Parametric multi-fidelity Monte Carlo estimation with applications to extremes

In a multi-fidelity setting, data are available from two sources, high- and low-fidelity. Low-fidelity data has larger size and can be leveraged to make more efficient inference about quantities of interest, e.g. the mean, for high-fidelity variables. In this work, such multi-fidelity setting is studied when the goal is to fit more efficiently a parametric model to high-fidelity data. Three multi-fidelity parameter estimation methods are considered, joint maximum likelihood, (multi-fidelity) moment estimation and (multi-fidelity) marginal maximum likelihood, and are illustrated on several parametric models, with the focus on parametric families used in extreme value analysis. An application is also provided concerning quantification of occurrences of extreme ship motions generated by two computer codes of varying fidelity.

💡 Research Summary

The paper addresses the problem of estimating parametric distribution parameters for a high‑fidelity (HF) random variable when both HF and low‑fidelity (LF) data are available. Classical multi‑fidelity Monte Carlo (MFMC) techniques reduce the variance of a mean estimator by using the LF mean as a control variate, but they do not directly address the estimation of distribution parameters that are needed for extreme‑value analysis, reliability assessment, or tail‑risk quantification. To fill this gap, the authors propose and analyze three multi‑fidelity parameter‑estimation strategies:

1. Joint Maximum Likelihood (JML) – assumes a fully specified joint distribution (F_{\eta}(y_1,y_2)) for ((Y^{(1)},Y^{(2)})). The estimator maximizes the combined likelihood of the (n) paired HF–LF observations and the additional (m) LF‑only observations. Because it exploits the complete dependence structure, JML attains the lowest asymptotic variance among the three methods, provided the joint model is correctly specified.

2. Moment Multi‑Fidelity (MoM) – assumes that the HF parameters (\theta_1) can be expressed as a deterministic function (g) of population moments of a transformation (h(Y^{(1)})). The method replaces the unknown moments by sample averages from the HF data and then applies an MFMC‑type control‑variates correction using the LF data. MoM requires only marginal information about the HF distribution and no joint model, but its efficiency depends on the linearity of (g) and on the strength of correlation between HF and LF variables. For non‑linear moment‑parameter relationships (e.g., variance or shape parameters of extreme‑value families) the efficiency can be substantially lower than JML.

3. Marginal Maximum Likelihood (MML) – assumes separate marginal models for HF and LF variables, (F^{(1)}{\theta_1}) and (F^{(2)}{\theta_2}), without specifying their joint dependence. First, the HF marginal parameters are estimated by standard maximum likelihood on the paired HF data; second, the LF marginal parameters are estimated on the LF‑only data. A linear correction term (\alpha = \operatorname{Cov}(Y^{(1)},Y^{(2)})/\operatorname{Var}(Y^{(2)})) is then estimated from the paired data and applied to the HF estimator, yielding a variance reduction analogous to a control‑variates adjustment. MML occupies a middle ground: it is easier to implement than JML (no joint density needed) and typically more efficient than MoM because it uses a calibrated linear correction.

The authors derive the asymptotic covariance matrices for each estimator using Fisher‑information theory and the joint covariance structure of ((Y^{(1)},Y^{(2)})). They illustrate the results on three families of distributions:

• Gaussian – where the joint distribution can be taken as bivariate normal. All three methods achieve the same asymptotic efficiency for the mean, while MoM is slightly less efficient for the variance because the variance is a non‑linear moment.
• Gumbel (a special case of the Generalized Extreme Value family) – here the joint distribution is not Gaussian, and the shape parameter is linked to higher‑order moments. JML shows the greatest variance reduction, MML is moderately efficient, and MoM suffers when the correlation between HF and LF is modest.
• Bernoulli – used to demonstrate the methods for discrete data; similar patterns of efficiency emerge.

A cost‑allocation analysis is also presented. Assuming a total computational budget (C) and per‑sample costs (c_1) (HF) and (c_2) (LF), the optimal numbers of HF and LF samples ((n,m)) that minimize the asymptotic variance satisfy \