A geometric interpretation of the Metropolis-Hastings algorithm

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The Markov chain Monte Carlo revolution (2008, background): “In [8] it is shown to be the L(1) projection of the proposal distribution to the p self-adjoint kernels on Ω.”
Proto-value Functions: A Laplacian Framework for Learning Representation and Control in Markov Decision Processes (2007, methodology): “20 To ensure that the decomposed matrices are not only stochastic, but also diagonalizable, which the Kronecker factorization procedure does not guarantee, we incorporate an additional step using the Metropolis Hastings algorithm (Billera and Diaconis, 2001) to make the smaller matrices Pa and Pb re”
Compact Spectral Bases for Value Function Approximation Using Kronecker Factorization (2007, background): “This transformation was proven (Billera & Diaconis 2001) to minimize the distance in an L 1 metric between the original matrix B and the space of reversible stochastic matrices with stationary distribution π.”
Use of exchangeable pairs in the analysis of simulations (2004, background): “See Diaconis and Zabell [8] and Hudson [15] for more characterizations.”
Fastest Mixing Markov Chain on a Graph (2004, background): “(See [8] for a nice geometric interpretation of the Metropolis–Hastings algorithm.”