  We propose an adaptive independent Metropolis--Hastings algorithm with the
ability to learn from all previous proposals in the chain except the current
location. It is an extension of the independent Metropolis--Hastings algorithm.
Convergence is proved provided a strong Doeblin condition is satisfied, which
essentially requires that all the proposal functions have uniformly heavier
tails than the stationary distribution. The proof also holds if proposals
depending on the current state are used intermittently, provided the
information from these iterations is not used for adaption. The algorithm gives
samples from the exact distribution within a finite number of iterations with
probability arbitrarily close to 1. The algorithm is particularly useful when a
large number of samples from the same distribution is necessary, like in
Bayesian estimation, and in CPU intensive applications like, for example, in
inverse problems and optimization.
