Optimal scaling of random-walk metropolis algorithms on general target distributions

Yang, Jun; Roberts, Gareth O.; Rosenthal, Jeffrey S.

doi:10.1016/j.spa.2020.05.004

Cited by 25 publications

(31 citation statements)

References 55 publications

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“…In particular, the acceptance rate of the adapted KGE is around 20% for a system that we have good input and observations (case study 2). This is similar to the formal likelihood function and is also close to the theoretically optimal acceptance rate (0.234) in Metropolis algorithms with random walk (Yang et al, 2020).…”

Section: Discussionsupporting

confidence: 79%

Pitfalls and a feasible solution for using KGE as an informal likelihood function in MCMC methods: DREAM(ZS) as an example

Liu

Fernández-Ortega

Mudarra

et al. 2021

Preprint

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Abstract. The Kling-Gupta Efficiency (KGE) is a widely used performance measure because of its advantages in orthogonally considering bias, correlation and variability. However, in most Markov chain Monte Carlo (MCMC) algorithms, error-based formal likelihood functions are commonly applied. Due to its statistically informal characteristics, using the original KGE in MCMC methods leads to problems in posterior density ratios due to negative KGE values and high proposal acceptance rates resulting in less identifiable parameters. In this study we propose adapting the original KGE using a gamma distribution to solve these problems and to apply KGE as an informal likelihood function in the DiffeRential Evolution Adaptive Metropolis DREAM(ZS), which is an advanced MCMC algorithm. We compare our results with the formal likelihood function to show whether our approach is robust and plausible to explore posterior distributions of model parameters and to reproduce the discharge behaviors. For that, we set three case studies that contain different uncertainties. Our results show that model parameters cannot be identified and the uncertainty of discharge simulations is large when directly using the original KGE. Our approach finds similar posterior distributions of model parameters compared to the formal likelihood function. Even though the acceptance rate of the adapted KGE is lower than the formal likelihood function for some systems, the convergence rate (efficiency) is similar between the two approaches for the calibration of real hydrological systems showing generally acceptable performances. We also show that both the adapted KGE and the formal likelihood function provide low performances for low flows, with the larger overestimations obtained from using the formal likelihood function. Furthermore, the adapted KGE approach behaves closely to the formal likelihood function in terms of the correlation between simulations and observations. Thus, our study provides a feasible way to use KGE as an informal likelihood in the MCMC algorithm and provides possibilities to combine multiple data for better and more realistic model calibrations.

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Section: Discussionsupporting

confidence: 79%

Pitfalls and a feasible solution for using KGE as an informal likelihood function in MCMC methods: DREAM(ZS) as an example

Liu

Fernández-Ortega

Mudarra

et al. 2021

Preprint

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show abstract

“…[55] compares the ergodicity properties of a delayed-acceptance algorithm with those of the parent MH algorithm, while [28] compares the asymptotic variance of the ergodic average from a delayed-acceptance algorithm with the variance of an importance-sampling estimator which takes as its proposal a sample from an MCMC algorithm targeting a surrogate. Historically, insights into the performance and tuning of MCMC algorithms have been obtained by examining the limiting behaviour of a rescaled version of the Markov chain as the dimension of the statespace increases to infinity [6,8,44,45,46,57,62,63]. In this article, we focus on random-walk proposals since this class of methods has the advantage of not requiring further information about the target, such as the local gradient or Hessian.…”

Section: Introductionmentioning

confidence: 99%

Efficiency of delayed-acceptance random walk Metropolis algorithms

2021

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“…Accepting that σ 2 n is the optimal decay rate for proposal variances, we turn our attention to choosing the that maximises the ESJD, equivalently (as it will turn out) determining the optimal average acceptance rate. Revisiting (34), W ( ) takes the form…”

Section: −→mentioning

confidence: 99%

“…Furthermore, it proves optimal as n → ∞ to choose the constant of proportionality so as to obtain average acceptance rates of 0.234 of the proposed moves for RWM and 0.574 for MALA. These results were originally proved only for the toy example of product targets given above; nevertheless simulation evidence suggests that they should hold in much greater generality and notable progress has been made to generalise the theory towards more general targets, especially in the case of RWM: see, for example, Yang, Roberts and Rosenthal [34]. Consequently the theory does indeed provide very practical and useful guidelines for practitioners, and additionally provides an important context for motivating and assessing adaptive MCMC methods.…”

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confidence: 99%

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

Vogrinc

Kendall

2021

Ann. Appl. Probab.

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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 of the order of C 1 (R) for RWM and C 3 (R) for MALA are indeed required if these scaling rates are to hold. The counterexamples identify classes of marginal targets for which these guidelines are violated, obtained by perturbing a standard normal density (at the level of the potential for RWM and the second derivative of the potential for MALA) using roughness generated by a path of fractional Brownian motion with Hurst exponent H . 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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Optimal scaling of random-walk metropolis algorithms on general target distributions

Cited by 25 publications

References 55 publications

Pitfalls and a feasible solution for using KGE as an informal likelihood function in MCMC methods: DREAM(ZS) as an example

Pitfalls and a feasible solution for using KGE as an informal likelihood function in MCMC methods: DREAM(ZS) as an example

Efficiency of delayed-acceptance random walk Metropolis algorithms

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

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