen A. When using core-sensitive seismic observational constraints, these can rise above ∼40 per cent. The comparison of results to previous studies also indicates that these seismic constraints are of critical importance for our knowledge of the structure of this star.