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arxiv: 2511.20183 · v2 · pith:KTGFTZA2new · submitted 2025-11-25 · 📊 stat.AP

Multi-fidelity Gaussian process regression for noisy outputs and non-nested experimental designs: a comparison between the recursive and non-recursive formulations

Nils Baillie , Baptiste Kerleguer , Cyril Feau , Josselin Garnier This is my paper

Pith reviewed 2026-05-21 18:35 UTC · model grok-4.3

classification 📊 stat.AP
keywords multi-fidelity Gaussian processesrecursive formulationexpectation-maximizationnon-nested designsnoisy outputssurrogate modelingscaling factor
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The pith

The recursive multi-fidelity Gaussian process trains faster than the non-recursive version by using decoupled EM optimization on noisy non-nested data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper compares a recursive formulation of auto-regressive multi-fidelity Gaussian process regression against the classical non-recursive approach of Kennedy and O'Hagan. It develops a decoupled optimization strategy that applies the expectation-maximization algorithm and derives closed-form update formulas when the scaling factor is a parametric linear predictor. Benchmarks on applications of increasing complexity show that the recursive strategy cuts training time substantially when low-fidelity data sets are large, while delivering competitive predictive accuracy and uncertainty estimates. Readers working with surrogate models for expensive simulations may care because faster training makes it practical to incorporate abundant cheap data without sacrificing reliability.

Core claim

The recursive auto-regressive multi-fidelity Gaussian process model admits a decoupled optimization strategy based on the expectation-maximization algorithm; when the scaling factor between fidelity levels is modeled as a parametric linear predictor this yields closed-form update formulas, and benchmark experiments demonstrate that the resulting procedure reduces training time relative to the fully coupled likelihood maximization of the non-recursive formulation while maintaining competitive predictive accuracy and uncertainty estimation on noisy non-nested data.

What carries the argument

Decoupled expectation-maximization optimization of the recursive auto-regressive multi-fidelity Gaussian process model that supplies closed-form updates for the parametric linear scaling predictor.

If this is right

  • Training time decreases significantly when large low-fidelity data sets are available.
  • Predictive accuracy remains competitive with the non-recursive formulation.
  • Uncertainty estimation quality is preserved across the tested applications.
  • The method handles both noisy outputs and non-nested experimental designs without requiring special nesting assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decoupled EM structure might be adapted to other autoregressive multi-fidelity surrogate models that currently use coupled likelihoods.
  • As low-fidelity data volumes continue to increase in practice the relative speed advantage would grow and could enable more frequent model retraining.
  • The approach could be combined with active learning loops that decide when to acquire additional high-fidelity points.

Load-bearing premise

The scaling factor between fidelity levels can be modeled as a parametric linear predictor that permits closed-form EM update formulas and decoupled optimization.

What would settle it

A benchmark experiment on noisy non-nested data in which the recursive method shows no reduction in training time or substantially lower predictive accuracy and uncertainty quality would falsify the performance claims.

Figures

Figures reproduced from arXiv: 2511.20183 by Baptiste Kerleguer, Cyril Feau, Josselin Garnier, Nils Baillie.

Figure 1
Figure 1. Figure 1: Box-plots of 1 − Q2 values for both models: Single-fidelity GP on HF data (HP-only-GP, red), Multi-fidelity GP (MF-GP, blue) for increasing values of NH and different values of the HF noise variance. Analogous plots for the MF-PCE are given in [11, p. 9]. 10 20 30 40 50 NH 10 2 10 1 IAE CI 2 , H = 0.008 2 HF only MF 10 20 30 40 50 NH IAE CI 2 , H = 0.166 2 HF only MF [PITH_FULL_IMAGE:figures/full_fig_p014… view at source ↗
Figure 2
Figure 2. Figure 2: Box-plots of IAECI values for both models: HF-only-GP (red) and MF-GP (blue) for increasing values of NH and different values of the HF noise variance. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Box-plots of IAEPI values for both models: HF-only-GP (red) and MF-GP (blue) for increasing values of NH and different values of the HF noise variance. 4.2 Analytical 4D case: the Park function We are interested in assessing the performance of the auto-regressive model on a more complicated, multi-dimensional test case. We consider the Park function with: yH(x1, x2, x3, x4) = x1 2 r 1 + (x2 + x 2 3 ) x4 x… view at source ↗
Figure 4
Figure 4. Figure 4: Predicted values versus true function values for the 3 GPs of interest: the LF [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: α-CI (left) and α-PI (right) plots for the 3 GPs of interest: the LF-GP of the MF model (green, solid), the HF-GP of the MF model (blue, dotted) and the single￾fidelity GP trained on HF data only (red, dash-dot). The respective IAE values are also given. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Box-plots of 1 − Q2 values for all models: single-fidelity GP on HF data (HF￾only-GP, red), multi-fidelity GP with NL = 75 (royal blue) and multi-fidelity GP with NL = 150 (turquoise) for different values of NH. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Box-plots of IAECI values (left) and CIW95% values (right) for all models: single-fidelity GP on HF data (HF-only-GP, red), multi-fidelity GP with NL = 75 (royal blue) and multi-fidelity GP with NL = 150 (turquoise) for different values of NH. 20 40 60 NH 10 2 10 1 IAE PI HF only MF (NL = 75) MF (NL = 150) 20 40 60 NH 2 × 10 0 3 × 10 0 4 × 10 0 6 × 10 0 Width 95% PI HF only MF (NL = 75) MF (NL = 150) [PIT… view at source ↗
Figure 8
Figure 8. Figure 8: Box-plots of IAEPI values (left) and PIW95% values (right) for all models: single-fidelity GP on HF data (HF-only-GP, red), multi-fidelity GP with NL = 75 (royal blue) and multi-fidelity GP with NL = 150 (turquoise) for different values of NH. 4.3 Real-world case: sea surface temperature dataset We now apply the noisy AR(1) recursive multi-fidelity GP model to a real dataset of sea surface temperatures (SS… view at source ↗
Figure 9
Figure 9. Figure 9: Multi-fidelity predictions and confidence intervals at each day of the 2015 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Multi-fidelity mean predictions (left) and standard deviations (right) com [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
read the original abstract

This paper investigates a recursive formulation of auto-regressive multi-fidelity Gaussian process regression in the challenging setting of noisy and non-nested high- and low-fidelity data. We propose a decoupled optimization strategy based on the expectation-maximization algorithm, which exploits the structure of the recursive model. In particular, we derive closed-form update formulas when the scaling factor is modeled as a parametric linear predictor. This approach is compared with the fully coupled likelihood maximization of the classical non-recursive formulation introduced by Kennedy and O'Hagan. A series of benchmark experiments, covering applications of increasing complexity, highlights the performance of both approaches. The results demonstrate that the proposed recursive strategy significantly reduces training time, especially when large low-fidelity datasets are available, while maintaining competitive predictive accuracy and uncertainty estimation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a recursive auto-regressive formulation for multi-fidelity Gaussian process regression suited to noisy outputs and non-nested designs. It introduces a decoupled EM optimization strategy that yields closed-form M-step updates when the inter-fidelity scaling factor is represented by a parametric linear predictor, and compares this approach against the fully coupled Kennedy-O'Hagan non-recursive formulation. Benchmark experiments on problems of increasing complexity are used to demonstrate that the recursive method substantially reduces training time (particularly with large low-fidelity sets) while preserving competitive predictive accuracy and uncertainty calibration.

Significance. If the reported performance parity holds under the stated modeling assumptions, the work supplies a practical, computationally scalable route for multi-fidelity GP modeling when abundant low-fidelity data are available. The derivation of closed-form EM updates for the linear scaling case is a clear technical contribution that could be adopted in engineering and scientific applications where high-fidelity evaluations remain expensive.

major comments (2)
  1. [EM derivation and abstract] The central performance claims rest on the assumption that the scaling factor between fidelity levels admits a parametric linear predictor, which supplies the closed-form M-step and permits independent optimization of the low-fidelity GP and the scaling parameters. The manuscript does not derive approximation error bounds relative to the joint likelihood nor present a controlled misspecification study (e.g., synthetic nonlinear or input-dependent scaling) that would quantify bias in predictive accuracy or uncertainty calibration when this linearity is violated.
  2. [Benchmark experiments section] Benchmark experiments are invoked to support the time-accuracy trade-off, yet the description lacks explicit details on data splits, exact implementation of the non-recursive baseline, hyperparameter initialization, or the number of Monte Carlo repetitions used for uncertainty metrics. Without these, the strength of evidence for “maintaining competitive predictive accuracy and uncertainty estimation” cannot be fully verified.
minor comments (2)
  1. [Model formulation] Clarify the precise definition of the linear predictor coefficients for the scaling factor and their relation to the free-parameter count listed in the supplementary material.
  2. [Results] Add a short table summarizing wall-clock times, RMSE, and negative log predictive density for both methods across all benchmark problems to facilitate direct comparison.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment point by point below, proposing revisions where they strengthen the manuscript while maintaining its scope and focus on the linear scaling case for closed-form updates.

read point-by-point responses

  1. Referee: [EM derivation and abstract] The central performance claims rest on the assumption that the scaling factor between fidelity levels admits a parametric linear predictor, which supplies the closed-form M-step and permits independent optimization of the low-fidelity GP and the scaling parameters. The manuscript does not derive approximation error bounds relative to the joint likelihood nor present a controlled misspecification study (e.g., synthetic nonlinear or input-dependent scaling) that would quantify bias in predictive accuracy or uncertainty calibration when this linearity is violated.

    Authors: The paper deliberately restricts attention to the parametric linear predictor case for the scaling factor precisely because it yields closed-form M-step updates and enables the decoupled EM procedure described in Section 3. This modeling choice is stated explicitly in the abstract and is the basis for the computational advantages claimed relative to the fully coupled Kennedy-O'Hagan formulation (which can also employ a linear scaling). We agree that neither approximation error bounds nor a controlled misspecification study appear in the current manuscript. Adding rigorous bounds would constitute substantial new theoretical work outside the present scope; we therefore cannot supply them in a revision. We will, however, expand the discussion to include a qualitative assessment of the consequences of mild nonlinearity and to recommend more flexible scaling representations when the linearity assumption is in doubt. revision: partial

  2. Referee: [Benchmark experiments section] Benchmark experiments are invoked to support the time-accuracy trade-off, yet the description lacks explicit details on data splits, exact implementation of the non-recursive baseline, hyperparameter initialization, or the number of Monte Carlo repetitions used for uncertainty metrics. Without these, the strength of evidence for “maintaining competitive predictive accuracy and uncertainty estimation” cannot be fully verified.

    Authors: We accept that the current description of the benchmark experiments is insufficient for full reproducibility. In the revised manuscript we will augment the relevant section with: (i) explicit train/test splits and the precise construction of the non-nested designs for each example; (ii) implementation particulars of the non-recursive baseline, including the optimizer and any external libraries; (iii) the hyperparameter initialization protocol used for both the recursive EM and the coupled approaches; and (iv) the number of Monte Carlo repetitions performed to obtain the uncertainty calibration statistics. These additions will allow readers to verify the reported performance parity. revision: yes

standing simulated objections not resolved
  • Deriving approximation error bounds relative to the joint likelihood for the recursive formulation when the linear scaling assumption is violated.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The paper starts from the standard recursive auto-regressive multi-fidelity GP model, assumes a parametric linear form for the scaling factor between fidelities, and derives closed-form EM updates from that modeling choice. It then compares the resulting decoupled optimization against the fully coupled Kennedy-O'Hagan non-recursive formulation on benchmark data. No step reduces a claimed prediction or uniqueness result to a self-fit or self-citation by construction; the performance claims rest on external experimental comparison rather than internal redefinition. The linear-scaling assumption is an explicit modeling decision whose consequences are evaluated empirically, not smuggled in via prior self-work.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on standard Gaussian process priors and the auto-regressive multi-fidelity structure; the novel element is the algorithmic decoupling via EM rather than new theoretical primitives.

free parameters (1)
  • linear predictor coefficients for scaling factor
    Treated as parametric and updated in closed form within the EM procedure; these are fitted to data.
axioms (1)
  • domain assumption Auto-regressive structure linking high- and low-fidelity outputs
    Standard modeling choice for multi-fidelity GP regression invoked to enable recursive formulation.

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