Counterexamples for optimal scaling of Metropolis-Hastings chains with rough target densities

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 $n^{-\frac{1}{3}}$ 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 $\mathcal{C}^1(\mathbb{R})$ for RWM and $\mathcal{C}^3(\mathbb{R})$ 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 $n^{-\frac{1}{H}}$ and as $n^{-\frac{1}{2+H}}$ 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).