In Bayesian statistics, the selection of noninformative priors is a crucial issue. There have been discussions on theoretical justification and problems for the Jeffreys prior, as well as alternative objective priors. Among them, we will focus on the two types of matching priors consistent with frequency theory: the probability matching priors and the moment matching priors. In particular, there is no clear relationship between these two matching priors on non-regular statistical models, even though they have similar objectives. Considering information geometry on a one-sided truncated exponential family, a typical example of non-regular statistical models, we obtain the result that the Lie derivative along one vector field provides the conditions for the probability and moment matching priors. Note that this Lie derivative does not appear in regular models. This result promotes a unified understanding of probability and moment matching priors on non-regular models. Further, we discuss the relationship between the probability and moment matching priors and the \(\alpha \) -parallel priors.