Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posed

• URL: http://arxiv.org/abs/2203.09179v3
• Date: Tue, 25 Apr 2023 07:59:22 GMT
• Title: Maximum Likelihood Estimation in Gaussian Process Regression is Ill-Posed
• Authors: Toni Karvonen and Chris J. Oates
• Abstract summary: It remains an open problem to establish the circumstances in which maximum likelihood estimation is well-posed. This article identifies scenarios where the maximum likelihood estimator fails to be well-posed. Although the failure of maximum likelihood estimation is part of Gaussian process folklore, these rigorous theoretical results appear to be the first of their kind.
• Score: 7.018149356115115
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Gaussian process regression underpins countless academic and industrial applications of machine learning and statistics, with maximum likelihood estimation routinely used to select appropriate parameters for the covariance kernel. However, it remains an open problem to establish the circumstances in which maximum likelihood estimation is well-posed, that is, when the predictions of the regression model are insensitive to small perturbations of the data. This article identifies scenarios where the maximum likelihood estimator fails to be well-posed, in that the predictive distributions are not Lipschitz in the data with respect to the Hellinger distance. These failure cases occur in the noiseless data setting, for any Gaussian process with a stationary covariance function whose lengthscale parameter is estimated using maximum likelihood. Although the failure of maximum likelihood estimation is part of Gaussian process folklore, these rigorous theoretical results appear to be the first of their kind. The implication of these negative results is that well-posedness may need to be assessed post-hoc, on a case-by-case basis, when maximum likelihood estimation is used to train a Gaussian process model.

Related papers

• Relaxed Quantile Regression: Prediction Intervals for Asymmetric Noise [51.87307904567702]
  Quantile regression is a leading approach for obtaining such intervals via the empirical estimation of quantiles in the distribution of outputs. We propose Relaxed Quantile Regression (RQR), a direct alternative to quantile regression based interval construction that removes this arbitrary constraint. We demonstrate that this added flexibility results in intervals with an improvement in desirable qualities.
  arXiv  Detail & Related papers  (2024-06-05T13:36:38Z)
• Calibrating Neural Simulation-Based Inference with Differentiable Coverage Probability [50.44439018155837]
  We propose to include a calibration term directly into the training objective of the neural model. By introducing a relaxation of the classical formulation of calibration error we enable end-to-end backpropagation. It is directly applicable to existing computational pipelines allowing reliable black-box posterior inference.
  arXiv  Detail & Related papers  (2023-10-20T10:20:45Z)
• Robust Gaussian Process Regression with Huber Likelihood [2.7184224088243365]
  We propose a robust process model in the Gaussian process framework with the likelihood of observed data expressed as the Huber probability distribution. The proposed model employs weights based on projection statistics to scale residuals and bound the influence of vertical outliers and bad leverage points on the latent functions estimates.
  arXiv  Detail & Related papers  (2023-01-19T02:59:33Z)
• Posterior and Computational Uncertainty in Gaussian Processes [52.26904059556759]
  Gaussian processes scale prohibitively with the size of the dataset. Many approximation methods have been developed, which inevitably introduce approximation error. This additional source of uncertainty, due to limited computation, is entirely ignored when using the approximate posterior. We develop a new class of methods that provides consistent estimation of the combined uncertainty arising from both the finite number of data observed and the finite amount of computation expended.
  arXiv  Detail & Related papers  (2022-05-30T22:16:25Z)
• Learning to Estimate Without Bias [57.82628598276623]
  Gauss theorem states that the weighted least squares estimator is a linear minimum variance unbiased estimation (MVUE) in linear models. In this paper, we take a first step towards extending this result to non linear settings via deep learning with bias constraints. A second motivation to BCE is in applications where multiple estimates of the same unknown are averaged for improved performance.
  arXiv  Detail & Related papers  (2021-10-24T10:23:51Z)
• Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning [78.83598532168256]
  Marginal-likelihood based model-selection is rarely used in deep learning due to estimation difficulties. Our work shows that marginal likelihoods can improve generalization and be useful when validation data is unavailable.
  arXiv  Detail & Related papers  (2021-04-11T09:50:24Z)
• Learning Probabilistic Ordinal Embeddings for Uncertainty-Aware Regression [91.3373131262391]
  Uncertainty is the only certainty there is. Traditionally, the direct regression formulation is considered and the uncertainty is modeled by modifying the output space to a certain family of probabilistic distributions. How to model the uncertainty within the present-day technologies for regression remains an open issue.
  arXiv  Detail & Related papers  (2021-03-25T06:56:09Z)
• Distributionally Robust Parametric Maximum Likelihood Estimation [13.09499764232737]
  We propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric nominal distribution. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.
  arXiv  Detail & Related papers  (2020-10-11T19:05:49Z)
• Maximum likelihood estimation and uncertainty quantification for Gaussian process approximation of deterministic functions [10.319367855067476]
  This article provides one of the first theoretical analyses in the context of Gaussian process regression with a noiseless dataset. We show that the maximum likelihood estimation of the scale parameter alone provides significant adaptation against misspecification of the Gaussian process model.
  arXiv  Detail & Related papers  (2020-01-29T17:20:21Z)
• Robust Gaussian Process Regression with a Bias Model [0.6850683267295248]
  Most existing approaches replace an outlier-prone Gaussian likelihood with a non-Gaussian likelihood induced from a heavy tail distribution. The proposed approach models an outlier as a noisy and biased observation of an unknown regression function. Conditioned on the bias estimates, the robust GP regression can be reduced to a standard GP regression problem.
  arXiv  Detail & Related papers  (2020-01-14T06:21:51Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.