MCMC Methods for Functions: ModifyingOld Algorithms to Make Them Faster

By Simon L. Cotter, Gareth O. Roberts, Andrew M. Stuart, D. White. 2012

463 citations recorded by the source corpus.

Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the un- known function. We describe an approach to modifying a whole range of MCMC methods, applicable whenever the target measure has density with respect to a Gaussian process or Gaussian random field reference measure, which ensures that their speed of convergence is robust under mesh refinement. Gaussian processes or random fields are fields whose marginal distri- butions, when evaluated at any finite set of N points, are RN-valued Gaussians. The algorithmic approach that we describe is applicable not only when the desired probability measure has density with respect to a Gaussian process or Gaussian random field reference measure, but also to some useful non-Gaussian reference measures constructed through random truncation. In the applications of interest the data is often sparse and the prior specification is an essential part of the over- all modelling strategy. These Gaussian-based reference measures are a very flexible modelling tool, finding wide-ranging application. Examples are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method so that it is, in principle, applicable for functions; this may be achieved by use of proposals based on carefully chosen time-discretizations of stochas- tic dynamical systems which exactly preserve the Gaussian reference measure. Taking this approach leads to many new algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems.

Subfields: Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/74defd863af4579846e493a34aaf14bc62ca8984

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Citations in the corpus

Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “the expected value of hover the posterior distribution (4.3) E[h] = Z Z h(z)p(zjy obs)dz: MCMC methods are a popular way to sample from the posterior distribution, which have been studied extensively [178, 90, 139, 80, 50, 137, 53, 52]; see also the books [77, 130]. One example of an MCMC method is ”
A Survey of Optimization Methods From a Machine Learning Perspective (2019, methodology): “meters in the HMC include not only the step size ǫ and the length of iteration steps L, but also the mass M, etc. Optimizing these hyper-parameters can help improve sampling performance [199], [205], [206]. It is convenient and efﬁcient to tune the hyper-parameters automatically without cumbersome a”
Bayesian computation: a summary of the current state, and samples backwards and forwards (2015, background): “2.4), combinations with proximal operators (Pereyra 2015; Schreck et al. 2013), and applications and algorithm development for the inﬁnite-dimensional context (Pillai et al. 2012; Cotter et al. 2013).”
Fourier Neural Operator for Parametric Partial Differential Equations (2020, methodology): “In this experiment, we use a function space Markov chain Monte Carlo (MCMC) method (Cotter et al., 2013) to draw samples from the posterior distribution of the initial vorticity in Navier-Stokes given sparse, noisy observations at time T = 50 .”
The Geometric Foundations of Hamiltonian Monte Carlo (2014, background): “Many of the properties of smooth manifolds critical to the construction of Hamiltonian Monte Carlo do not immediately extend to the infinite-dimensional target distributions common to functional analysis, such as the study of partial differential equations [15].”
Bayesian computation: a perspective on the current state, and sampling backwards and forwards (2015, background): “…scaling and choice of the discretisation step h , adaptive versions (both discussed in Section 2.4), combinations with proximal operators Pereyra (2013); Schreck et al. (2013), and applications and algorithm development for the inﬁnite-dimensional context Pillai et al. (2012); Cotter et al. (2013).”
Data Assimilation: A Mathematical Introduction (2015, methodology): “The specific form of the pCN-MCMC method which we introduce here has been chosen to be particularly effective in high dimensions; see [31] for an overview, [16] for the introduction of pCN and other methods for sampling probability measures in infinite dimensions, in the context of conditioned diffu”
Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions (2011, methodology): “Numerical experiments in [11] demonstrate its favorable properties in comparison with the RWM algorithm.”
Earth System Modeling 2.0: A Blueprint for Models That Learn From Observations and Targeted High‐Resolution Simulations (2017, background): “Standard parameter estimation and inverse problem approaches may require O(105) function or derivative evaluations to learn O(100) parameters, especially if uncertainty in the estimates is also required [Cotter et al., 2013].”
Hyperpriors for Matérn fields with applications in Bayesian inversion (2016, methodology): “Metropolis-within-Gibbs can be made meshfree using preconditioned Crank-Nicholson [11], which allows chain mixing to be the same when changing the number of unknowns N .”
The Bayesian Approach to Inverse Problems (2013, unknown): “n [MRTT53] and led, via the paper [Has70], to the development of the more general class of Metropolis-Hastings methods. MCMC methods which are invariant with respect to the target measure µ.The paper [CRSW12] overviews this subject area, including the new RWM method. The speciﬁc idea of the new RWM ”
Rare event estimation with sequential directional importance sampling (2022, methodology): “2 Conditional sampling M-H (CSM-H) algorithm The second proposal distribution we used in this paper is the normal distribution conditioned on the current state (conditional M-H algorithm) [42, 43], i.”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, methodology): “While these measures of uncertainty can be computed by MCMC methods (see [32, 33, 13, 54, 12, 5] and below), few statistical guarantees are available for the validity of such posterior inferences in typical PDE settings where G is non-linear and θ is modelled as a Gaussian process.”
Estimation of the lower and upper bounds on the probability of failure using subset simulation and random set theory (2018, methodology): “It is worth mentioning that the CS algorithm can be regarded as a particular case of the MCMC sampler with preconditioned Crank-Nicolson proposal presented in Cotter et al. (2013), q(u|v) = N(u| √ 1 − σ2 v, σ(2)Σ), Σ = I and σ ∈ [0, 1].”
Neural Operator: Learning Maps Between Function Spaces (2021, methodology): “, 2009; Stuart, 2010), and see (Cotter et al., 2013) for MCMC methods adapted to the function-space setting.”
The correlated pseudomarginal method (2015, methodology): “Data-informed proposals such as the preconditioned Crank-Nicolson Langevin proposal of Cotter et al. (2013) and its extensions proposed by Titsias and Papaspiliopoulos (2018) could also be used to update the auxiliary random variates at the cost of more complex acceptance probabilities.”
Stacking for Non-mixing Bayesian Computations: The Curse and Blessing of Multimodal Posteriors (2020, methodology): “That ensures rapid mixing for random-walk Metropolis, scaling as O ( d ) (Roberts et al., 1997; Cotter et al., 2013; Dwivedi et al., 2019), and Hamiltonian Monte Carlo, scaling as O ( d 1 / 4 ) (Beskos et al., 2013; Bou-Rabee et al., 2020; Mangoubi and Smith, 2017, 2019).”
Analysis of the Gibbs Sampler for Hierarchical Inverse Problems (2013, methodology): “Similarly, conceiving of Metropolis–Hastings methods in the infinite-dimensional limit leads to algorithms with provably dimension-independent convergence properties, while standard methods have convergence properties which degenerate with increased refinement of the discretization; see [14] and dis”