Counterexamples for optimal scaling of Metropolis-Hastings chains with rough target densities
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For sufficiently smooth targets of product form it is known that the variance of a single coordinate of the proposal in RWM (Random walk Metropolis) and MALA (Metropolis adjusted Langevin algorithm) should optimally scale as n^-1 and as n^-1/3 with dimension n, and that the acceptance rates should be tuned to 0.234 and 0.574. We establish counterexamples to demonstrate that smoothness assumptions such as C^1(R) for RWM and C^3(R) for MALA are indeed required if these guidelines are to hold. The counterexamples identify classes of marginal targets (obtained by perturbing a standard Normal density at the level of the potential (or second derivative of the potential for MALA) by a path of fractional Brownian motion with Hurst exponent H) for which these guidelines are violated. For such targets there is strong evidence that RWM and MALA proposal variances should optimally be scaled as n^-1/H and as n^-1/2+H and will then obey anomalous acceptance rate guidelines. Useful heuristics resulting from this theory are discussed. The paper develops a framework capable of tackling optimal scaling results for quite general Metropolis-Hastings algorithms (possibly depending on a random environment).
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