  A general Bayesian method for $L^2$ calibration of a mathematical model is
presented. General Bayesian inference starts with the specification of a loss
function. Then, the log-likelihood in Bayes' theorem is replaced by the
negative loss. While the minimiser of the loss function is unchanged by, for
example, multiplying the loss by a constant, the same is not true of the
resulting general posterior distribution. To address this problem in the
context of $L^2$ calibration of mathematical models, different automatic
scalings of the general Bayesian posterior are proposed. These are based on
equating asymptotic properties of the general Bayesian posterior and the
minimiser of the $L^2$ loss.
