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Constrained Fiducial Inference for Gaussian Models

Hank Flury, Jan Hannig, Richard Smith

TL;DR

Addresses fitting parametric Gaussian models using a prior-free fiducial approach that accommodates non-i.i.d. time-series and spatial data. The core method reparameterizes the covariance with a Cayley transform, defines a general data-generating algorithm $Y=\mu(\theta)+\Sigma^{1/2}(\theta)U$, and constrains it to the model manifold to yield the Generalized Constrained Fiducial Distribution (GCFD) used in an MCMC. The paper derives fiducial quantities, provides a practical Pseudocode, proves that the MCMC targets the fiducial distribution, and demonstrates performance on MA(1) and Matérn models with publicly available Matlab code. The work extends fiducial inference to a broad class of Gaussian models, enabling Bayesian-like uncertainty without priors, and discusses computational strategies including composite-likelihood to improve scalability for larger arrays.

Abstract

We propose a new fiducial Markov Chain Monte Carlo (MCMC) method for fitting parametric Gaussian models. We utilize the Cayley transform to decompose the parametric covariance matrix, which in turn allows us to formulate a general data generating algorithm for Gaussian data. Leveraging constrained generalized fiducial inference, we are able to create the basis of an MCMC algorithm, which can be specified to parametric models with minimal effort. The appeal of this novel approach is the wide class of models which it permits, ease of implementation and the posterior-like fiducial distribution without the need for a prior. We provide background information for the derivation of the relevant fiducial quantities, and a proof that the proposed MCMC algorithm targets the correct fiducial distribution. We need not assume independence nor identical distribution of the data, which makes the method attractive for application to time series and spatial data. Well-performing simulation results of the MA(1) and Matérn models are presented.

Constrained Fiducial Inference for Gaussian Models

TL;DR

Addresses fitting parametric Gaussian models using a prior-free fiducial approach that accommodates non-i.i.d. time-series and spatial data. The core method reparameterizes the covariance with a Cayley transform, defines a general data-generating algorithm , and constrains it to the model manifold to yield the Generalized Constrained Fiducial Distribution (GCFD) used in an MCMC. The paper derives fiducial quantities, provides a practical Pseudocode, proves that the MCMC targets the fiducial distribution, and demonstrates performance on MA(1) and Matérn models with publicly available Matlab code. The work extends fiducial inference to a broad class of Gaussian models, enabling Bayesian-like uncertainty without priors, and discusses computational strategies including composite-likelihood to improve scalability for larger arrays.

Abstract

We propose a new fiducial Markov Chain Monte Carlo (MCMC) method for fitting parametric Gaussian models. We utilize the Cayley transform to decompose the parametric covariance matrix, which in turn allows us to formulate a general data generating algorithm for Gaussian data. Leveraging constrained generalized fiducial inference, we are able to create the basis of an MCMC algorithm, which can be specified to parametric models with minimal effort. The appeal of this novel approach is the wide class of models which it permits, ease of implementation and the posterior-like fiducial distribution without the need for a prior. We provide background information for the derivation of the relevant fiducial quantities, and a proof that the proposed MCMC algorithm targets the correct fiducial distribution. We need not assume independence nor identical distribution of the data, which makes the method attractive for application to time series and spatial data. Well-performing simulation results of the MA(1) and Matérn models are presented.
Paper Structure (5 sections, 1 theorem, 38 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 5 sections, 1 theorem, 38 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Derivation of the Fiducial Distribution
  3. Pseudocode
  4. Simulations
  5. Proof

Key Result

Proposition 1

Assume the stationary distribution The proposed algorithm satisfies detailed balance.

Figures (6)

  • Figure 1: Left: Estimate of $\rho = .5$ from 200 runs of our algorithm, compared to the MLE for the same data. Right: Estimate of $\sigma^2 = 6$ from 200 runs of our algorithm, compared to the MLE for the same data.
  • Figure 2: Scatter plot of estimated parameter values for each of the 200 runs on simulated MA(1) data. The red dot indicates the true parameter value.
  • Figure 3: Visualization of the modification of the data for the composite likelihood approach. Here a $2 \times 7$ matrix becomes a $6 \times 5$ matrix.
  • Figure 4: Trace plots for both parameters of the MA(1) for the full likelihood and composite likelihood approaches. Note that the mixing of the composite likelihood is good, while that of the full likelihood is subpar.
  • Figure 5: Estimates of Matérn parameters from 200 runs of our algorithm, compared to the MLE for the same data. From left to right, the estimated parameters are $\nu, \sigma^2,\rho$, with true values of $2,6$, and $1$, respectively.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Proposition 1