Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1405.4896

Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity

J. Kuntz·M. Ottobre·A. M. Stuart
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Random Walk Metropolis (RWM) algorithm is a Metropolis- Hastings MCMC algorithm designed to sample from a given target distribution πwith Lebesgue density on R^N. RWM constructs a Markov chain by randomly proposing a new position (the "proposal move"), which is then accepted or rejected according to a rule which makes the chain reversible with respect to π. When the dimension N is large a key question is to determine the optimal scaling with N of the proposal variance: if the proposal variance is too large, the algorithm will reject the proposed moves too often; if it is too small, the algorithm will explore the state space too slowly. Determining the optimal scaling of the proposal variance gives a measure of the cost of the algorithm as well. One approach to tackle this issue, which we adopt here, is to derive diffusion limits for the algorithm. Such an approach has been proposed in the seminal papers [RGG97, RR98]; in particular in [RGG97] the authors derive a diffusion limit for the RWM algorithm under the two following assumptions: i) the algorithm is started in stationarity; ii) the target measure $π$ is in product form. The present paper considers the situation of practical interest in which both assumptions i) and ii) are removed. That is a) we study the case (which occurs in practice) in which the algorithm is started out of stationarity and b) we consider target measures which are in non-product form. The target measures that we consider arise in Bayesian nonparametric statistics and in the study of conditioned diffusions. We prove that, out of stationarity, the optimal scaling for the proposal variance is O(N), as it is in stationarity. Notice that the optimal scaling in and out of stationatity need not be the same in general, and indeed they differ e.g. in the case of the MALA algorithm [KOS16].

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

J. Kuntz, M. Ottobre, A. M. Stuart. 2016-08-30. Diffusion Limit For The Random Walk Metropolis Algorithm Out Of stationarity. https://arxiv.org/abs/1405.4896

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Local well-posedness of general mean field game master equations

This paper presents a generic approach for establishing mean field game master equations, applicable whenever the mean field equilibrium can be characterized by a McKean-Vlasov forward-backward stochastic differential equation system. The core of our approach is a representation formula for the first-order Lions derivative of the decoupling field of this forward-backward SDE system. We then employ a bootstrap argument to recursively compute its higher-order derivatives. To demonstrate the method's versatility, we establish the local well-posedness for master equations in three distinct models: extended mean field games, mean field games with volatility control, and mean field games with a major player.

math.PR↗
arXiv2026

Uniqueness for nonlinear Fokker-Planck equations with general diffusion terms and their associated nonlinear Markov processes

This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type \begin{equation} u_{t}-\sum_{i,j=1}^{d} D^{2}_{ij}(a_{ij}(x)β(x,u))+ \text{div}(b(x,u)u)=0 \quad \text{in}\; (0, \infty) \times \mathbb{R}^{d} ,\notag \end{equation} with initial condition $u(0,x)\equiv u_{0}(x)$, where $a_{ij}$, $β$, and $b$ are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup $S(t): L^{1}(\mathbb{R}^{d}) \rightarrow L^{1}(\mathbb{R}^{d})$, and $u(t)=S(t)u_{0}$ is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case $a_{ij} \equiv δ_{ij}$. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new $L^{\infty}$ estimate for mild solutions starting from data in $L^{1}\cap L^{\infty}$ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.

math.PR↗
arXiv2026

Small-time annealed large deviations principle for one-dimensional diffusions in a random environment

In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-ρ(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{ρ(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $ρ$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The Itô--McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.

math.PR↗