On fixed-domain asymptotics, parameter estimation and isotropic Gaussian random fields with Matérn covariance functions

Loh, Wei-Liem; Sun, Saifei; Wen, Junhai

doi:10.1214/21-aos2077

Cited by 13 publications

(10 citation statements)

References 23 publications

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“…The denominator is lower bounded by exp(−n 1/2−τ )c π,0 on the event E 5 , similar to the proof of (48). The numerator can be upper bounded on the event E 6 , using the same derivation as in (50) and (49).…”

Section: A Proof Of Theorems 1 Andmentioning

confidence: 75%

“…Throughout the paper, we assume that the smoothness parameter ν > 0 is fixed and known. Estimation of the smoothness parameter ν is an important research topic with some recent developments in frequentist literature (Loh [47], Loh et al [48]), but is beyond the scope of the current paper. We let the true parameter values in the Matérn covariance function that generate X be (σ 2 0 , α 0 ).…”

Section: Bernstein-von Mises Theorem For Covariance Parametersmentioning

confidence: 99%

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Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

Li¹

2020

Preprint

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Gaussian process models typically contain finite dimensional parameters in the covariance functions that need to be estimated from the data. We study the Bayesian fixed-domain asymptotic properties of the covariance parameters in a Gaussian process with an isotropic Matérn covariance function, which has many applications in spatial statistics. Under fixeddomain asymptotics, it is well known that when the dimension of data is less than or equal to three, the microergodic parameter can be consistently estimated with asymptotic normality while the variance parameter and the range (or length-scale) parameter cannot. Motivated by the frequentist theory, we prove a Bernstein-von Mises theorem for the covariance parameters in isotropic Matérn covariance functions. We show that under fixed-domain asymptotics, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors as the sample size goes to infinity. The posterior of the microergodic parameter converges in total variation norm to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not necessarily converge to any degenerate distribution in general. Our theory allows an unbounded prior support for the range parameter on the whole positive real line. Furthermore, we propose a new property called the posterior asymptotic efficiency in linear prediction, and show that the Bayesian kriging predictor at a new spatial location with covariance parameters randomly drawn from their posterior distribution has the same prediction mean squared error as if the true parameters were known. In the special case of one-dimensional Ornstein-Uhlenbeck process, we derive an explicit form for the limiting posterior distribution of the range parameter and an explicit posterior convergence rate for the posterior asymptotic efficiency in linear prediction. We verify these asymptotic results in numerical examples.

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Section: A Proof Of Theorems 1 Andmentioning

confidence: 75%

Section: Bernstein-von Mises Theorem For Covariance Parametersmentioning

confidence: 99%

Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

Li¹

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Throughout the article, we assume that the smoothness parameter ν > 0 is fixed and known. Estimation of ν is a challenging problem with some recent advances (Loh [20], Loh et al [21]), but is beyond the scope of this article. Our main goal is to perform Bayesian inference on the parameters (θ, α, τ, β) in Model (1) based on the observed data (Y n , F n ).…”

Section: General Theory For Fixed-domain Posterior Contraction Ratesmentioning

confidence: 99%

“…Higher-order quadratic variation has been adopted in recent theoretical works of spatial Gaussian processes (Loh [20], Loh et al [21]). The basic idea is to use a carefully designed series of constants (c…”

Section: Higher-order Quadratic Variation Estimators For Isotropic Ma...mentioning

confidence: 99%

“…We illustrate this using the isotropic Matérn covariance function in Example 1. Higher-order quadratic variation has been successfully applied to parameter estimation in frequentist fixed-domain asymptotics for Gaussian process models without nugget (Loh [20], Loh et al [21]). We further generalize this powerful tool to Model (1) with nugget.…”

Section: Introductionmentioning

confidence: 99%

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Fixed-domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget

Li¹,

Sun²,

Zhu³

2022

Preprint

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Spatial Gaussian process regression models typically contain finite dimensional covariance parameters that need to be estimated from the data. We study the Bayesian estimation of covariance parameters including the nugget parameter in a general class of stationary covariance functions under fixed-domain asymptotics, which is theoretically challenging due to the increasingly strong dependence among spatial observations. We propose a novel adaptation of the Schwartz's consistency theorem for showing posterior contraction rates of the covariance parameters including the nugget. We derive a new polynomial evidence lower bound, and propose new consistent higher-order quadratic variation estimators that satisfy concentration inequalities with exponentially small tails. Our Bayesian fixed-domain asymptotics theory leads to explicit posterior contraction rates for the microergodic and nugget parameters in the isotropic Matérn covariance function under a general stratified sampling design. We verify our theory and the Bayesian predictive performance in simulation studies and an application to sea surface temperature data.

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Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term

Chen,

Eisenberg

2024

J Theor Probab

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On fixed-domain asymptotics, parameter estimation and isotropic Gaussian random fields with Matérn covariance functions

Cited by 13 publications

References 23 publications

Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

Fixed-domain Posterior Contraction Rates for Spatial Gaussian Process Model with Nugget

Invariant Measures for the Nonlinear Stochastic Heat Equation with No Drift Term

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