Abstract : The inverse problem considered here is to estimate the distribution of a non-observed random variable $X$ from some noisy observed data $Y$ linked to $X$ through a time-consuming physical model $H$. Bayesian inference is considered to take into account prior expert knowledge on $X$ in a small sample size setting. A Metropolis-Hastings within Gibbs algorithm is proposed to compute the posterior distribution of the parameters of $X$ through a data augmentation process. Since calls to $H$ are quite expensive, this inference is achieved by replacing $H$ with a kriging emulator interpolating $H$ from a numerical design of experiments. This approach involves several errors of different nature and, in this paper, we pay effort to measure and reduce the possible impact of those errors. In particular, we propose to use the so-called DAC criterion to assess in the same exercise the relevance of the numerical design and the prior distributions. After describing how computing this criterion for the emulator at hand, its behavior is illustrated on numerical experiments.