Page 677 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 677
PROBABILITY
Counterexamples for optimal scaling of Metropolis-Hastings chains with
rough target densities
Jure Vogrinc, Jure.Vogrinc@warwick.ac.uk
University of Warwick, United Kingdom
Coauthor: Wilfrid Kendall
For sufficiently smooth targets of product form it is known that the variance of a single coor-
dinate 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 demon-
strate that smoothness assumptions such as C1(R) for RWM and C3(R) for MALA are indeed
required if these guidelines are to hold. The counterexamples identify classes of marginal tar-
gets, 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 ex-
ponent H, for which these guidelines are violated. For such targets RWM and MALA proposal
variances should optimally be scaled as n−1/H and as n−1/(2+H) and will then obey anomalous
optimal acceptance rate guidelines. We will discuss useful heuristic implications of the results.
The talk is based on the preprint: https://arxiv.org/abs/1910.09485.
675
Counterexamples for optimal scaling of Metropolis-Hastings chains with
rough target densities
Jure Vogrinc, Jure.Vogrinc@warwick.ac.uk
University of Warwick, United Kingdom
Coauthor: Wilfrid Kendall
For sufficiently smooth targets of product form it is known that the variance of a single coor-
dinate 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 demon-
strate that smoothness assumptions such as C1(R) for RWM and C3(R) for MALA are indeed
required if these guidelines are to hold. The counterexamples identify classes of marginal tar-
gets, 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 ex-
ponent H, for which these guidelines are violated. For such targets RWM and MALA proposal
variances should optimally be scaled as n−1/H and as n−1/(2+H) and will then obey anomalous
optimal acceptance rate guidelines. We will discuss useful heuristic implications of the results.
The talk is based on the preprint: https://arxiv.org/abs/1910.09485.
675
