Parallel tempering is a generic Markov chain Monte Carlo sampling method
which allows good mixing with multimodal target distributions, where
conventional Metropolis-Hastings algorithms often fail. The mixing properties
of the sampler depend strongly on the choice of tuning parameters, such as the
temperature schedule and the proposal distribution used for local exploration.
We propose an adaptive algorithm which tunes both the temperature schedule and
the parameters of the random-walk Metropolis kernel automatically.

The adaptive Metropolis (AM) algorithm of Haario, Saksman and Tamminen
[Bernoulli 7 (2001) 223-242] uses the estimated covariance of the target
distribution in the proposal distribution. This paper introduces a new robust
adaptive Metropolis algorithm estimating the shape of the target distribution
and simultaneously coercing the acceptance rate. The adaptation rule is
computationally simple adding no extra cost compared with the AM algorithm.

This paper describes sufficient conditions to ensure the correct ergodicity
of the Adaptive Metropolis (AM) algorithm of Haario, Saksman, and Tamminen
(2001) [9], for target distributions with a non-compact support. The conditions
ensuring a strong law of large numbers and a central limit theorem require that
the tails of the target density decay super-exponentially and have regular
contours.