For a given Markov chain Monte Carlo (MCMC) algorithm, we define distance between configurations, which quantifies difficulty of transition from one configuration to the other. This distance gives a universal form for a class of MCMC algorithms which generate local moves of configurations. The introduction of distance enables us to investigate a relaxation process in a MCMC simulation from a geometrical point of view. We here consider a system whose equilibrium distribution is highly multimodal with a large number of degenerate classical vacua. We show that, when we implement the simulated tempering method for such a system, the anti-de Sitter (AdS) geometry emerges in the extended configuration space. This talk is based on the work with M. Fukuma and N. Umeda [JHEP12(2017)001, work in preparation].