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Title: Counterexamples for optimal scaling of Metropolis-Hastings chains with rough target densities Authors:  Jure Vogrinc - University of Warwick (United Kingdom) [presenting]
Wilfrid Kendall - University of Warwick (United Kingdom)
Abstract: 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 having a continuous derivative for RWM and having three continuous derivatives 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. We will briefly discuss useful heuristics resulting from this theory.