arxiv: 2605.08018 · v1 · submitted 2026-05-08 · 📊 stat.ME

Recognition: no theorem link

BAMIFun: Bayesian Multiple Imputation for Functional Data

Eric F. Lock, Erjia Cui, Lei Xuan, Ziren Jiang

Pith reviewed 2026-05-11 02:39 UTC · model grok-4.3

classification 📊 stat.ME

keywords Bayesian multiple imputationfunctional data analysismissing datapenalized splineslow-rank modelsfunctional principal componentstensor decompositionGibbs sampler

0 comments

The pith

Bayesian multiple imputation for functional data provides accurate reconstructions and reliable uncertainty estimates by drawing from posterior distributions rather than single point estimates.

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

The paper develops BAMIFun, a Bayesian framework for multiple imputation of missing values in functional datasets that are often observed sparsely or irregularly. It replaces single-imputation methods based on functional principal component analysis with draws from a posterior distribution, using a low-rank model with penalized splines to enforce smoothness of eigenfunctions and an efficient Gibbs sampler for computation. The approach extends to multiway functional data via a low-rank functional tensor singular value decomposition model. A sympathetic reader would care because single imputation typically produces overly optimistic inferences in downstream analyses, while multiple imputation better captures estimation uncertainty and yields improved coverage. Simulations and applications to physical activity and infant gut microbiome datasets under severe missingness illustrate these gains.

Core claim

BAMIFun imposes a Bayesian low-rank model that incorporates penalized spline representations to enforce smoothness of eigenfunctions and derives an efficient Gibbs sampler algorithm for posterior computation. For single-level functional data this enables multiple imputations that properly account for estimation uncertainties in downstream analysis. The framework extends to multiway functional data using a low-rank Functional Tensor Singular Value Decomposition model. Simulation studies show that compared to existing methods BAMIFun achieves accurate imputation while providing substantially improved coverage and more reliable downstream inference.

What carries the argument

Bayesian low-rank model with penalized spline representations for eigenfunctions and Gibbs sampling for posterior draws, extended via low-rank Functional Tensor Singular Value Decomposition for multiway data.

If this is right

Multiple imputations drawn from the posterior replace single point estimates and thereby avoid overconfident downstream inferences.
Coverage probabilities for parameters in subsequent analyses improve relative to existing single-imputation functional principal component methods.
The same framework applies directly to multiway functional data where no prior multiple-imputation methods existed.
Case studies on physical activity trajectories and infant gut microbiome data confirm practical advantages when missingness is severe.

Where Pith is reading between the lines

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

If the penalized spline low-rank representation remains adequate for other classes of smooth trajectories, the method could be applied to longitudinal biomedical studies that record irregular time courses.
Efficiency of the Gibbs sampler may allow scaling to larger numbers of subjects or denser grids once computational cost is profiled.
The multiway extension suggests a route to handle tensor-valued functional observations with missing entries in fields such as neuroimaging.

Load-bearing premise

The low-rank structure with penalized splines sufficiently captures the true underlying functional variation and the Bayesian model correctly represents the posterior uncertainty under the chosen priors and sampling scheme.

What would settle it

In a simulation study where true functional curves are generated from a model whose effective rank exceeds the low-rank assumption, the empirical coverage of 95 percent intervals constructed from BAMIFun imputations falls substantially below the nominal level.

Figures

Figures reproduced from arXiv: 2605.08018 by Eric F. Lock, Erjia Cui, Lei Xuan, Ziren Jiang.

**Figure 2.** Figure 2: Simulation results for the performance of downstream Scalar-on-Function Regression [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Multiple imputation results using the infant gut microbiome dataset. The top panel [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

read the original abstract

Missing data are pervasive in modern functional datasets, where trajectories are often sparsely or irregularly observed. Although Functional Principal Component Analysis (FPCA) is widely used to reconstruct incomplete curves, existing FPCA-based approaches typically employ single imputation, leading to overly optimistic inferences in downstream analyses. To address these challenges, we develop a novel Bayesian multiple imputation framework for functional data (BAMIFun). For single-level functional data, we impose a Bayesian low-rank model that incorporates penalized spline representations to enforce smoothness of eigenfunctions and derive an efficient Gibbs sampler algorithm for posterior computation. In addition, we demonstrate and validate how to properly account for the estimation uncertainties in downstream analysis. Furthermore, we extend the framework to multiway functional data using a low-rank Functional Tensor Singular Value Decomposition (FTSVD) model, enabling Bayesian multiple imputation in settings not supported by existing methods. Simulation studies show that, compared to existing methods, BAMIFun achieves accurate imputation while providing substantially improved coverage and more reliable downstream inference. Case studies using a physical activity dataset and an infant gut microbiome dataset further demonstrate the practical advantages of our proposed methods under severe missingness. Code for our algorithms is available at https://github.com/ZirenJiang/BAMIFun.

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.

Desk Editor's Note private letter to a colleague

BAMIFun gives a Bayesian multiple-imputation route for missing functional data via low-rank penalized splines and an FTSVD extension, but the reported coverage gains rest on simulations that match the model's own assumptions.

read the letter

The paper's core move is to replace single-imputation FPCA with a Bayesian low-rank model that uses penalized splines for smoothness and draws multiple imputations from the posterior. They supply a Gibbs sampler for the single-level case and extend the same logic to multiway data through FTSVD. This directly targets the over-optimism problem when downstream analyses treat imputed curves as fixed.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces BAMIFun, a Bayesian multiple imputation framework for functional data with missingness. For single-level data it specifies a low-rank model using penalized splines for smooth eigenfunctions together with a Gibbs sampler; the approach is extended to multiway data via a low-rank Functional Tensor Singular Value Decomposition (FTSVD) model. The central claims are that the method yields accurate imputations, substantially improved coverage relative to existing single-imputation FPCA methods, and more reliable downstream inference, supported by simulation studies and two case studies on physical activity and infant gut microbiome data.

Significance. If the low-rank penalized-spline and FTSVD assumptions hold for the target data, the framework supplies a coherent mechanism for propagating imputation uncertainty into downstream functional-data analyses, addressing a recognized limitation of single-imputation approaches. The public release of code is a positive feature for reproducibility.

major comments (3)

[§4] §4 (Simulation studies): Data are generated from the same low-rank penalized-spline model that BAMIFun assumes; the reported coverage gains and downstream-inference improvements are therefore conditional on correct model specification and do not address performance under higher-rank or non-smooth trajectories, which is load-bearing for the claim of 'substantially improved coverage'.
[§3.2] §3.2 (Downstream inference): The procedure for combining multiple imputations with a subsequent analysis model is outlined but lacks an explicit derivation or theorem establishing frequentist coverage or Bayesian calibration of the resulting intervals; without this, the assertion of 'more reliable downstream inference' rests only on empirical simulation results.
[§3.3] §3.3 (Multiway extension): The FTSVD low-rank model is introduced without reported sensitivity checks on the chosen tensor rank or spline penalty parameters; because these are free parameters in the model, their misspecification directly affects the posterior uncertainty used for imputation.

minor comments (2)

[Abstract and §3.2] The abstract states that the method 'properly account[s] for the estimation uncertainties'; the corresponding section should include a short algorithmic box or pseudocode showing the exact steps for propagating the imputed draws into a generic downstream estimator.
[Tables in §4] Table captions for simulation results should report the number of Monte Carlo replications and the exact missingness mechanisms used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed and constructive comments. We address each major point below, indicating planned revisions where appropriate.

read point-by-point responses

Referee: §4 (Simulation studies): Data are generated from the same low-rank penalized-spline model that BAMIFun assumes; the reported coverage gains and downstream-inference improvements are therefore conditional on correct model specification and do not address performance under higher-rank or non-smooth trajectories, which is load-bearing for the claim of 'substantially improved coverage'.

Authors: We agree that the primary simulations are conducted under the assumed model. To evaluate robustness, we will add new simulation scenarios in the revised manuscript that generate data from higher-rank models and non-smooth trajectories, reporting imputation accuracy and coverage under these misspecification settings. revision: yes
Referee: §3.2 (Downstream inference): The procedure for combining multiple imputations with a subsequent analysis model is outlined but lacks an explicit derivation or theorem establishing frequentist coverage or Bayesian calibration of the resulting intervals; without this, the assertion of 'more reliable downstream inference' rests only on empirical simulation results.

Authors: We will revise Section 3.2 to include a clearer step-by-step derivation of the combining rules based on standard Bayesian multiple imputation theory (Rubin 1987), along with references to existing results on posterior calibration. A new general theorem on frequentist coverage is outside the scope of the current work, but the expanded explanation and simulation evidence will better support the downstream inference claims. revision: partial
Referee: §3.3 (Multiway extension): The FTSVD low-rank model is introduced without reported sensitivity checks on the chosen tensor rank or spline penalty parameters; because these are free parameters in the model, their misspecification directly affects the posterior uncertainty used for imputation.

Authors: We will incorporate sensitivity analyses for the multiway extension, varying the tensor rank and penalty parameters in both the simulation studies and the infant gut microbiome case study, and report their impact on imputation uncertainty and downstream results. revision: yes

Circularity Check

0 steps flagged

Bayesian hierarchical model for functional imputation is self-contained

full rationale

The paper defines a standard Bayesian low-rank penalized-spline model for single-level functional data and an FTSVD extension for multiway data, then derives a Gibbs sampler for posterior sampling and multiple imputation. Simulation studies evaluate coverage and imputation accuracy under the model's own generative assumptions, which is standard practice and does not constitute a reduction of any claimed result to its inputs by construction. No load-bearing step relies on self-citation of an unverified uniqueness theorem or ansatz; the framework applies established Bayesian hierarchical modeling techniques to functional data without tautological equivalences.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The method rests on standard functional data analysis assumptions plus Bayesian modeling choices; hyperparameters for rank and penalties are typical but not enumerated in the abstract.

free parameters (2)

low-rank dimension
Rank of the approximation is a modeling choice that must be selected or estimated.
spline penalty parameters
Smoothing penalties are hyperparameters that control eigenfunction smoothness.

axioms (2)

domain assumption Functional trajectories admit a low-rank representation with smooth eigenfunctions.
Core modeling assumption invoked for the Bayesian low-rank model.
standard math The Gibbs sampler produces draws from the target posterior distribution.
Standard MCMC convergence assumption required for posterior inference.

pith-pipeline@v0.9.0 · 5517 in / 1295 out tokens · 59083 ms · 2026-05-11T02:39:33.049007+00:00 · methodology

discussion (0)

Reference graph

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