  Multimodal structures in the sampling density (e.g. two competing phases) can
be a serious problem for traditional Markov Chain Monte Carlo (MCMC), because
correct sampling of the different structures can only be guaranteed for
infinite sampling time. Samples may not decouple from the initial configuration
for a long time and autocorrelation times may be hard to determine.
  We analyze a suitable modification (C. J. Geyer and E. A. Thompson, J. Amer.
Statist. Assoc., 90, 909, 1995) of the simulated tempering idea (E. Marinari
and G. Parisi, Europhys. Lett. 19, 451, 1992), which has orders of magnitude
smaller autocorrelation times for multimodal sampling densities and which
samples all peaks of multimodal structures according to their weight. The
method generates exact, i.e. uncorrelated, samples and thus gives access to
reliable error estimates. Exact tempering is applicable to arbitrary
(continuous or discreet) sampling densities and moreover presents a possibility
to calculate integrals over the density (e.g. the partition function for the
Boltzmann distribution), which are not accessible by usual MCMC.
