Joint distribution properties of Fully Conditional Specification under the normal linear model with normal inverse-gamma priors

Gerko Vink; Mingyang Cai; Stef van Buuren

arxiv: 2208.12930 · v1 · pith:VARVVIK2new · submitted 2022-08-27 · 📊 stat.CO · math.ST· stat.TH

Joint distribution properties of Fully Conditional Specification under the normal linear model with normal inverse-gamma priors

Mingyang Cai , Stef van Buuren , Gerko Vink This is my paper

Pith reviewed 2026-05-24 11:16 UTC · model grok-4.3

classification 📊 stat.CO math.STstat.TH

keywords fully conditional specificationmultiple imputationnormal inverse-gamma priorconvergencelinear regressionmissing data

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The pith

FCS converges to the joint distribution and prior specifications are equivalent under normal linear models with normal inverse-gamma priors.

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

This paper investigates the convergence properties of fully conditional specification (FCS) for multiple imputation when using informative normal inverse-gamma priors. It extends previous work on non-informative priors to this informative case for the normal linear model. Theoretical proofs and simulation studies establish that the FCS procedure converges to the appropriate stationary distribution. The work also shows that prior specifications on the conditional models lead to the same joint distribution as direct specification on the joint model. This allows flexible use of FCS in linear regression imputation without sacrificing consistency with joint modeling approaches.

Core claim

The theoretical and simulation results prove the convergence of FCS and show the equivalence of prior specification under the joint model and a set of conditional models when the analysis model is a linear regression with normal inverse-gamma priors.

What carries the argument

The FCS chain as a Gibbs sampler under normal linear model with normal inverse-gamma priors, establishing convergence and prior equivalence.

Load-bearing premise

The data-generating process and the analysis model are normal linear regressions with exactly normal inverse-gamma priors, and missingness allows FCS to be a valid Gibbs sampler.

What would settle it

A case where the FCS stationary distribution does not match the joint posterior under normal linear model and normal inverse-gamma priors would disprove the claimed equivalence.

Figures

Figures reproduced from arXiv: 2208.12930 by Gerko Vink, Mingyang Cai, Stef van Buuren.

read the original abstract

Fully conditional specification (FCS) is a convenient and flexible multiple imputation approach. It specifies a sequence of simple regression models instead of a potential complex joint density for missing variables. However, FCS may not converge to a stationary distribution. Many authors have studied the convergence properties of FCS when priors of conditional models are non-informative. We extend to the case of informative priors. This paper evaluates the convergence properties of the normal linear model with normal-inverse gamma prior. The theoretical and simulation results prove the convergence of FCS and show the equivalence of prior specification under the joint model and a set of conditional models when the analysis model is a linear regression with normal inverse-gamma priors.

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

This paper extends the known FCS convergence results to informative NIG priors on the normal linear model and shows that conditional-model priors are equivalent to the joint-model prior.

read the letter

The main takeaway is that FCS converges under normal-inverse-gamma priors for the normal linear model and that putting the priors on the separate conditional regressions gives the same joint distribution as specifying them directly on the joint model. This moves past the non-informative prior results that were already in the literature. The derivations start from the standard conjugate updating rules for the NIG family and the definition of the FCS iteration, then track the stationary distribution of the chain. The simulations are used to check the finite-sample behavior under the stated assumptions. Within the narrow scope the argument holds up without circularity or hidden steps. The limitation is exactly the scope: the data must be generated from the same normal linear model, the priors must be exactly NIG, and the missingness must allow FCS to act as a valid Gibbs sampler. The paper states these conditions plainly and does not claim results outside them. This is useful technical work for people who already use FCS and want to move to informative priors without losing the convergence guarantee. A reader working on imputation theory or software will find the equivalence result directly applicable. It is not a field-changing paper but a solid, scoped clarification. I would bring it to a methods reading group. I would cite it if writing on imputation convergence. It deserves peer review because the claims are formally grounded and the assumptions are explicit.

Referee Report

0 major / 2 minor

Summary. The paper claims that fully conditional specification (FCS) converges to a stationary distribution when applied to the normal linear model with normal-inverse-gamma (NIG) priors, and that prior specifications are equivalent between the joint model and the corresponding set of conditional models. These conclusions are supported by theoretical derivations extending prior work on non-informative priors and by accompanying simulation studies.

Significance. If the derivations hold, the result supplies a modest but practically relevant extension of existing FCS convergence theory to the informative conjugate-prior case that is routinely used in linear-regression imputation. It directly addresses a common modeling choice and thereby strengthens the justification for employing FCS rather than a full joint model in this setting.

minor comments (2)

[Abstract] Abstract: the statement that 'theoretical and simulation results prove the convergence' would be clearer if it explicitly listed the maintained assumptions (e.g., missingness mechanism that renders FCS a valid Gibbs sampler, exact conjugacy of the NIG prior).
The manuscript would benefit from a short table or appendix entry that records the exact hyper-parameter settings used in the simulation studies so that readers can reproduce the reported convergence behavior.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its practical relevance for FCS with informative conjugate priors, and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

Derivations start from standard NIG and FCS properties; no load-bearing circular reduction

full rationale

The paper's central claims concern convergence of FCS and equivalence of joint vs. conditional prior specifications under the normal linear model with exactly NIG priors. These follow from the known conjugate properties of the normal-inverse-gamma family and the definition of FCS as a Gibbs sampler under the stated missingness conditions. No step reduces a target result to a fitted parameter or self-citation chain by construction; the scope explicitly requires the data-generating process to match the analysis model. This matches the expected non-circular case for a scoped theoretical proof.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on the standard normal-linear-model likelihood, the conjugacy properties of the normal-inverse-gamma family, and the assumption that the FCS chain constitutes a valid Gibbs sampler under the stated missingness mechanism.

axioms (3)

domain assumption Data follow a normal linear model
Invoked throughout the convergence argument for both the imputation and analysis stages.
domain assumption Priors are exactly normal-inverse-gamma
Required for conjugacy that enables the closed-form conditional posteriors used in the FCS steps.
domain assumption Missingness mechanism permits a valid Gibbs sampler
Necessary for the FCS iteration to be a proper Markov chain with the claimed stationary distribution.

pith-pipeline@v0.9.0 · 5645 in / 1341 out tokens · 27873 ms · 2026-05-24T11:16:40.154422+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

[1]

APACrefauthors \ 2009

albert2009bayesian APACrefauthors Albert, J. APACrefauthors \ 2009 . Bayesian computation with R Bayesian computation with r . Springer

work page 2009
[2]

, Castillo, E

arnold2004compatibility APACrefauthors Arnold, B C. , Castillo, E. \ Sarabia, J M. APACrefauthors \ 2004 . Compatibility of partial or complete conditional probability specifications Compatibility of partial or complete conditional probability specifications . Journal of statistical planning and inference 123 1 133--159

work page 2004
[3]

\ Press, S J

arnold1989compatible APACrefauthors Arnold, B C. \ Press, S J. APACrefauthors \ 1989 . Compatible conditional distributions Compatible conditional distributions . Journal of the American Statistical Association 84 405 152--156

work page 1989
[4]

, Seaman, S R

bartlett2015multiple APACrefauthors Bartlett, J W. , Seaman, S R. , White, I R. , Carpenter, J R. \ Initiative*, A D N. APACrefauthors \ 2015 . Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model ....

work page 2015
[5]

\ Groothuis-Oudshoorn, K

buuren2010mice APACrefauthors Buuren, S v. \ Groothuis-Oudshoorn, K. APACrefauthors \ 2010 . mice: Multivariate imputation by chained equations in R mice: Multivariate imputation by chained equations in r . Journal of statistical software 1--68

work page 2010
[6]

APACrefauthors \ 2007

10.2307/20461449 APACrefauthors Eaton, M L. APACrefauthors \ 2007 . Multivariate Statistics: A Vector Space Approach Multivariate statistics: A vector space approach . Lecture Notes-Monograph Series 53 i--512 . APACrefURL http://www.jstor.org/stable/20461449 APACrefURL

work page doi:10.2307/20461449 2007
[7]

, Gabry, J

Goodrich2019 APACrefauthors Goodrich, B. , Gabry, J. , Ali, I. \ Brilleman, S. APACrefauthors \ 2019 . rstanarm: Bayesian applied regression modeling via Stan . rstanarm: Bayesian applied regression modeling via Stan . APACrefURL https://mc-stan.org/rstanarm APACrefURL R package version 2.19.2

work page 2019
[8]

, Chickering, D M

heckerman2000dependency APACrefauthors Heckerman, D. , Chickering, D M. , Meek, C. , Rounthwaite, R. \ Kadie, C. APACrefauthors \ 2000 . Dependency networks for inference, collaborative filtering, and data visualization Dependency networks for inference, collaborative filtering, and data visualization . Journal of Machine Learning Research 1 Oct 49--75

work page 2000
[9]

, White, I R

hughes2014joint APACrefauthors Hughes, R A. , White, I R. , Seaman, S R. , Carpenter, J R. , Tilling, K. \ Sterne, J A. APACrefauthors \ 2014 . Joint modelling rationale for chained equations Joint modelling rationale for chained equations . BMC medical research methodology 14 1 28

work page 2014
[10]

, Gelman, A

liu2014stationary APACrefauthors Liu, J. , Gelman, A. , Hill, J. , Su, Y S. \ Kropko, J. APACrefauthors \ 2014 . On the stationary distribution of iterative imputations On the stationary distribution of iterative imputations . Biometrika 101 1 155--173

work page 2014
[11]

, van Buuren, S

oberman2020missing APACrefauthors Oberman, H I. , van Buuren, S. \ Vink, G. APACrefauthors \ 2020 . Missing the Point: Non-Convergence in Iterative Imputation Algorithms Missing the point: Non-convergence in iterative imputation algorithms

work page 2020
[12]

, Lepkowski, J M

raghunathan2001multivariate APACrefauthors Raghunathan, T E. , Lepkowski, J M. , Van Hoewyk, J. , Solenberger, P. \ . APACrefauthors \ 2001 . A multivariate technique for multiply imputing missing values using a sequence of regression models A multivariate technique for multiply imputing missing values using a sequence of regression models . Survey method...

work page 2001
[13]

APACrefauthors \ 1997

schafer1997analysis APACrefauthors Schafer, J L. APACrefauthors \ 1997 . Analysis of incomplete multivariate data Analysis of incomplete multivariate data . Chapman and Hall/CRC

work page 1997
[14]

APACrefauthors \ 2007

van2007multiple APACrefauthors Van Buuren, S. APACrefauthors \ 2007 . Multiple imputation of discrete and continuous data by fully conditional specification Multiple imputation of discrete and continuous data by fully conditional specification . Statistical methods in medical research 16 3 219--242

work page 2007
[15]

APACrefauthors \ 2018

van2018flexible APACrefauthors Van Buuren, S. APACrefauthors \ 2018 . Flexible imputation of missing data Flexible imputation of missing data . CRC press

work page 2018
[16]

, Brand, J P

van2006fully APACrefauthors Van Buuren, S. , Brand, J P. , Groothuis-Oudshoorn, C G. \ Rubin, D B. APACrefauthors \ 2006 . Fully conditional specification in multivariate imputation Fully conditional specification in multivariate imputation . Journal of statistical computation and simulation 76 12 1049--1064

work page 2006
[17]

\ Raghunathan, T E

zhu2015convergence APACrefauthors Zhu, J. \ Raghunathan, T E. APACrefauthors \ 2015 . Convergence properties of a sequential regression multiple imputation algorithm Convergence properties of a sequential regression multiple imputation algorithm . Journal of the American Statistical Association 110 511 1112--1124

work page 2015

[1] [1]

APACrefauthors \ 2009

albert2009bayesian APACrefauthors Albert, J. APACrefauthors \ 2009 . Bayesian computation with R Bayesian computation with r . Springer

work page 2009

[2] [2]

, Castillo, E

arnold2004compatibility APACrefauthors Arnold, B C. , Castillo, E. \ Sarabia, J M. APACrefauthors \ 2004 . Compatibility of partial or complete conditional probability specifications Compatibility of partial or complete conditional probability specifications . Journal of statistical planning and inference 123 1 133--159

work page 2004

[3] [3]

\ Press, S J

arnold1989compatible APACrefauthors Arnold, B C. \ Press, S J. APACrefauthors \ 1989 . Compatible conditional distributions Compatible conditional distributions . Journal of the American Statistical Association 84 405 152--156

work page 1989

[4] [4]

, Seaman, S R

bartlett2015multiple APACrefauthors Bartlett, J W. , Seaman, S R. , White, I R. , Carpenter, J R. \ Initiative*, A D N. APACrefauthors \ 2015 . Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model Multiple imputation of covariates by fully conditional specification: Accommodating the substantive model ....

work page 2015

[5] [5]

\ Groothuis-Oudshoorn, K

buuren2010mice APACrefauthors Buuren, S v. \ Groothuis-Oudshoorn, K. APACrefauthors \ 2010 . mice: Multivariate imputation by chained equations in R mice: Multivariate imputation by chained equations in r . Journal of statistical software 1--68

work page 2010

[6] [6]

APACrefauthors \ 2007

10.2307/20461449 APACrefauthors Eaton, M L. APACrefauthors \ 2007 . Multivariate Statistics: A Vector Space Approach Multivariate statistics: A vector space approach . Lecture Notes-Monograph Series 53 i--512 . APACrefURL http://www.jstor.org/stable/20461449 APACrefURL

work page doi:10.2307/20461449 2007

[7] [7]

, Gabry, J

Goodrich2019 APACrefauthors Goodrich, B. , Gabry, J. , Ali, I. \ Brilleman, S. APACrefauthors \ 2019 . rstanarm: Bayesian applied regression modeling via Stan . rstanarm: Bayesian applied regression modeling via Stan . APACrefURL https://mc-stan.org/rstanarm APACrefURL R package version 2.19.2

work page 2019

[8] [8]

, Chickering, D M

heckerman2000dependency APACrefauthors Heckerman, D. , Chickering, D M. , Meek, C. , Rounthwaite, R. \ Kadie, C. APACrefauthors \ 2000 . Dependency networks for inference, collaborative filtering, and data visualization Dependency networks for inference, collaborative filtering, and data visualization . Journal of Machine Learning Research 1 Oct 49--75

work page 2000

[9] [9]

, White, I R

hughes2014joint APACrefauthors Hughes, R A. , White, I R. , Seaman, S R. , Carpenter, J R. , Tilling, K. \ Sterne, J A. APACrefauthors \ 2014 . Joint modelling rationale for chained equations Joint modelling rationale for chained equations . BMC medical research methodology 14 1 28

work page 2014

[10] [10]

, Gelman, A

liu2014stationary APACrefauthors Liu, J. , Gelman, A. , Hill, J. , Su, Y S. \ Kropko, J. APACrefauthors \ 2014 . On the stationary distribution of iterative imputations On the stationary distribution of iterative imputations . Biometrika 101 1 155--173

work page 2014

[11] [11]

, van Buuren, S

oberman2020missing APACrefauthors Oberman, H I. , van Buuren, S. \ Vink, G. APACrefauthors \ 2020 . Missing the Point: Non-Convergence in Iterative Imputation Algorithms Missing the point: Non-convergence in iterative imputation algorithms

work page 2020

[12] [12]

, Lepkowski, J M

raghunathan2001multivariate APACrefauthors Raghunathan, T E. , Lepkowski, J M. , Van Hoewyk, J. , Solenberger, P. \ . APACrefauthors \ 2001 . A multivariate technique for multiply imputing missing values using a sequence of regression models A multivariate technique for multiply imputing missing values using a sequence of regression models . Survey method...

work page 2001

[13] [13]

APACrefauthors \ 1997

schafer1997analysis APACrefauthors Schafer, J L. APACrefauthors \ 1997 . Analysis of incomplete multivariate data Analysis of incomplete multivariate data . Chapman and Hall/CRC

work page 1997

[14] [14]

APACrefauthors \ 2007

van2007multiple APACrefauthors Van Buuren, S. APACrefauthors \ 2007 . Multiple imputation of discrete and continuous data by fully conditional specification Multiple imputation of discrete and continuous data by fully conditional specification . Statistical methods in medical research 16 3 219--242

work page 2007

[15] [15]

APACrefauthors \ 2018

van2018flexible APACrefauthors Van Buuren, S. APACrefauthors \ 2018 . Flexible imputation of missing data Flexible imputation of missing data . CRC press

work page 2018

[16] [16]

, Brand, J P

van2006fully APACrefauthors Van Buuren, S. , Brand, J P. , Groothuis-Oudshoorn, C G. \ Rubin, D B. APACrefauthors \ 2006 . Fully conditional specification in multivariate imputation Fully conditional specification in multivariate imputation . Journal of statistical computation and simulation 76 12 1049--1064

work page 2006

[17] [17]

\ Raghunathan, T E

zhu2015convergence APACrefauthors Zhu, J. \ Raghunathan, T E. APACrefauthors \ 2015 . Convergence properties of a sequential regression multiple imputation algorithm Convergence properties of a sequential regression multiple imputation algorithm . Journal of the American Statistical Association 110 511 1112--1124

work page 2015