# Locally Scaled and Stochastic Volatility Metropolis-Hastings Algorithms
###### Abstract
These new methods are based on different formulations of the scale matrix in the MH algorithm, with the proposal Gaussian distribution still having the mean being the current state. In practice, typically, the scale matrix is treated as being diagonal, with all the dimensions having the same standard deviation \(\sigma\)[22]. The MH algorithm can be made arbitrarily poor by making \(\sigma\) either very small or very large [22; 23]. [PERSON] and [PERSON] [22] attempt to find the optimal \(\sigma\) based on assuming a specific form for the distribution [23], and targeting specific acceptance rates--which are related to the efficiency of the algorithm. [PERSON] et al. [21] extend this to more general target distributions. Within MH sampling, the use of gradient and Hessian information has been successfully employed to construct intelligent proposal distributions [4; 24]. In the MALA [13; 24], a drift term that includes first-order gradient data is used to control the exploration of the parameter space. In the manifold MALA [4], the local structure of the target distribution is acknowledged through the use of the Hessian or different metric tensor [4; 24]. One typically has to conduct pilot runs in order to tune the parameters of MH and MALA respectevklyc [24]. Scaling the proposal distribution with a metric tensor that takes into account the curvature of the posterior distribution can simplify the tuning of the method such that expensive pilot runs are no longer needed [24]. [PERSON] et al. [24] introduce the particle MH algorithm, which uses gradient and Hessian information of the approximated target posterior using a Laplace approximation. In this work, we propose Locally Scaled Metropolis-Hastings (LSMH), which uses the local geometry of the target to scale MH, and Stochastic Volatility Metropolis-Hastings (SVMH), which randomly scales MH. These two algorithms deviate from the particle MH algorithm in that we focus solely on designing the scale matrix, so we do not consider first-order gradient information. Secondly, for the LSMH algorithm--the Hessian is based on the exact unnomalised posterior and not on the approximated target posterior. Both the LSMH and SVMH algorithms are easier to implement, needing only minor adjustments to MH, when compared to the particle MH algorithm of [PERSON] et al. [24]. The particle MH algorithm is more complex when compared to the vanilla MH algorithm as the method was specifically developed to perform Bayesian parameter deduction in nonlinear state-space models, which typically have intractable likelihoods [24]. Thus, the particle MH method allows for both intractable likelihoods and posteriors, while the MH algorithm and the proposed methods use tractable likelihoods. The LSMH algorithm extends the MH method by designing the scale matrix so that it depends on the local geometry of the target distribution in the same way as in manifold MALA and Riemannian Manifold Hamiltonian Monte Carlo [4]. The use of the local curvature of the target reduces the correlations in the samples generated by MH. However, this comes at a cost of a substantial increase in the computation cost due to the calculation of the Hessian matrix of the posterior distribution. The SVMH method sets the scaling matrix in MH to be random with a user-specified distribution, similarly to setting the mass matrix in Quantum-Inspired Magnetic Hamiltonian Monte Carlo [19; 25] to be random to mimic the behavior of quantum particles. This approach has the benefit of improving the exploration of MH without any noticeable increase in computational cost. This strategy has been shown in the Quantum-Inspired Hamiltonian Monte [25] Carlo to improve sampling of multi-modal distributions. The weakness of this approach is that it requires the user to specify the distribution of the scale matrix and possibly tune the parameters of the chosen distribution.