A Fair Bayesian Inference through Matched Gibbs Posterior
Authors: Jihu Lee
Kunwoong Kim
Sehyun Park
Insung Kong
Dongyoon Yang
Yongdai Kim
Venue: ICLR 2026 Poster (2026)
Keywords: Algorithmic fairness
Bayesian inference
Gibbs posterior
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: With the growing importance of trustworthy AI, algorithmic fairness has emerged as a critical concern. 
Among various fairness notions, group fairness - which measures the model bias between sensitive groups - has received significant attention. 
While many group-fair models have focused on satisfying group fairness constraints, model uncertainty has received relatively little attention, despite its importance for robust and trustworthy decision-making. 
To address this, we adopt a Bayesian framework to capture model uncertainty in fair model training. 
We first define group-fair posterior distributions and then introduce a fair variational Bayesian inference. 
Then we propose a novel distribution termed matched Gibbs posterior, as a proxy distribution for the fair variational Bayesian inference by employing a new group fairness measure, the matched deviation. 
A notable feature of matched Gibbs posterior is that it approximates the posterior distribution well under the fairness constraint without requiring heavy computation. 
Theoretically, we show that the matched deviation has a strong relation to existing group fairness measures, highlighting desirable fairness guarantees. 
Computationally, by treating the matching function in the matched deviation as a learnable parameter, we develop an efficient MCMC algorithm.
Experiments on real-world datasets demonstrates that matched Gibbs posterior outperforms other methods in balancing uncertainty–fairness and utility–fairness trade-offs, while also offering additional desirable properties.

Published as a conference paper at ICLR 2026
A FAIR BAYESIAN INFERENCE THROUGH MATCHED GIBBS POSTERIOR
Jihu Lee', Kunwoong Kim K Park', Insung Kong', Dongyoon Yang', Yongdai Kim 19
Department of Statistics, Scoul National University KAIST
DDepartment of Applied Mathematics, University of Twente
AI Advanced Technology, SK Hynix Inc.
superstringl155, twkim.onling fegmail. com, ps.hyendsnu.ace kr. insung. www.kkktentteeeeeeeeeeee dengyoon.yande sk. com, (aa.m000030mma....
ABSTRACT
With the growing importance of trustworthy AI, algorithmic fairness has emerged as a critical concern. Among various fairness notions, group fairness which mea- sures the model bias between sensitive groups has received significant attention. While many group-fair models have focused satisfying group faimess con- straints, model uncertainty has received relatively little attention, despite its im- portance for robust and trustworthy decision-making To address this, we adopt a Bayesian framework to capture model uncertainty in fair model training. We first define group-fair posterior distributions and then introduce a fair variational Bayesian inference. Then we propose a novel distribution termed matched Gibbs posterior, a proxy distribution for the fair variational Bayesian inference by employing a new group fairness measure, the matched deviation. A notable fea- ture of matched Gibbs posterior is that approximates the posterior distribution well under the fairness constraint without requiring heavy computation. Theoret- ically, we show that the matched deviation has strong relation to existing group fairness measures, highlighting desirable fairesss guarantees. Computationally, by treating the matching function in the matched deviation as a learnable parameter, we develop an efficient MCMC algorithm. Experiments on real-world datasets demonstrates that matched Gibbs posterior outperforms other methods in balane- ing uncertainty- fairness and utility- fairness trade-offs, while also offering addi- tional desirable properties.
INTRODUCTION
Artificial intelligence (AI) technologies have been incredibly successful and are now widely used as essential decision-making tools in variety of fields, such as college admissions, criminal risk assessment, and credit scoring. However, when observed data contain unfair biases, the resulting trained models may have biases favoring specific groups, such as white individuals or males (Angwin) ctt al., 2016 Ingold & Soper 2016 Dua & Graff (017). which leads us to consider algorithmic fairness in Al-based decisions a crucial mission (Corbett-Davies al. 2017 Starke et al. 2022
Among various notions of algorithmic fairness, group fairness is the most widely studied, which re- quires that certain statistics across protected groups remain similar. For example, the ratio of positive predictions should be similar across each protected group (Calders et al. 2009 Barocas Selbst, Zafar et al.] 2017 Donini et al. 2018 Agarwal al. 2018b). In turn, large amount of re- search have been conducted to develop algorithms for ensuring group fairness in various supervised learning tasks (Zafar et al. 2017 Donini al. 2018 Agarwal et al. 2018h Quadrianto al. 2019
2016
Jiang et al. 2020b).
Work done while at Seoul National University. "ooresponning author.

---

Published as a conference paper at ICLR 2026
Existing algorithms for group faimess focus finding accurate prediction model under fairness constraints. However, learning deep neural networks (DNNs) which are highly parametttetee and susceptible to overfitting to lead over-confident predictions. Proper quantification of uncer- tainties in prediction of DNNs becomes a crucial mission to make Al algorithms more reliable and trustworthy (Gal et al.) 2016)] Malinin & Gales 2018 Marret et al. (02T). For example, performing medical decision making such as disease diagnosis, uncertainty quantification is essential for sup- porting a physician's decision making while the datasets are biased with respect sensitive attributes; e.g. skin cancer diagnosis with biased dataset (Wei et al. 2024), Alzheimer's 's disease diagnosis with biased dataset (Bercea et al. 2023).
Bayesian inference turns out to be a useful approach for this purpose due to its ability to quantify uncertainties in prediction models (Neal 2012, Gal & Ghahramani 2016; Kendall & Gal, 2017
Abdar et al. 202T; Gawlikowski et al.) 2023). For example. |McAllister et al. (2017); Michel
jt al. 2020)] Ding et al. (202Tb) highlight the importance of uncertainty quantification (UQ) in
autonomous vehicles, and suggest Bayesian deep learning to handle it. Bayesian inference is also
well utilized in medical image segmentation, where UQ is considered important (Sedai et al. 2019
Yu et al. 2019 Wang et al., 2020 2021 Shi al. 2021 Wang & Lukasiewicz 2022).
Nonetheless, Bayesian inference under group fairness has received little attention, which is the main theme of this paper. Modifying the single accurate fair prediction model for Bayesian inference would not be easy, since Bayesian inference needs a (posterior) distribution of prediction models. For fair Bayesian inference, we first propose the definition of group fairness of posterior distri- butior Then, we develop variational inference ( (VI) method for searching a group-fair posterior distribution approximating the full Bayesian posterior distribution well without heavy computation. Note that a simple modification of the standard VI, such as mean-field Gaussian model (Graves) 2011 Blundell et al. 2015) for group fairness, would not be practically feasible due to the compu- tational burden of calculating the group-fair constraint at each gradient update.
We resolve this problem by employing the idea of Gibbs posterior, which is a useful tool quantify uncertainties in a prediction model learned by minimizing an objective function (e.g., the empirical
risk) (Zhang] 2006 Tiang & Tanner [2008] [Bissiri et al., 2016 Grünwald & Mehta 2020; Miller]
2021 Martin & Syring Syring & Martin 2023). It provides posterior distribution by treating
the objective function as the ligliikelhood. Gibbs posterior is popularly used when specifying the
likelihood is difficult, with applications of clustering (Rigon et al. 2023). PCA (Winter al.) 2023),
and medical image registration (Wang et al. (002)). The main contribution of this paper is to propose specially designed penalized log-likelihood for group fairness and develop an efficient MCMC algorithm for the corresponding Gibbs posterior. By the proposed Gibbs posterior, we can explore the posterior distribution without any constraint on the parameter space and thus standard MCMC algorithm can be directly used without much hamper. By numerical experiment, we illustrate that the proposed Gibbs posterior outperforms the mean-field Gaussian VI and other baseline methods, in terms of prediction accuracy and uncertainty quantification.
We can summarize our contributions as follows:
To enable uncertainty quantification of group-fair models, we consider variational Bayesian approaches. We define the level of group fairness of given posterior, and propose varia- tional Bayesian inference methods under the group fairness.
We propose matched Gibbs posterior, novel distribution based on the penalized log- likelihood with the matched deviation form the fair variational inference. In addition, we provide a novel theoretical relationship between the matched deviation and other group fairness measures, which explains the ben- efit of employing the matched deviation in controlling group fairness.
penalty for group fairness, to successfully per-
We empirically demonstrate that the proposed matched Gibbs posterior outperforms base- line methods by analyzing multiple weal-world datasets, in terms of trade-off between utility and group fairness well as trade-off between uncertainty and group fairness. We show that matched Gibbs posterior also has a desirable property of improved individual fairness as a by-product.
We mean by 'a posterior distribution' any distribution on the parameter space. For the posterior distribution proportional to the product of the likelihond and the prior, we call it 'the full Bayesian posterior'.
2

---

Published as a conference paper at ICLR 2026
2 PRELIMINARIES
Let = (X,Y, be random vector, where X E X is the input, Y E 3 is the output and S {0,1] is the sensitive attribute. Let 2 = (zt, Vi, 8;), = 1....,n n be given data, which are independent copies of Z, and write D,, { Let F be the set of prediction models f(x, s) where the domain is X' x {0, with the codomain being R for regression problems and {p E [0, 1]s : ||el| }} for e-class classification problems (i.e., y are IR and (l)...,c). respectively).
For given f, let e(f) be the log-likelihood of ʃ given Dn.-
2.1 NOTION OF GROUP FAIRNESS
Three representative criteria for group fairness are independence, separation and sufficiency (Castel- novo et al., 2022 Barocas et al. 2023). In this paper, we focus on independence, so-called Demo- graphic Parity he formal definition of demographic parity, often called statistical parity, is as follows AAgarwal et al. 2018a).
Definition 2.1 (Demographic Parity, DP). A prediction model ff satisfies demographic parity under the distribution of = (X, Y, S), if f(X, S) S holds. In other words, 1/,00 =\ Fy,1, where Py,, is the conditional distribution of [(X, given S =
From the above definition, we naturally define the demographic parity gap follows. Let ()) be a given deviation measure of two probability measures.
Definition 2.2 (DP gap). The (d->)demographic parity (DP) gap of a prediction model f is defined as
Ax(0)= =
Py,1).
The empirical DDDP gap of f is defined as An, (f) == (P P7,1), where P is the empirical
distribution of (f( 8) h:e= We say that f is r fa-group-fair with level 8 if A An, (J) S 8, and we call (f. < 8 the (D-)DPs constraint. For notational simplicity, in the followings we drop the subscript in A,, (f) and denote (f) as the empirical DD gap unless there is any confusion.
Note that the original measure for the violation of DP (Agarwal et al. 2018a; 2019) is ADP == [Pr(f(X,0) 0) r|S = 0) (x,
r|5 = 1)] for prespecified threshold However, dependency on the specific value of r limits the reliability of this measure (Silvia et al., 2020). To resolve the limitation, the DD gap has been studied through line of research (Jiang et al. [020aa]
Chzhen et al. 2020 Silvia et al. 2020 Barata et al., (021). Corresponding examples of xD gaps TV(F, y.. y.o.P..). (iii) (xs(f) := KS(PP) f.. Pya)) Here, W, TV and KS is the Wasserstein distance, the total variation distance. and the Kolmogorov- Smirnov distance, respectively. See Appendix [A.]] for the definitions of each. From now on, we slightly abuse notation and use (f) and Ag (B) interchangeably. when f is parameterized by 0.
are: (i) Aw( == (F f.o. Py1)) (ii) ATv(f)
2.2 VARIATIONAL BAYESIAN INFERENCE
For a given prior N on F, the full Bayesian posterior distribution is given as T(()Dx)) x exp()()))=()). When it is difficult to calculate (or generate samples from) πT (()),,,, a well known remedy is to search a distribution u on F that approximates > r([l,,) well and is easy to be handled (e.g., easy to generate samples), which is often called as variational inference (VI) (Graves) 201T Blei et al. 2017). For VI, we first choose a variational class V of distributions, called variational distributions, which are easy to be handled. Then, we find the optimal variational distribution μ that minimizes the KL. divergence between v and π (-((,..) as stated formally:
(f) == arg min (x()))
NEV
Finally, all of downstream Bayesian inference tasks such as uncertainty quantification and estima- tion of the predictive distribution are done with D² instead of (-I1,,). For example, the predictive distribution of Y given X = x is obtained by p(Y = y X = r) = f = y = xx
We extend our discussion and methodology to the Equalized Odds criteria. See Appendix F.1 for details.
3

---

Published as a conference paper at ICLR 2026
3 VARIATIONAL BAYESIAN INFERENCE UNDER GROUP FAIRNESS
3.1 CHALLENGES IN THE FULL BAYESIAN INFERENCE
The posterior distribution under the DP; constraint could be defined as the full Bayesian posterior T (()) truncated on the set of 1/ group-fair prediction models with level 6, i.e. {f ( S 63. However, this constrained full Bayesian posterior is hard to compute, because we cannot formulate the constrained space analytically, especially when using complex prediction models such as deep neural networks. Also, since π {f (+) S 2(2,) is usually very small, acceptance-rejectie sampling is not practically feasible. See Appendix and Appendix C.3 for further explanations.
3.2 DEFINITION OF GROUP-FAIR POSTERIOR DISTRIBUTIONS
We say that a given posterior v is strongly f-fair with level if v{f (o( S 63 = 1. Our aim is to find strongly v--fiir posterior which approximates the full Bayesian posterior well. The definition of strongly f--fair posterior distributions, however, is very restrictive and hard to be satisfied. Even for linear models, variational inference with the mean-field Gaussian models could not be applied to find a strongly ti-fair posterior since none of variational distributions are strongly vi-fair. See Appendix C.5 for details.
A remedy is to relax the definition of fairness. We say that v is th-fair with level 5 when By A (f) 6. Here, we call Ax()) the average DP gap. Our strategy is to find good fair posterior y(() whose average DP gap is upper bounded, and make it to strongly fi-fair by the rejection sampling. That is, for given fair posterior (ww) with level 1/ 6, obtain a strongly fair posterior y(s) by letting (-) v)")())
8). The Markov inequality implies that ((w) [ (J) < 0} > n//5 and thus the rejection sampling is expected work well. See Appendix C.5 for empirical evidences.
3.3
FAIR VARIATIONAL INFERENCE
We explain how to modify the standard VI for fairnesa.rness. Suppose that F is parameterized by
C Rdd That is, = (Jo 0 E ) for == (0,,...,00) 02) For a proxy distribution on θ in the
variational inference, let {,(-)),, E r} be given class of proxy distribution parameterized by 7 E I. Then, we search y which maximizes the ELBO under the constraint Ay (fe) S 8. As usual, we maximize the penalized ELBO formularized as: ELBO(>) An 4. (fo), where À is the Lagrangian multiplier. Specifically, the ELBO term is defined as:
ELBO(7) == Ee- [[())] - DRL (v()
(3)
3.4 CHALLENGES FAIR VARIATIONAL INFERENCE
When v(-ly) is tractable distribution such mean-field Gaussian model which assumes that v(-h) is a product of Gaussian distributions, the ELBO can be calculated easily: the first term can be calculated through samples generated from w(-h), while the second term is analytically tractable when the prior is also product of Gaussian distribution.
A critical problem is calculate the average DP since computation of A (fe) computationally demanding. In particular, when vó is the KL divergence or integral probability metrics (IPMs), com- putation of the average DP gap requires adversarial learning which is computationally demanding and numerically unstable. To be more specific, consider the IPM with discriminator class G as v defined as:
/ 9(f(2,0)) - o(f
AIPMo = sup geg
(4)
To calculate AIPM, (f) for each f. we have to find the discriminator gs that maximizes Is f (((()) (dx)) which is computationally demanding. In particular, when calculating the average DP gap, we have to find 9f, for all 0 ~ v(-lr), which would be practically infeasible. One may argue that certain deviation measures such MMD (Gretton al.,
4

---

Published as a conference paper at ICLR 2026
2012) do not require adversarial learning, but the computation of MMD is still demanding when 72 is large since the computational complexity is On²)).
In the following section, propose specially designed Gibbs posterior approach to resolve the problem in fair VI. We first develop a novel deviation measure called matched deviation and then the Gibbs posterior based the li-llikli penalized by the matched deviation distributions fair VI, which we call matched Gibbs posterior. Its computation practically feasible since adversarial learning is required and an efficient MCMC algorithm can be implemented whose computational complexity of each update is (n)) instead of O(1²).
proxy
4 MATCHED GIBBS POSTERIOR FOR FAIR VARIATIONAL. INFERENCE
Gibbs posterior is a tool to obtain a posterior without specifying the full likelihood (Zhang 2006 Jiang & Tanner 2008 Bissiri et al.) 2016). Suppose that we estimate 0 by minimizing certain objective function R. (0.. Then, the Gibbs posterior with respect to Ra and prior π is defined as (0) x exp( R,, (0))
Our aim is to develop a class of objective functions (0; A E A} to use the corresponding Gibbs posteriors vn (=: A) the class of proxy distributions and estimate X by maximizing the ELBO under the fairness constraint.
In general, we estimate A by minimizing the negative bog-likelihod -E(O) subject to A()) < å or
cquivalently by minimizing the penalized negative log- lig-liielhood -((0) + (0) for X 2 0. We
use this penalized negative log-like eeeliikl to define the fair Gibbs posterior. That is,
(0; A) exp (E(O) (n)( (0)
(5)
variational distributions and find À that maximizes the ELBO subject to the average DP3
straint. Since À is a scalar, a grid search could be used to estimate See Appendix A.2 for the detailed calculation of ELBO.
4.1 MATCHED DEVIATION
In this section, we propose a deviation so-called matched deviation. A key advantage of the matched deviation is that the corresponding Gibbs posterior does not require adversarial learning.
A function T Xi + No is called matching function if T Po, where T& P1 is the push- forward measure of P1 induced by T. For a given matching function T, the matched deviation of is defined as
(0 T) := Ex, ~P, (X s = 1) (T(T))), 0)113)
(6)
where ||-|| is the Euclidean norm. Note that when P. and iP; are empirical distributions, the matching function T reduces a mapping between individual samples.
A notable property of the matched deviation is that it is an upper bound of the Wasserstein norm regardless of the choice of matching function T. and there exists matching function T making the matched deviation upper bounded when the total variation norm is bounded. These are formally stated in the following Theorems 4.1 4.2, whose proofs are deferred to Appendix
Theorem 4.1 (AM = Aw). For all T X1 + No. if (0, T) < 8 holds for 8 2 0, then we have (+ < 8. Theorem 4.2 (Arv = AM). If ATv (0) < 8 holds for some ō E [0, 1]. then there exists matching function T Xi - Xy satisfying Am (6, T) < 2cō for some not depending on T.
4.2 MATCHED GIBBS POSTERIOR
Motivated by the above theorems, we consider the following penalized log-likelihood {(O) A0, T). In particular, Theorem 4. implies that we can control the group fairness of the min- imizer of the above penalized log- log-likelihood by controlling À accordingly. On the other hand, The- orem 4.2 implies that any group-fair prediction model f in terms of the total variation norm has a matching function T with small (f T). These interesting results suggest the following Gibbs posterior:
(J TJA) x exp (I()) AnnA( TT)==)))(((
5

---

Published as a conference paper at ICLR 2026
which we call matched Gibbs posterior. A notable feature is that, treat T as well f as the pa- rameter to be inferred. By doing so, we can avoid an adversarial learning even though additional computation is required for treating T. For generating samples from (M(f.T)) it is natural to use the Gibbs sampler algorithm where f ~ (M[[[T X) and T ~ (T],, X) are repeated until con- vergence. Note that generating samples using Metropolis-Hasting algorithm is computationally much easier than the optimization for the adversarial learning.
An additional advantage of matched Gibbs posterior is that PM (M(/ A) can have simple form for certain problems. For example, consider a regression problem Y; = -(X.))+) where e; ~ N(0 x2) are independent noises. When f is Gaussian process priori and o² is known, then UM TT, A) also becomes a Gaussian process and hence a sample can be generated easily (see Appendix A.4 for the detailed derivation). When either the likelihood is not Gaussian or f is not Gaussian a priori, a sample from PM (M([T, X) can be generated by use of well known techniques such as Hamilto- nian Monte-Carlo (HMC) (Neal et al., 201T). Generating T from its conditional posterior needs a specially designed algorithm, which is the topic for the next subsection.
Remark 4.3. Instead of learning together with S, we may fix a certain matching function to save computation needed to generate T. An example is the optimal transport between {X; : = 0} and {X; S; = }} if there is a reasonable metric on X. Kim et al. (2025a) showed that the minimizer of the penalized mikeeehodoood with the optimal transport for T has many desirable properties such as improvement on individual fairness metrics without hampering group faireess. Our numerical studies in Appendix C.6 implies that such desirable properties are still valid for matched Gibbs posterior. Note, however, that this approach is not applicable when A' is not a metric space (e.g.. categorical data, text data) and yields suboptimal results.
4.3 MCMC ALGORITHM
For the posterior inference, we consider an MCMC (Markov Chain Monte-Carlo algorithm. Simply, we can utilize some well-known sampling methods as HMC, yield samples 0 from v(0: A). How- ever, for matched Gibbs posterior, we need an additional step to generate T from its conditional pos- terior. Here, we adopt a Gibbs sampler to iteratively sample 0 p(0 D,,) and T p (T Dn).
Sampling T, D,) is equivalent with ((t) (0; A) for the given T. We utilize HMC for this step. For sampling of T, we consider a Matstngs (MH) algorithm with regard T.
To infer T, we have specify the prior. Motivated by Volkovs Zemel (2012), we consider e(T), the energy of T which is defined as:
T(T) = e=:r)) = exp
Σ d(x
,(X T(x("))/no))
(8)
with pre-specified distance measure d and temperature O. We let π (T) x e(T).
For the proposal T > T' in the MH algorithm, we randomly select & indices i,,..., ig from [ni],
and define as:
TG) for jorje & [m1] \ fii...., 4x) (((n,()))
T
for j it,
where I1 is a random permutation of [k]. Here, is pre-specified hyperparameter. The acceptance probability can be easily calculated by the posterior ratio, since the proposals are totally random. See Appendix C.I for a visualization of the proposal T' Fig. and the acceptance probability.
4.4 SELECTION OF À
Since À is a real-valued constant, we find the optimal À by a grid-search. As usual, we first choose the set Acand of finitely many candidates for X. Then, for each À E Açands we generate posterior samples of 0 and T and calculate the ELBO and average DP. Finally, we choose À which maximizes the ELBO among those whose average DP is less than 8. See Appendix for values that we use to provide the results in our numerical experiments.

---

Published as a conference paper at ICLR 2026
5 EXPERIMENTS
We conduct numerical experiments on various benchmark datasets For the prediction models, we use DNNs since DNNs are vulnerable to overfitting and thus uncertainty quantification necessary. See Appendix B for more details of the experimental setting. See Appendix C D for omitted results and ablation studies, respectively.
F7
5.1 SETTINGS
Datasets We analyze the following five benchmark datasets that are commonly used for group-fair classification: ADULT, DUTCH, CRIME, CELEBA, and CIVIL. Brief explanations for each dataset are given below, with details in Appendix B.I B.T
Tabular datasets:
(1) ADULT: Adult dataset (Becker Kohavi 1996) has label whether income from an in- dividual is larger than S50K/yr based on census data. We consider 'gender' sensitive variable.
(ii) DUTCH: Dutch census dataset (Van der Laan 2000: Quy et al. 2022) has label whether a person's occupation can be categorized as high-level or lowelevel. We consider 'gender' sensitive variable.
(iii) CRIME: Communities & Crime dataset (Redmond & Baveja 2002) includes a label of the number of violent crimes per 100,000 population, based socio-economic, law forcement, and crime data from communities. We binarize 'number of violent crimes' with its median and consider as label. We also binarize 'percentage of population that is African- Ameraicn" with its median and consider it as sensitive variable.
Image dataset:
(iv) CELEBA: CelehAMask--O dataset (Lee et al. 2020) is face image dataset, where facial attributes are annotated with binary labels. We consider 'Male' sensitive variable, and predict 'Attractive' as target variable.
Text dataset:
(v) CIvIL: CivilComments-WW Wilds dataset (Borkan et al. 2019 Koh er al. (021) is a text classi- fication data identify whether the comment is toxic not. We consider two race groups: 'black' and 'asian', since those groups show the largest gap in the proportion of toxic com- ments.
Learning algorithms We consider 3 fair-Bayesian methods: mean-field Gaussian with MMD, Gibbs posterior with MMD, and matched Gibbs posterior. We also consider most popular state- of-the-art algorithms for demographic parity mitigation, GapReg (Donini et 2018 Chuang & Mrouch 2021). Reduction (Agarwal et al.) 2018a) and Adv (Zhang et al., 2018) as deterministic baseline methods. These 6 methods are annotated in the figures and tables as: variational mmdd, gibbs mmd, matched, gapreg, reduction and adv. Note that the computation of MMD can- not be done by mini-batches, hence we consider full-batch for the MMD based methods. Due to computational burden in calculating MMD which requires O(n²) operations, variational and gibbssmmdd are only applicable when the dataset size moderate, hence we use them only for CRIME. Also. performed ady only for tabular datasets, due to its numerical instability. See Appendix B.2 for more details about these learning algorithms.
We measure the prediction performance of the posteriors or estimators each learning algorithm on test data in terms of prediction utility as well uncertainty quantification. For prediction utility, we consider the classification accuracy the Bayes estimator (Acc). For uncer- tainty quantification, 1950), and expected calibration error (Ece) (Guo et al. 2017) the predictive distribution. Higher values of Acc mean better prediction performances while lower values of N11, brier and Ece mean better uncertainty quantification. For group fairness measure, we use A1/2 (TP f.o, J.)))
Evaluation metrics
consider the negative lig-likelhood (N11), Brier score (brier) (Brier
Our code is available at http:://tttbbbboooo/uuullleeee/e

---

Published as a conference paper at ICLR 2026
We take square root Aw, to preserve the original scale of Wasserstein distance W2. See Ap- pendix B.2 for the definitions of the performance measures.
poptng
D42150
0.20
pep og
5.05
0.22
0.18
mid mww.hhtthhheeeeccc
globe mnnd
9.00
gitdo, marchedd
mwnched
0.20
0.16
0.18
0.14
2.50
0.16
6:10
p1prog
I9SLction
6:04
1.90
0.12
2000, mootchod
0.09
64
0.1
32
0.
A
(a) CRIME-AGC
(b) CRIME-N11
(c) CRIME-brier
(d) CRIME-RGe
Figure 1: Tabular classification. Pareto-front lines between level of AW (on the x-axis) and the predictive performance (on the y-axis), on CRIME.
5.2 RESULTS
We plot the Pareto-from lines between the level of group fairness measured by 1/2 and the evalu- ation metries for predictive performance. We the averages of the fairness levels and metrics for predictive performance on 5 repeated experiments, along with their error bars. We can compare the efficiency of each trade-off using them.
Fig. I provides the results on CRIME. Results for other tabular datasets (ADULT and DUTCII) are provided in Appendix C.4 We can clearly observe that the trade-off of both utility (Acc) and uncer- tainty (N11, brier) is superior to that of other algorithms, in all 3 datasets. An interesting point is that we can observe that gibbs N11 and brier. Hence, we conclude that even in the sole practical case of moderate-size dataset, both methods are not feasible competitors to gibbs matched. For Ecce, the results of gibbs matched on ADULT show the best trade-offs. Results on CRIME and DUTCH, show the best performance when the group fairness level is not too strict, and becomes less favorable when the group fairness level becomes strict. In this regard, we argue that low Bce alone does not guarantee a good model. For example, a model that assigns a constant confidence of 0.5 to every test sample would yield an Ece close to zero, yet its accuracy would be poor. This can be supported by the result of adv DUTCH in Fig. 7 placed in Appendix C.4 where its performance on (Acc, N11, brier) are not favorable than others, while maintaining relatively good trade-off in Ece.
and variational _mnd does not perform well in terms of Acc,
Fig. 2 provides the result on CELEBA. We can observe that battche shows the best perfor- mance in terms of Acc with large margins. A similar behavior is observed in the results on CIVIL in Fig. 3 Also, for both datasets, the uncertainty measures of gibbs matched consistently achieve the lowest (best) compared to deterministic baselines, while maintaining high Acc trade-off. This is because Bayesian inference generally enhances uncertainty measures compared to deterministic al- gorithms (Gal al. 2016 Lakshminarayanan al. 2017). Note that the increase in N11 of gapreg and reduction as the fairness level increases, is due to their overfitting, even with the best validation selection.
Another benefit of the use of matched Gibbs posterior is that, it also improves individual fairness metrics, similar to the result from Kim et al. (025a)). This is because the matched deviation di- rectly minimizes the gap between the output from different individuals. We empirically validate that matched Gibbs posterior shows superior performance on Con (consistency score (Zemel et al. 2013) Yurochkin et al. 2020 Yurochkin & Sun| 2021)) that is metric for individual faireess, than that of other baselines. See Appendix [C.6 for for detailed results. We also investigate the convergence of the proposed MCMC algorithm for matched Gibbs posterior in Appendix C.7 through the traceplots
8

---

Published as a conference paper at ICLR 2026
200
gepneg
0.775
3.3
0.752
0.36
0.24
3.4
0.243
0.22
Shpreg
recustion
EC
ECR
0.15
recuution
1800 matched
0.572
0.28
68
0.22
64
0.56
Aw
4
1/2
(a) CELEBA-ACC
(b) CELEBA-NL1
(c) CELEBA-brier
(d) CELEBA-Ece
Figure 2: Image classification. Pareto-front lines between level of 1/2 (on the x-axis) and the pre- dictive performance (on the y-axis). The trade-off of proposed matched Gibbs posterior is superior to that of other competitors.
related to and acceptance probability of T. The results amply suggest that the proposed MCMC algorithm converges well.
In summary, from the experiments on various real-world datasets, we can conclude that matched Gibbs posterior does improve the utility- ffirness trade-off and incertainty-faimmest trade-offs. Specifically, matched Gibbs posterior shows (i) substantially better trade-offs in (Acc, N11 and brier), and (ii) superior trade-off's in Ece, on ADULT, CELEBA and CIVIL. Also, matched Gibbs posterior (iii) exhibits some an additional benefit of improved individual fairness, and (iv) does con- verge well. Therefore, we conclude that matched Gibbs posterior favorable proxy distribution for the fair variational Bayesian inference.
DJ
0.77
recuction
0.14
0.33
0.28
0.12
0.55
0.28
0.0
gapreg
brier
0.28
0.38
0.04
Iodection
044
0.34
0.05
2.10
0.12
0.20
0.5
0.35
0.20
0.15
5.20
0.25
0.55
0.10
0.15
0.20
0.25
6.05
0.30
0.13
0.20
0.25
41/
A1/
4
41/2
(a) CIVIL-ACC
(b) CIVIL-N11
(c) CIVIL-brier
(d) CIvIL-Ece
Figure 3: Text classification. Pareto-front lines between level of A (on the x-axis) and the pre- dictive performance (on the y-axis). The trade-off of proposed matched Gibbs posterior is superior that of other competitors.
5.3 ABLATION STUDIES
We perform several ablations studies, including (i) effects of т. (ii) effect of pretrain epochs, (iii) effect of the change of the direction of T by flipping the label of sensitive attributes, and (vi) effect of prior choice. See Appendix for details and results. each study, we still observe that matched Gibbs posterior maintains its superior trade-offs under varying conditions.

---

Published as a conference paper at ICLR 2026
5.4 ADDITIONAL RESULTS
We also provide additional results including (i) extension of matched Gibbs posterior to Equal- ized Odds (EO), (ii) extension multinary sensitive attribute case, (iii) performance evaluated with Strong Demographic Parity (SDP) (Jian er al. 2020a; Siiliaa et al. 2020). (iv) measurement on computational costs, and (v) additional experiment on ACSIncome dataset (Han al. 2023). See Appendix E for details and results. These results provide additional support for the effectiveness of matched Gibbs posterior.
CONCLUDING REMARKS
We introduced a variational Bayesian inference framework for learning group-fair posteriors. Then, we proposed novel proxy distribution named matched Gibbs posterior, based on the matched de- viation that can effectively control the group fairness of the posteriors. We show that the matched deviation possesses several theoretical advantages and that matched Gibbs posterior achieves supe- rior performance over other baseline methods along with some additional benefits such individual fairness.
We mainly focus on binary sensitive attributes in this work. Extension of matched Gibbs posterior to multinary sensitive attributes could be done similarly to what (Kim et al. 2025b) has done in Section A.3. More detailed explanation for this extension is described in Appendix E.2 We will report related algorithms and results in a near future.
Study of theoretical properties of matched Gibbs posterior such posterior consistency would be worth pursuing. In addition, Bayesian analysis for other group fairness aware tasks such as repre- sentation learning (Zemel al. 2013 Madras al. 2018: Kim et al. 2022) would be promising future work.
ETHICS STATEMENT
This paper studies group fairness in classification using only public datasets; no new human-subject data were collected. The fairness notions and performance measures are also widely used in re- cent works. We believe the framework helps prevent discrimination in classification supported by theoretical guarantees, and would not raise new critical societal concerns.
REPRODUCIBILITY STATEMENT
We have made significant efforts to ensure the reproducibility of our findings in this study. For theoretical results, present complete proofs in Appendix. The source codes for implementing proposed model and running conducted experiments are provided the supplementary material. Detailed information for the hyperparameters, datasets, experimental setup are given in Appendix
B.2
ACKNOWLEDGMENTS
This work was partly supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea governmen((MSIT) (No. NN--2-22022--12201184 Devel- opment and Study of AI Technologies Inexpensively Conform to Evolving Policy on Ethics), the National Research Foundation of Korea(NRF) grant funded by the Korea (MVer MSSTT) the National Research Foundation of Korea(NRF) grant funded by the Korea and Institute of Information & communications Technol- ogy Planning Evaluation (IITP) grant funded by the Korea government(MSIT) [NO.RS-200 11211343, Artificial Intelligence Graduate School Program (Scoul National University)].
REFERENCES
Moloud Abdar, Farhad Pourpanah, Sadiq Hussain, Dana Rezazadegan, Li Liu, Mohammad Ghavamzadeh, Paul Fieguth, Xiaochun Cao, Abbas Khosravi, U Rajendra Acharya, et al. A review of uncertainty guantification in deep learning: Techniques, applications and challenges. Information fusion, 76:243-297, 2021.
10

---

Published as a conference paper at ICLR 2026
Alekh Agarwal, Alina Beygelzimer, Miroslav Dudik, John Langford, and Hanna Wallach. A reduc- tions approach to fair classification. In International conference on machine learning, pp. 60-69. PMLR. 2018a.
Alekh Agarwal, Alina Beygelzimer, Miroslav Dudik, John Langford, and Hanna Wallach. A reduc- tions approach to fair classification. In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 60-69, PMLR, 10-15 Jul 2018b. URL https::/
mir.press v80 htt.///gaawaalB... wttpp//ss//////aaaaaalllBBB.. html
Alekh Agarwal, Miroslav Dudík, and Zhiwei Steven Wu. Fair regression: Quantitative definitions and reduction-based algorithms. In International Conference on Machine Learning, pp. 120-129. PMLR, 2019.
Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner. Machine bias. ProPublica, May, 23: 2016, 2016.
António Pereira Barata, Frank W. Takes, H. Jaap van den Herik, and Cor J. Veenman. Fair tree classifier using strong demographic parity, 2021.
Solon Barocas and Andrew D Selbst. Big data's disparate impact. Calif. L. Rev., 104:671, 2016.
Solon Barocas, Moritz Hardt, and Arvind Narayanan. Fairness and machine learning: Limitations and opportunities. MIT press, 2023.
Barry Becker and Ronny Kohavi. Adult. UCI Machine Learning Repository, 1996. DOI: http://doiorg//1224422///XXX
Cosmin I Bercea, Esther Puyol-Antón, Benedikt Wiestler, Daniel Rueckert, Julia A Schnabel, and Andrew P King. Bias in unsupervised anomaly detection in brain mri. In Workshop Clinical Image-Based Procedures, pp. 122-131. Springer, 2023.
Michael Betancourt. A conceptual introduction to hamiltonian monte carlo. arXiv preprint arXiv:1701.02434, 2017.
Pier Giovanni Bissiri, Chris C Holmes, and Stephen G Walker. A general framework for updating belief distributions. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(5): 100-11300 2016.
David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisti- cians. Journal of the American statistical Association, 1126518):59--87,200 2017.
Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In International conference on machine learning, pp. 1613-1622. PMLR, 2015.
Daniel Borkan, Lucas Dixon, Jeffrey Sorensen, Nithum Thain, and Lucy Vasserman. Nuanced metrics for measuring unintended bias with real data for text classification. In Companion Pro- ceedings The 2019 World Wide Web Conference, 2019.
Glenn Brier. Verification forecasts expressed terms of probability. Monthly weather review, 78(1):1-3, 1950.
Toon Calders, Faisal Kamiran, and Mykola Pechenizkiy. Building classifiers with independency constraints. In 2009 IEEE International Conference on Data Mining Workshops, pp. 13-18.
IEEE, 2009.
Alessandro Castelnovo, Riccardo Crupi, Greta Greco, Daniele Regoli, Iariaa Giuseppina Penco, and Andrea Claudio Cosentini. A clarification of the nuances in the faimess metrics landscape. Sci- entific Reports, 12(1):4209, 2022.
Ching Yao Chuang and Youssef Mrouch. Fair mixup: Fairness via interpolation. arXiv preprint arXiv:2103. 00.00603,, 2021.
11

---

Published as a conference paper at ICLR 2026
Evgenii Chzhen, Christophe Denis, Mohamed Hebiri, Luca Oneto, and Massimiliano Pontil. Fair regression with wasserstein barycenters. Advances in Neural Information Processing Systems, 33: 7321-7331, 2020.
2020.
Adam Cobb, Atilim Güneş Baydin, Andrew Markham, and Stephen Roberts. Introducing an explicit symplectic integration scheme for riemannian manifold hamiltonian monte carlo. arXiv preprint arXiv:/910.000 10.0624,, 2019.
Sam Corbett- Davies, Emma Pierson, Avi Feller, Sharad Goel, and Aziz Huq. Algorithmic decision making and the cost of fairness. In Proceedings of the 23rd acm sigkdd international conference on knowledge discovery and data mining, pp. 797-806, 2017.
Frances Ding, Moritz Hardt, John Miller, and Ludwig Schmidt. Retiring adult: New datasets for fair machine learning. Advances in neural information processing systems, 34:6478-6490, 2021a
Liuhui Ding, Dachuan Li, Bowen Liu, Wenxing Lan, Bing Bai, Qi Hao, Weipeng Cao, and Ke Pei. Capture uncertainties in deep neural networks for safe operation of autonomous driving vehicles. In 2021 IEEE Intl Conf Parallel & Distributed Processing with Applications. Big Data & Cloud Computing, Sustainable Computing Communications, Social Computing & Networking JSSP///DCCuu/SsscclCC///ssssaai
pp. 826-835. IEEE, 2021b
Michele Donini, Luca Oneto, Shai Ben-David, John Shawe-Taylor and Massimiliano Pontil. Em- pirical risk mininization under faimesss constraints. Advances in neural information processing systems, 31, 2018.
Dheeru Dua and Casey Graff. UCI machine learning repository, 2017. URL htttt:///archive ics. uci. edu/ml
Vincent Fortuin. Priors in bayesian deep learning: A review. International Statistical Review, 90(3): 566-551, 2022.
Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In international conference machine learning, pp. 1050-1059. PMLR, 2016.
Yarin Gal et al. Uncertainty in deep learning. 2016.
Jakob Gawlikowski, Cedrique Rovile Njieutcheu Tassi, Mohsin Ali, Jongseok Lee, Matthias Humt, Jianxiang Feng, Anna Kruspe, Rudolph Triebel, Peter Jung, Ribana Roscher, et al. A survey of uncertainty in deep neural networks. Artificial Intelligence Review, 56(Supp1 1):1513-1589, 2023.
Andrew Gelman, Walter R Gilks, and Gareth O Roberts. Weak convergence and optimal scaling of random walk metropolis algorithms. The annals of applied probability, 7(1):110-120 10-120, 1997.
Alex Graves. Practical variational inference for neural networks. Advances in neural information processing systems, 24, 2011.
Arthur Gretton, Karsten M Borgwardt, Malte Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. The Journal of Machine Learning Research, 13(1):723-773, 2012.
Peter D Grünwald and Nishant A Mehta. Fast rates for general unbounded loss functions: from erm to generalized bayes. Journal of Machine Learning Research, 21-5)):1-80, 2020.
Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger. On calibration of modern neural networks. In International conference on machine learning, pp. 1321-1330. PMLR, 2017.
Xiaotian Han, Zhimeng Jiang, Hongye Jin, Zirui Liu, Na Zou, Qifan Wang, and Xia Hu. Retiring dp: New distribution-level metrics for demographic parity. arXiv preprint arXiv:2301.1344/ 2023.
David Ingold and Spencer Soper. Amazon doesn't consider the race its customers. should it. Bloomberg, April, 1, 2016.
12

---

Published as a conference paper at ICLR 2026
Ray Jiang, Aldo Pacchiano, Tom Stepleton, Heinrich Jiang, and Silvia Chiappa. Wasserstein fair classification. In Uncertainty in artificial intelligence, pp. 272...2.27 PMLR, 2020a.
Ray Jiang, Aldo Pacchiano, Tom Stepleton. Heinrich Jiang, and Silvia Chiappa. Wasserstein fair classification. In Ryan P. Adams and Vibhav Gogate (eds.), Proceedings of The 35th Uncer- tainty in Artificial Intelligence Conference, volume 115 Proceedings of Machine Learning
Research, pp. 862-872. PMLR, 22-25 Jul 2020b. URL press/v115/ (aapiiisjianng2 jiang20a.. html
Wenxin Jiang and Martin A. Tanner. Gibbs posterior for variable selection in high-dimensional classification and data mining. The Annals of Statistics, 36(5):2207 2231, 2008. doi: 10.1214/ 07-AOS547. URLhttps: //doi. wwt.dii.oggg...o 121p://00..1110070555554
Alex Kendall and Yarin Gal. What uncertainties do we need in bayesian deep learning for computer vision? Advances in neural information processing systems, 30, 2017.
Dongha Kim, Kunwoong Kim, Insung Kong, Ilsang Ohn, and Yongdai Kim. Learning fair repre- sentation with parametric integral probability metrie. In International conference on machine learning, pp. 11074-11101. PMLR, 2022.
Kunwoong Kim, Insung Kong, Jongjin Lee, Minwoo Chae, Sangchul Park, and Yongdai Kim. Fair- ness through matching. Transactions on Machine Learning Research, 2025a.
Kunwoong Kim, Jihu Lee, Sangchul Park, and Yongdai Kim. Fair clustering via aligmment. In Forty- second International Conference on Machine Learning, 2025b. /olLhtttps://pperevvie net. formm? 7 id= j Tm K8 3NmV
Pang Wei Koh, Shiori Sagawa, Henrik Marklund, Sang Michael Xie, Marvin Zhang, Akshay Bal- subramani, Weihua Hu, Michihiro Yasunaga, Richard Lanas Phillips, Irena Gao, et al. Wilds: A benchmark of in-the-wild distribution shifts. In International conference on machine learning. pp. 5637-5664. PMLR, 2021.
Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017.
Cheng-Han Lee, Ziwei Liu, Lingyun Wu, and Ping Luo. Maskgan: Towards diverse and interactive facial image manipulation. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv: /907. 1692, 2019.
David Madras, Elliot Creager, Toniann Pitassi, and Richard Zemel. Learning adversarially fair and transferable representations. In International Conference on Machine Learning, pp. 3384-3393. PMLR, 2018
Andrey Malinin and Mark Gales. Predictive uncertainty estimation via prior networks. Advances in neural information processing systems, 31, 2018
Zelda E Mariet, Rodolphe Jenatton, Florian Wenzel, and Dustin Tran. Distilling ensembles improves uncertainty estimates. In Third sympossium on advances in approximate bayesian inference, 2021.
Ryan Martin and Nicholas Syring. Direct gibbs posterior inference on risk minimizers: Construction, concentration, and calibration. In Handbook of Statistics, volume 47, pp. 1 1-41. Elsevier, 2022.
Rowan Thomas McAllister, Yarin Gal, Alex Kendall, Mark Van Der Wilk, Amar Shah, Roberto Cipolla, and Adrian Weller. Concrete problems for autonomous vehicle safety: Advantages of bayesian deep learning. International Joint Conferences on Artificial Intelligence, Inc., 2017.
Rhiannon Michelmore, Matthew Wicker, Luca Laurenti, Luca Cardelli, Yarin Gal, and Marta Kwiatkowska. Uncertainty quantification with statistical guarantees in end-to-end autonomous driving control. In 2020 IEEE international conference on robotics and automation (ICRA), pp- 7344-7350. IEEF, 2020.
13

---

Published as a conference paper at ICLR 2026
Jeffrey W Miller. Asymptotic normality, concentration, and coverage of generalized posteriors. Journal of Machine Learning Research, 22(168): 2021.
Radford M Neal. Bayesian learning for neural networks, volume 118. Springer Science & Business Media, 2012.
Radford M Neal et al. Meme using hamiltonian dynamics. Handbook of markov chain monte carlo, 2(11):2, 2011.
Novi Quadrianto, Viktoriia Sharmanska, and Oliver Thomas. Discovering fair representations in the data domain. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 8227-8236, 2019.
Tai Le Quy, Arjun Roy, Vasileios Iosifidis, Wenbin Zhang, and Eirini Ntoutsi. A survey on datasets for fairness-aware machine learning. WIREs Data Mining and Knowledge Discovery, 12(3), mar 2022. doi: 10.1002/widm. 1452. URLhttpss http::::/doo.org////// http://....r....00002000ooo..o.. 100...00222ii.m.... 1452
Michael Redmond and Alok Baveja. A data-driven software tool for enabling cooperative infor- mation sharing among police departments. European Journal of Operational Research, 141(3): 660-678, 2002.
Tommaso Rigon, Amy H Herring, and David B Dunson. A generalized bayes framework for prob- abilistic clustering. Biometrika, 1 110(3):559-577 2023.
Gareth O Roberts and Jeffrey S Rosenthal. Optimal scaling for various metropolis-hastings algo- rithms. Statistical science, 16(4):351-36) 2001.
Suman Sedai, Bhavna Antony, Ravneet Rai, Katie Jones, Hiroshi Ishikawa, Joel Schuman, Wollstein Gadi, and Rahil Garnavi. Uncertainty guided semi-supervised segmentation of retinal layers in oct images. In International conference medical image computing and computer-assisted intervention, pp. 282-290. Springer, 2019.
Yinghuan Shi. Jian Zhang, Tong Ling, Jiwen Lu, Yefeng Zheng, Qian Yu, Lei Qi, and Yang Gao. Inconsistency-awar uncertainty estimation for semi-supervised medical image segmentation. IEEE transactions on medical imaging, 41(3):608-633 2021.
Chiappa Silvia, Jiang Ray, Stepleton Tom, Pacchiano Aldo, Jiang Heinrich, and Aslanides John. A general approach to fairness with optimal transport. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pp. 3633-3640 2020.
Christopher Starke, Janine Baleis, Birte Keller, and Frank Marcinkowski. Fairness perceptions of algorithmiic decision-making: A systematic review of the empirical literature. Big Data & Society, 902:2053951122111151888 2022. doi: 10.11772053951722111155 URL htttt:://
Nicholas Syring and Ryan Martin. Gibbs posterior concentration rates under sub-exponential type losses. Bernoulli, 1080-1008 2023.
Paul Van der Laan. Integrating administrative registers and household surveys. Netherlands Official Statistics, 15(2):7-15, 2000.
C. Villani. Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften. Springer Berlin Heidelberg, 2008. ISBN 9783540710509, URL httpt:t//ooookss googll.....................o....
co.
Maksims Volkovs and Richard Zemel. Efficient sampling for bipartite matching prob- lems. In F. Pereira, C.J. Burges, L. Bottou, and K.Q. Weinberger (eds.), Ad- vances in Neural Information Processing Systems, volume 25. Curran Associates. Inc., 2012. URL https::/proceeddggg www.ips..sessssscco.oommm ccpppppppccpppppr..pppppppppp paper files/paper/20127 4lee/422ceea
838c930333550 foc.ee B 83cc9605-PapPr .pof
Guoqing Wang, Abhirup Datta, and Martin A Lindquist. Bayesian functional registration fmri activation maps. The annaly of applied statistics, 16(3):1676, 2022.
14

---

Published as a conference paper at ICLR 2026
Jianfeng Wang and Thomas Lukasiewicz. Rethinking bayesian deep learning methods for semi- supervised volumetric medical image segmentation. In Proceedings of the IEEE/CVE Conference Computer Vision and Pattern Recognition, pp. 182-190, 2022.
Kaiping Wang, Bo Zhan, Chen Zu, Xi Wu, Jiliu Zhou, Luping Zhou, and Yan Wang. Tripled- uncertainty guided mean teacher model for semi-supervised medical image segmentation. In International Conference on Medical Image Computing and Computer- Intervention, pp- 450-460. Springer, 2021.
Yixin Wang, Yao Zhang, Jiang Tian, Cheng Zhong, Zhongehao Shi, Yang Zhang, and Zhiqiang He. Double-uncertainty weighted method for semi-supervised learning. In International conference medical image computing and computer-assiste intervention, pp. 542-551. Springer, 2020.
Maria L Wei, Mikio Tada, Alexandra So, and Rodrigo Torres. Artificial intelligence and skin cancer. Frontiers in medicine, 1 11:1331895, 2024.
Steven Winter, Omar Melikechi, and David B Dunson. Sequential gibbs posteriors with applications to principal component analysis. arXiv preprint arXiv:2310.12882, 2023.
Lequan Yu, Shujun Wang, Xiaomeng Li, Chi-Wing Fu, and Pheng-Ann Heng. Uncertainty-aware self-ensembling model for supervusee 3d left atrium segmentation. In International con- ference on medical image computing and computer-assiste intervention, pp. 605-613. Springer, 2019.
Mikhail Yurochkin and Yuekai Sun. Sensei: Sensitive invariance for enforcing individual
fairness. In International Conference on Learning Representations, 2021. URL https:
//openzeview. http://www..et/tommmadddddd-eeee
Fx
Mikhail Yurochkin, Amanda Bower, and Yuekai Sun. Training individually fair ml models with
sensitive subspace robustness. In International Conference on Learning Representations, 2020.
URL httpps://ppperrvieew net. wtr.foorrmi/ id iu
Muhammad Bilal Zafar, Isabel Valera, Manuel Gomez Rogriguez, and Krishna P Gummadi. Fairness constraints: Mechanisms for fair classification. In Artificial intelligence and statistics, pp. 962- 970. PMLR, 2017.
Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. In International conference on machine learning, pp. 333-333. PMLR, 2013.
Brian Hu Zhang, Blake Lemoine, and Margaret Mitchell. Mitigating Unwanted Biases with Ad- versarial Learning. In Proceedings of the 2018 AAAI/ACM Conference on AI, Ethics, and Soci- New York, NY, USA, 2018. Association for Computing Machinery. doi: 10. 10.1145/2778711327777777
ety, AIES '18, pp.
ISBN 978- 1 778-1-4503-6012
URL htttht::: //dl acm. org/
Tong Zhang. Information-theorrticc upper and lower bounds for statistical estimation. IEEE Trans- actions on Information Theory, 1204:113771132,, 2006.
15

---

Published as a conference paper at ICLR 2026
APPENDIX
A THEORETICAL DETAILS
A. DEFINITIONS OF DEVIANCES
Consider two random variables z and y, which follows the probability distribution and Q, respec-
tively. The formerly mentioned deviances of P and Q can be defined as follows.
The Wassesstnnnneinn distance defined as:
W2(P, 2)
inf (aww)>yl1 - y1/2-
(10)
7ER(P,2)
Here, I(P, Q) is the set of joint probability distributions, whose marginals are and Q.
The total variation distance is defined as:
TV(P. Q) = sup (P(A) Q(4)).
(11)
AEA
Here, A is the set of all measurable subsets.
The Kolmogorov- -emnooov distance is defined as:
KS(P, Q) suppFr(t) - Fo(()).
(12)
rER
Here, (p(() = (xx S t) is the cumulative distribution of P.
A.2 CALCULATION ELBO
The evidence lower-bound (ELBO) is defined as:
ELBO( : Eg-wwlog L(D,: f)] DKL (xL((((])
(13)
In this section, provide the calculation of ELBO when considering v., (0; A). This calculation can be used to the selection of x, under certain level of averaged DP. Note that the first term can be easily calculated using samples yielded from 1/, by Monte-Carle approximation.
We can also calculate the ELBO in general cases. We have:
vn(0;A) A) (0)e (R:(0:))
ZA
T(0)e -R.00:) = Ea-z (o:2))))
(14)
Zx
where (0; = -((0) + An) Hence, we can simplify the KL term as follows:
(K (-)(((()) = Eg (::A) (log un (0: A) - log T()] = Egser, (+A))- (0; A)] log Zx.
(15)
Now, using samples (0(9)T= (n and (g(r) T(-). we can calculate the KL term by Monte-Carlo approximation as follows:
s expl Rn (0()) (
(KL(Pn(; = - Σ
1
(0(0 log
1
Σ
(16)
S
t=1
s=1
For the numerical stability, one could calculate the last term by utilizing log-sum-exp.
A.3 PROOFS OF THE BOUNDS RELATED TO THE MATCHED DEVIANCE
Proof of Theorem 4.1 Recall the definitions of the deviances that
Aw (0) W2(P/o.. PJo.), (m T) EE -
f6((()))))
(17)
for given matching T 2r 2n satisfying T IP IPo.
16

---

Published as a conference paper at ICLR 2026
For simplicity, let μ4 Law(P/o.e) for E {0,1). Hence, by the Kantorovich formulation of the Wasserstein distance W2.
(P Pfo,o) - (X,, (T(
(18)
for any T. Taking the infimum over on the right- side yields:
(11: - inf E -
(19)
T
Therefore,
Aw (0) v inf (0 T)
(20)
T
In partieular, if there exists T with Am T) S 6, then Aw()) S 5.
Proof of Theorem 4.2 Fix 7 and partition the cube |0. 1J into finite family of disjoint cubes {B& k= of length for each side. Then, we have diam (Ba) van and diam 1 = Vc.
Let A == fo (Bx) {S = 1] c X1 and Ao. == f/ (Bx) n {5 = 0} C No. Write Pk ==
(1(A11)) = P/o, (B&) and q: = Po(A),)) (t..o Define T == (pt- qk +, 8k == (gk
pa)4. and mg == min
fain (p. qu Pk P3 9k Sh. Let = Σx= Pk Σ Sb S
3 D Pk gk S (V(P) PJo.) S Here, note that Σ rk =E 8k since Σ Pk Σ qk 1.
Note also that Dk=1 my 2x (Pk- rK)) = - >x Tk
R.
Then, for each k there exist measurable subsets A S AI, and Amaa C Ao, with Pi (AA.))
0.k (A(A)))) my, and a bijection T A 0,k satisfying Po(B) = PP - for all
B C Amar
Let AT = X1 Up mat and AT == Xo \ UR A0, be the residual domains. By construction,
P1 (A(pm) = (o() = R < 8. Then, there exists a bijection (3)), AT + Apmm.
Define T: 11 - No piecewise by
T(r) = T (x), (T), AT
x A for E [m].
(21)
T is well-defined as a countable union of maps on disjoint domains. Moreover, for any C Xy that
is a finite union of pieces from 0,k la and A we have (B) = (T-) and T#P: = Po-
If x Ama then fo(æ, 1) and fo (T()), 0) both lie in the same cube Bx, l|fo 1)
fo(T))).)) 0)1B < (Ver)² crj² If Apm, we trivial bound that |1- s diam (0, 1]9/2 G, that we have (0,, T) Ex, (X,, (T(()))) S R)ei? Re s c(11-8)m7 - 8) + 6), since R 5<a
c(28- Taking 7 == v [0, 1). we have (m T) < c((1 6)8 + c(28 - 82) S 2co, which concludes the proof. (Note that the proof also holds for regression problems, when the output is scaled into [0, 1).)
A.4 MATCHED GIBBS POSTERIOR FOR GAUSSIAN REGRESSION PROBLEMS
In Section we noted that matched Gibbs posterior can be computed easily for a certain problems. In this section, we provide the explicit calculation for Gaussian regression problems.
Consider a simple regression problem of:
Y, = f(X,, St) + 6, 6 ~ 0²)
(22)
Assume that f ~ GP((,, ((,, )), where gP denotes a Gaussian process. Let y = (yi...., BB.) and K be matrix with Kij = k((2,,, si), 8;))
17

---

Published as a conference paper at ICLR 2026
Then the log- ligllikelhhod is given as:
f() = log D()
20 11y -f11
1h
log log( (2TT²),
(23)
2
where f
The constraint is calculated as:
mA T) |1 PT) f1fl2
(24)
CTS
Here, S1 E Rmxx selects the coordinates with 1 and PT is the permutation matrix induced by the matching map ' on those coordinates. CT is defined as ST (T PT) (
Hence, the matched Gibbs posterior simplified as:
(M(f, T; A) exp(E(f) \J, ()) exp (-}s (K t T 2XCT) f + o y f) π (T)
(25)
x exp (-2 ATF - f) (T).
Here, AT := = K- II, + 2XCT. That is, the conditional posterior of f given T and o² is again a Gaussian process.
18

---

Published as a conference paper at ICLR 2026
B EXPERIMENTAL DETAILS
B.1 DATASETS
We first explain about benchmark datasets that used in experiments.
(i) ADULT: Adult dataset (Becker & Kohavi 1996) has a label whether income from an individual is larger than S50K/yr based on census data. We consider sensitive variable as gender. Preprocessing procedure follows from (Yurochkin et al. 2020). removing some features in the dataset, and one-hot encode the discrete and quantized continuous variables.
(ii) DUTCH: Dutch census dataset Van der Laan 2000, Quy et al. 2022) has a label whether person's occupation
be categorized high-level low-level. We consider sensitive variable as gender. We choose categorical features ('age', hhousehold.positiioo 'householdssize', 'citizenship', 'country.birth', 'edu.level', "economi c.status', 'cur.eco.activity" 'maritalstatus/ ) and expanded them by one-hot encoding.
(iii) CRIME: Communities & Crime dataset (Redmond & Baveja 2002) has label of the number of violent crimes per 100,000 population, based on socio-economic, law enforcement, and crime data about communities. The label number of violent crimes is divided into binary label by its median, and the sensitive variable percentage of population that is african american is also divided into bi- nary sensitive groups by its median. We dropped ( ('state', countryy ccomuniity' 'fodd' "communit taannnttttaamee yname') )
(iv) CELEBA: CelebAMask-HQ is face image dataset with 30,000 images, annotated with 19 binary facial attributes (e.g., eyegle 1aglaa asses, smil mmiling, wavy hair...). We consider the attribute sensitive variable (Female I, Male 0). We utilize the pretrained ResNet18 to yield the image embeddings of dimension 512.
attracti ve target label and male
(v) CIVIL: CivilComments Wilds (Koh et al. 2021) is processed version of the original CivilCom- ments dataset (Borkan et al.) 2019). We consider two sensitive variables from race 'black' and 'asian', since the proportion of toxic comments exhibits the largest gap between them. We utilize the pretrained Roberta (Liu et al. 2019) to yield a sentence embedding of dimension 768.
We split datasets into two partitions with proportion 8 corresponding to train and test datasets, respectively, except CRIME. For CRIME, we randomly pick one from predefined train test split candidates. We ReLU network with 2 hidden layers of width 200 for tabular datasets. For image and text datasets, we use ReLU network with 2 hidden layers of width 512. Table provides the basic descriptions of the 5 datasets.
Table 1: Summary statistics for each dataset.
Dataset
Features (d)
Size
Y =
S
train samples
test samples
ADULT
120
45,222
11,208
30,527
36,177
9,045
DUTCH
60,420
28,763
30,147
48,336
12,084
CRIME
121
1,994
1,042
1,038
1,794
200
CELEBA
512
30,000
17,218
18.943
24,000
6,000
CIVIL
768
33,613
7,714
12,059
26,890
6,723
B.2 TRAINING DETAILS
Learning algorithms For deterministic baselines (gapreg, reduction and adv), we randomly di- vided the train set in to two, with proportions 0.8 and 0.2 for train set and validation set, respectively. Then we save the model with via the performance on the validation set.
Evaluation metrics As mentioned, we provide utility measure Acc and uncertainty quan- tification measures N1l, Ece, and Brier. As mentioned in the main body, Acc is accuracy of the Bayes estimator. For uncertainty quantification measure, we measure the performance of the predictive distribution. Specifically for binary classification, we can calculate the predictive proba-
bility of test data (Ii, Vi, 82) as Pt: p(y 1|2;, 8t, Da) + - (Ni, 82). Considering
19

---

Published as a conference paper at ICLR 2026
binary classification tasks, N1 1 can be caleulated by the binary cross-entropy with {p:)--1 Ece is a measure for calibration scores (Guo et al. 2017). where higher score implies better calibration. The formal definition is as follows:
M
Bm |ace Bm) conf (Bm)l-
ECE
(26)
Here, Bm's 's are M bins of predictions, with sizes Bml. The accuracy and the confidence of the bins are defined follows: aCc( [Bmm Lien... I1(i; yı), conf( Ta.l Eie Big Pi-
Brier is a strictly proper scoring rule for measuring the accuracy of probabilistic predictions (Brier) 1950). commonly used in binary classification tasks. The formal definition on function is as fol- lows:
Brier-score Σ
fi y:)
(27)
Other details We use a python package hami ttorch Cobb et al., 2019) perform HMC. The followings are the detailed values for several experimental settings.
For gapreg, reduction, adv and variational mmd, we trained them with batch size of 1, 020 for 100 epochs, with learning rate 0.001 and Adam optimizer. For variational ummd, we used 5 samples for train and evaluation, where the prediction probabilities from the samples are averaged and used in the performance calculation. For Gibbs posteriors, also used 5 samples where the prior of parameters are set to the standard Gaussian distributions. Following hyperparameters are organized in the order of (CRIME, ADULT, DUTCH, CELEBA and CIVIL). For MCMC chains, we utilized (5, 1,,1000,,0,,0000000 burn-in epochs with 10 thinning intervals, respectively, and used HMC with step size of 0.001. To give an efficient initial value, we utilized pretrained DNN models as initial parameter values, with pretrain epochs of (50, 30, 10, 20, 70), respectively. For the permutation size of IIş (larger & implies use (10, 10, 10, 10) for k. respectively. For faster convergence, we initialized T with based on the optimal transport theory (Villani) 2008), along with the former insights of Kim et al. (2025a). Specifically, we utilized the joint OT map from Kim et al. (2025a) where the computation of a cost matrix is as follows: C? = |c",p] E Ra X where c3 X x
faster mixing),
,r,;") -y with some constant 7 > 0. Here, (x.'a) xx(*)) are observations from X and are the number of observations for each. Sufficiently large y suppress to match individuals that are differently labeled. We utilized POT package to solve optimal transport. The values of À for matched Gibbs posterior that we used are provided in Table
All our experiments are conducted on an Intel(R) Xeon(R) Silver 4410Y CPU with 128GB RAM and NVIDIA GeForee RTX 4090 GPUs
Table 2: À values that are used in the main experiments.
Dalasel
CRIME
00.0, 1.0. 2.0. 3.0.3.3..000
9.0, 10.0
ADULT
[0.0. 0.0, 3.00
DUTCH
[0.0, 2.0,
8.0, 9.0, 10.0
CELEBA
0.5.
1.0,
6.5,
CIVIL
(000, 1.0, 2.5,
5.5, 7.0, 9.5, 10.0
hponhhttppss/pryrcnnnr..iiihhi Itorch
20

---

Published as a conference paper at ICLR 2026
C OMITTED RESULTS AND DETAILS
In this section, we provide additional details and results omitted due to the shortage space, to support the results in the main manuscript.
C.1 DETAILS OF MATCHED GIBBS POSTERIOR
X1
X-
X1
T
x
(3)
1
T(maa))
T(5)
T(5)
T(6)
1/-
T(6)
T(2)
4
T(iu)))
T(4)
5
T(4)
T(1)
6
T(1)
T(8)
1/
T(fu()))
T(7)
14
T((n,())
Figure 4: visualization of the proposal construction of T T
The acceptance probability proposal T can be calculated as:
a (T') = min a (T')).
(28)
Denote L as the likelihood function. Here, a' is defined as:
a' (T (T 00 D, ) ((T (T' + T)) PT D,, ) + T'
( 8 T T)
(29)
C(D) , 0, (T
<(T)) O T
C(T)
The last equality holds since:
(T' + T> -T TT).
(30)
C.2 ABOUT GROUP-FAIR CONSTRAINED POSTERIOR
In this subsection, we provide the table to show n versus x <(0 (0) S (D,),, which was men- tioned in Section 3.1 Here, App is defined with the threshold = 0. Since the posterior distribution o (0 (,,) is intractable, we use Monte Carlo approximation to report those values. Specifically,
3.1
T(10 (0) S n))D.))
(0)
Jo:
(31)
T( ((()) s n).
where (Q()) are sampled from (O)) 0) using HMC with a learning rate 0.001. We used train dataset of ADULT as observations, and used a MLP with two hidden layers width 200, as the model. We sampled 1,200 times, where first 200 samples are burn-in, and we thin the remaining samples at intervals of 10, resulting in 100 final samples. Here, original App in the train dataset is 0.1991 iee difference on the proportion of positive labeled individuals from each groups.
The results are given in Table 3howingg that: when 77 drops below 0.10, the probability decreases to 0.03, making acceptance-rejection sampling practically infeasible.
21

---

Published as a conference paper at ICLR 2026
Table 3: i((0 ADr(O) S (n,) values for varying 7).
17 Prob.
0.08
0.09
0.10
0.11
0.12
0.13
0.14
0.00
0.02
0.03
0.05
0.07
0.08
0.10
1 Prob.
0.15
0.16
0.17
0.18
0.19
0.20
0.21
0.12
0.14
0.16
0.20
0.53
0.85
1.00
C.3 RANDOM-WALK METROPOLIS-HASTINGS
In Section 3.1 we explained why existing posterior sampling methods are challenging imple- faimess constraints. To provide empirical evidence for
ment with complex constraints such as group- group-faimess this statement, we conduct a simple experiment using the random-waid Metropolis-Hastinn Hastings with constraints, which accepts only samples that satisfy given group-fairness constraint, on ADULT. Using the train dataset, we propose a random walk proposals with Gaussian noise of learning rate 0.001. Then, accepted proposals if random uniform variable Uniform 1) is lower than the acceptance ratio a and the constraint is satisfied.
In Table 4 can observe that as the constraint becomes stricter, the acceptance ratio decreases. Due to this low acceptance ratio, the efficiency of MCMC samples is significantly reduced. Fig. is corresponding illustration. Random proposals (yellow arrows) wander through regions of high posterior density, which could not satisfy the constraint level.
Table 4: Acceptance ratio of random-walk Metsopolis--asttnngs,, only accepting samples that satisfies given constraint threshold.
Constraint threshold
0.18
0.175
0.17
0.165
0.16
0.155
Acceptance ratio
0.58
0.40
0.30
0.26
0.02
0.02
Constraint
High density
Low density
Posterior w/o constraint
Figure 5: An illustration of naive posterior sampling on constrained parameter space. Yellow rows indicate steps from naive Netropolis-Hastinne with random proposals, which wander through regions of high posterior density.
22

---

Published as a conference paper at ICLR 2026
C.4 OMITTED RESULTS
In this section, we provide all omitted Pareto-front lines of the main result from Section 5 From Figs. Glad 7 we can observe that gibbs_maiched shows better performance than other competitors in terms of performance-fairess trade-offs. For example, bobbss_ttthed achieves the best trade-off on all measures in ADULT with large margin. As mentioned in the main body, better Ece does not guarantee a good model. This is clearly observed with the result of adv, where its performance is clearly not favorable than others in terms of (Acc, N11, brier), while maintaining good trade- off in Ece. To mitigate the increase of Ece under strict fairess levels, one can apply calibration techniques such as temperature scaling (Guo et al. 2017) without degrading Acc. We leave this to the future work.
gapreg reduction
0.500
0.84
gibbs matched
0.475
0.450
0.82
ACC
NLL
0.425
0.400
0.80
0.375
0.79
gapreg reduction
0.350
0.78
gibbss_matched
matched /
0.325
0.77
0.05
0.10
0.15 W
0.20
0.25
0.05
0.10
0.15
0.20
0.25
1/2 W
(a) ADULT-ACC
(b) ADULT-N11
0.17
gapreg
reduction
0.12
gapreg
Bdv gibbs matched
adv gibbs, matched
0.15
0.15
brier
0.14
ECE
0.06
0.13
0.04
0.11
0.05
0.10
0.20
0.05
0.10
1/2
AW
(c) ADULT-brier
(d) ADULT-Ece
Figure 6: ADULT. Pareto-front lines between level of A (on the r-axis) and the predictive per- formance (on the y-axis). The trade-off of proposed matched Gibbs posterior is superior to that of other competitors.
1/2
23

---

Published as a conference paper at ICLR 2026
0.82
reduction
0.51
gibbs baabs_matched
0.80
0.52
0.78
ACC
0.76
0.46
0.44
gapreg
reduction
glabs mottcsemteheede
0.40
1/2
1/22
(a) DUTCH-AcC
(b) DUTCH-NLL
gapreg
0.18
gepreg
0.19
adv gibbs_matched gibbs matched
adv
gibbs_matched matched
0.18
0.14
0.12
brier
ECE
0.38
0.06
0.14
0.34
0.13
0.02
0.050 0.075
0.100
0.125 0.150
0.175
0.200
0.050 0.075 0.100 0,125 0.150 0.175 0.200
A
(c) burcr
(d) DUT
1/2 Figure 7: DUTCH. Pareto-front lines between level of A (on the a-iiss) and the predictive per- formance (on the y-axis). The trade-off of proposed matched Gibbs posterior is superior to that of other competitors.
24

---

Published as a conference paper at ICLR 2026
C.5 ABOUT STRONGLY U-FAIRNESS
Strongly fair VI Here, we empirically validate that the strongly fair variational inference with mean-field Gaussian does not work well, even for linear models. The experimental setup similar to that of Appendix C.3) We simply optimize ELBO with a mean-field Gaussian distribution without any group-faimess aware constraints, on CRIME. For the model, we use (linear) logistic regression model, with 100 epochs of training. We report the proportion of samples f that are fair with level 6, i.e. (fr S 6. Note that the original App difference in the train dataset 0.2642. Table shows that when 5 becomes slightly smaller, the proportion of samples f that satisfies the group fairness constraint vanishes. For the level of 0.24 the proportion is only 1% of the samples, and it totally vanishes to zero for the level under 0.22. Hence, mentioned in Section 3.2, variational inference with the mean-field Gaussian models cannot be directly applied to find a strongly fair distributions.
Table 5: Proportion of samples from mean-field Gaussian distribution optimized with standard ELBO, under varying level of d.
5 Prop.
0.27 0.014
0.26
0.25
0.24
0.23
0.22
0.007
0.003
0.001
0.001
0.000
Rejection sampling To show that the aforementioned rejection sampling in Section 3.2 works well, we conduct an experiment on ADULT. Note that the rejection sampling of v(() through v((r) can be performed simply by accepting the samples from w(w) that satisfies (+(+) 5. We take
1000 samples from matched Gibbs posterior with the same settings of the main result on ADULT, and perform rejection sampling.
Assume that we want samples that satisfies A/22 < 0.10, which is quite strict condition as shown in Fig. 6 By sampling 1, 000 samples from matched Gibbs posterior with À 4.0, we can accept 897 samples that satisfies the condition, yielding 0.897 of acceptance ratio in the rejection sampling. Hence, we can effectively utilize matched Gibbs posterior to directly obtain strongly fair models.
C.6 ADVANTAGE MATCHED GIBBS POSTERIOR: IMPROVED INDIVIDUAL FAIRNESS
As previously noted, the matched deviation resembles the concept of individual faimess, since mini- mizing it is equal making outputs from two (matched) individuals. In this section, we empirically demonstrate how matched Gibbs posterior achieves better individual faimess metrics than other al-
gorithms, when the 1/22 level is similar. We use the consistency score (Con) (Zemel et al. 2013 Yurochkin et al. 2020 Yurochkin & Sun 2021), which measures the similarity on the labels for similar individuals. For the computation, we utilized aif360 package
From Tables 6 and 7 we can observe that the corresponding Con of matched Gibbs posterior achieves the best along with the best Acc, supporting the advantage of using matched Gibbs poste- rior. (In CRIME, we report Con of adv with its lowest A due to its instability.)
Table 6: CRIME. With a fixed level of A reported. The bold faced values implies the best values.
1/2 2 0.09, the level of Con and corresponding Acc are
Con (Acc) variational
gapreg
reduction
adv
gibbs_mmd
gibbs
0.767 (0.718) 0.768 (0.724) 0.834 (0.684)
0.790 (0.729)
0.785 (0.696)
0.837 (0.736)
C.7 MCMC DIAGNOSIS
We show that the MCMC from matched Gibbs posterior well-behaves in practice, which also con- siders T as random variable. Following Betancourt (2077), we provide several plots of the en-
ht : / /ai /ttt:ss//aa1300..eeetteeeoo www.treddds.ss.o http:///wwestee//e///eee/ae///ee
sk 1kleamn metri cs. consi wennssstency_scorrre html
25

---

Published as a conference paper at ICLR 2026
1/2 Table ADULT and DUTCH. With fixed level of Aw" (approximately 0.09 for ADULT and 0.07 for DUTCH, the level of Con and corresponding Acc are reported. The bold faced values implies the best Con.
Con (Acc) adv
Dataset
gapreg
reduction
gibbs matched
ADULT
0.927 (0.824)
0.931 (0.824)
0.954 (0.813)
0.955 (0.826)
DUTCH
0.939 (0.785)
0.939 (0.788)
0,960 (0.786)
0.965 (0.809)
engy from each MCMC samples, that are commonly used to diagnose whether the MCMC is well- behaved. We also provide the energy Bayesian fraction of missing information (E-BFMI), value that is larger than 0.3 indicates that the MCMC is not problematic (Betancourt) 2017). The definition of E-BFMI is as follows:
N ΣN m== (E,, En (En 13
E-BFMI
(32)
Here, En, is an energy (U + K in HMC) values of the n-th sample. For the diagnosis, we provide 3 energy related plots traceplot of En. histogram of A En and lag plot (En-1, En) in Fig. g with results ADULT. To validate the convergence, we provide the plots when the fairness level of the predictive distribution (R.g Aw 1/2
strict (=v 0.09). We can observe that the F,, trace reduces first and then become stationary. Also, A F., is well-distributed (not too sharp wide, symmetry with center 0) and the lag plot is well-distributed as a point cloud of circle, indicating low autocorrelation, even when the Soiun-firness constraint is strict. In this case, E-BFMI 2 1.646 > 0.3 holds, indicating that the MCMC well-behaves.
Energy trace
AE histogram
Lag plot of energy
137000
100000
13:000
coint
100700
136-100
500000
130500
30600
135000
125500 1D0000 109300
Iborabion
Ex-
(a) Traceplot of E..
(b) Histogram of A
(c) Lagged scatterplot of Em.
Figure 8: Energy related plots of MCMC samples of matched Gibbs posterior on ADULT, when 1/2
E, ~ 0.09.
We also measure the acceptance probability T with different levels of A. For (0.0, 0.5, 2.0, 3.5, 4.0), corresponding acceptance ratio is (0.38, 0.32, 0.3, 0.36, 0.38, 0.36), where the range of [0.2, 0.5] is generally recommended in practice (Gelman et al. 1997 Roberts & Rosenthal 2001) Hence, conclude that our proposal of T for the Metropolis- Hetsopglis-Hastings is reasonable.
26

---

Published as a conference paper at ICLR 2026
D ABLATION STUDIES
D.1 EFFECT OF
In this section, study the effect of performances defined in the prior of T. in Eq. ( We report the performance of matched Gibbs posterior when T is varying, on ADULT. Fig. represents the trade-offs when the fairness level is strict, using т 00.01,0.1,1.00 values of prediction performances. Note that smaller r implies more powerful prior. We can observe that the values results in minimal variation. In other words, we can conclude that the choice of
Table g is the detailed
1.0 does not significantly impact the main result. Hence, we can say that the belief from the observation strong enough to overcome the effect of the prior, the practical use of matched Gibbs posterior on ADULT.
182-01301
0.200
Dai- 2.01
0.00
1.300
6.324
ACD
1.22%
0.00
0.300
0.000
0300
0.00
0.40
0,12
Ap
Figure 9: Pareto-from Proeto-frontto lines between level of group fairness 1/2 matched Gibbs posterior, with different values of r, on ADULT.
and prediction performance of
Table 8: Detailed values of the prediction performance with varying when Aw s 0.09 on ADULT.
Mcasures
T
0.01
0.1
1.0
Acc (x10²)
82.892
82.888
82.888
N11 (x10?)
36.462
36.463
36.463
brier (x10%)
11.860
11.860
11.860
Ece (x10²)
4.392
4.395
4.395
D.2 EFFECT OF PRETRAIN EPOCHS
To induce the faster convergence, we utilized pretrained DNN parameters initialization of the model parameters. We conduct an additional experiment by varying pretrain epochs on ADULT, to observe the effect on the performances. With the same setting from the main result, we report the Pareto-front lines with varying pretrain epochs as [10, 20, 30, 40), and use the same values for remaining hyperparameters such as burn-in epochs. Note that we utilized 30-epoch case for the initialization in the main result.
Fig. is the corresponding Pareto-front lines. Although quality MCMC samples largely depends on the chain configuration, we can observe that the number of pretrain epochs does not significantly affect the predictive performance of matched Gibbs posterior when the chain is sufficiently iterated, which is not for the case for 10 epochs. The 10-epoch case shows large performance gaps, primarily because the MCMC was not run long enough. We can also observe that more pretrain epochs results better performance, primarily because the MCMC converges more quickly.
D.3 EFFECT OF FLIPPED SENSITIVE ATTRIBUTES
Matched Gibbs posterior utilizes the matched deviation defined in Eq. (6) When the number of individuals from each sensitive group are different, the matched deviation changes when the labels
27

---

Published as a conference paper at ICLR 2026
032
C31
NLL
brier
0.04
C31
0.44
Figure 10: Pareto-front lines between level of group faimess 1/2 and prediction performance of matched Gibbs posterior, with varying pretrain epochs, on ADULT.
of sensitive attributes are flipped. Practically, we construct the matching function T from the smaller sensitive group to the larger sensitive group. To observe the effect of the flipping, we conduct an experiment on ADULT with the same settings from the main result.
In Fig. 11, we can observe that flipping the sensitive attribute (orange line) does not lead to sig- nificant change performance. The slight decrease in performance can likely be explained by the fact that matching from the larger group naturally permits many-to-one matching, which may overly reduce the distance between individuals from and = 1. contrast, matching from the smaller group does not exhibit this issue.
otri acu 18:22d
(ipped)
(ippad)
6.33
330
9125
ACC
C.3
8:122
1.00
02
3.5
0.55
8116
8.08
8.25
3.20
9.15
0.10
0:15
6.32
0.75
4.0
450
4,5
Figure 11: Pareto-front lines between level of group faimess A and prediction performance of matched Gibbs posterior, with flipped sensitive attributes, on ADULT.
D.4 IMPACT OF PRIOR CHOICE
The proposed matched Gibbs posterior does allow the different priors be considered. We investi- gate how variations in the prior distribution influence the model's inference, While the main analysis focuses on the Gaussian prior, we additionally examine alternative priors to show the robustness of method. We conduct experiments using the CRIME dataset and employ Gaussian, Cauchy, and Student-t priors to evaluate how different prior specifications affect the model's performance. Note that Cauchy and Student-t priors are alternatively considered when heavy-tailed priors needed (Fortuin 2022).
In Fig. 12 can observe that the performances does not significantly change when different priors are considered.
28

---

Published as a conference paper at ICLR 2026
natened (Cauche giobe matened (Student
matched (Cascho) matched (Studant
0.18
0.55
1.50
0.18
0.12
git6s. matched (Casso un!
gltcc matched (Cauchy)
gibe (Statent
6.55
0.10
2.40
0.09
A32
432
1
A
(a) Ace
(b) N11
(c) brier
Ece
Figure 12: Pareto-front lines between level of group faimesss matched Gibbs posterior with varying priors, CRIME.
AW and prediction performance of
29

---

Published as a conference paper at ICLR 2026
E ADDITIONAL RESULTS
E EXTENSION TO EQUALIZED ODDS
We mainly focus on demographic parity as a fairness measure. In this section we show that the extension with regard to EO (equalized odds) can be done similar.
Specifically, we consider two matching functions To Xo.1 - No,o and T1 11,1 + Xi,0, where Xy.s is the domain of X|Y = y, S = s. Now, we can modify the matched deviation as
(M To: Tn) Po (x (1 1) (To
0)l1²)
(33)
+ Pin1 P1n ( = 1o - (Xi), = = 0)1
where Pyl= = Pr(Y = =y = 5) and Ip ((.,s) is conditional distribution of X| y. = s. Now we can define matched Gibbs posterior with respect to the modified matched deviation.
For the MCMC, we consider MH for (To, T1) pl (o. T, Dr). We can consider a prior
e(To, Tr) = exp 1 Ee d(xx!0)) (0.0) To(x! (x))) 1 E d(x;! (100 (x((X"")))/=
For the proposal, we can reuse the partially permuted proposal of Eq. (9) for y E {0, as:
TT))) (T
for $ [my1] to , 1 for = a
T₂O)=
(34)
Using this formulation, additionally perform experiment CRIME, further show that the trade-offs from matched Gibbs posterior with regard to EO are better than the baseline methods, in Fig. 13 The results similar to the case of DP, strengthening the ability of matched Gibbs posterior.
We can continuously observe that matched Gibbs posterior shows better performances than other baselines. Therefore, we believe that the proposed matched Gibbs posterior can also be well applied
EO.
0.18
0.8
reductionn
redlction
0.37
adv
0.7
0.58
0.54
0.13
0.78
0.24
0.79
0.30
0.33
0.74
32prin
0.08
0.32
reduction
0.72
04
Ago
Ago
(a) ACC
(b) N11
(c) bricr
(d) Ece
Figure 13: Pareto-front lines between level of group fairness AEO and prediction performance of matched Gibbs posterior, on CRIME.
E.2 EXTENSION TO MULTINARY SENSITIVE ATTRIBUTE CASE
We can easily extend matched Gibbs posterior into the case of multinary sensitive attribute, S E (0, 1.... .M} for an integer M > 1. Specifically, we may choose one sensitive group (for example, X)) as an anchor group and define matching functions T, : X; + Xo that map each of the other sensitive groups to this anchor group. In this setting, we can define the matched deviation
(0, (0, (T.) EM, Ex,,8, (X,, 1) (T) () and then construct
30

---

Published as a conference paper at ICLR 2026
matched Gibbs posterior using this quantity. For MCMC procedure, the extension is also straight- forward: we can simply apply the Gibbs sampler iteratively for each T,, using same MH proposal with Eq. p9)
E.3 PERFORMANCE BY STRONG DEMOGRAPHIC PARITY
For an additional support of the performance of matched Gibbs posterior, we provide the trade-offss measured with other DP metric, Strong Demographic Parity (SDP). The definition of SDP gap is as follows (Jiang et al. 2020a Silvia et al. 2020):
(35) This is called strong DP, since this equivalent with (p which enforces DP to be hold regardless of the threshold value r. In other word, enforcing SDP does not depend on the threshold value, and thus not sensitive to the choice of the threshold (Jiang et al., 2020a Silvia et al. 2020).
ASDP (f) E, [rr(f(X, 0) 2 Pr(f(X, 1)
We evaluate the utility- / uncertainty-fairees trade-offs, using the original DP as fairness measure, on CRIME dataset on the same setting of the main experiment. The results are in Fig. [4 We consistently observe that the trade-offs of matched Gibbs posterior are better than the bascline methods
0.24
Dap reductio
peprog
2.55
DeSuvtion
resluctoon
0.22
0.18
0.20
1.55
0.18
0.14
0.12
22proy
1.45
0.14
0.00
0.99
Appp
Aspp
AspP
AspP
(a) Acc
(b) N1N
(c) prier
(d) Ece
Figure 14: Pareto-front lines between level of group fairness ASDP and prediction performance of matched Gibbs posterior, on CRIME.
E.4 ABOUT COMPUTATIONAL COSTS
The computational complexity of the MH step for T does not depend on the input dimension, and its scale on each step is O(1) since it is a random permutation on a fixed window size of k. We provide running time table of the main experiment
CIVII. dataset, in Table 9 We observe that the time spent on the actual sampling (for joint inference of f and T) is relatively small.
This because (i) the scale on each step is small as mentioned, and (ii) since we designed the procedure to use pretrained model as a good initial point for f and OT-based initialization for T, the MCMC converges quickly and reliably. See Appendix C.7 for additional supporting evidences of the reliability of the proposed MCMC.
E.5 ADDITIONAL DATASET
There are some concerns regarding the use of ADULT dataset (Ding et al. 2021a). Therefore, we additionally employed the ACSINCOME dataset which is an alternative to ADULT, to reproduce the experiments presented in the main text. Following the experimental setup (Han et al. 2023), employed MLP with four hidden layers of width 128, and used gapreg and reduction for 10 epochs. To give an efficient initial value, we utilized pretrained DNN models as initial parameter values, with pretrain epoch 10. The remaining experimental settings provided in Appendix B.2
31

---

Published as a conference paper at ICLR 2026
Table 9: running time table on CIVIL dataset. Etc. denotes the time measurement on initializations.
Method
Time (sec.)
Etc. (sec.)
gapreg reduction
48.85
47.32
pretrain: 46.87 ot initial: 48.56
gibbs matched
24.49
We split the dataset into partitions with proportion 8 2, corresponding train and test datasets. Table 10 provides the basic descriptions of the ACSINCOME dataset.
Table 10: Summary statistics for ACSINCOME dataset.
Dataset
Features (d)
Size
S =
train samples
test samples
ACSINCMME
120
1,664,500
614,067
866,735
1,331,600
332,900
The corresponding results in Fig. 15 Since the adversarial debiasing method (Adv) (Zhang et al, 2018) showed worse performance compared the baseline and our method, we excluded from this experiment. We can consistently observe that the trade-off's of gibbs matched is better than the other algorithms.
0.992
gapreg
gapray redlction
0.3350
0.802
0.424
0.3965
0.016
0.793
0.320
0.224
0.3355
0.012
0.423
0.797
0.013
0.732
6419
Qeprog
0.260
0.04
0.335
0.03
A
4
A
Acc
(b) N11
(c) brier
Rce
Figure 15: Pareto-front lines between level of group faimess A and prediction performance of matched Gibbs posterior, on ACSINCOME.
32


Reviews:
[{"review_id": "24DKT0OWme", "reviewer": "Reviewer_26og", "summary": "This paper addresses the underexplored challenge of integrating model uncertainty into algorithmic fairness. To this end, the authors propose a group-fair posterior distribution and develop a fair variational Bayesian inference framework that embeds fairness constraints into probabilistic learning. To improve computational efficiency, they introduce a novel matched Gibbs posterior, which approximates the fair posterior under fairness constraints using a newly defined metric, matched deviation. This measure is theoretically shown to be closely related to established fairness notions, thereby offering strong fairness guarantees. Empirical evaluations on real-world datasets demonstrate that the proposed approach achieves superior trade-offs between fairness, model uncertainty, and predictive utility compared to baseline methods.", "questions": "See weaknesses part.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. The topic of group fairness is highly relevant and socially significant, making this study timely and impactful.\n\n2. The authors effectively integrate group fairness and model uncertainty within a Bayesian inference framework and provide solid theoretical justification for their approach.\n\n3. Experiments conducted on three distinct modalities of datasets demonstrate the robustness and practical applicability of the proposed method.", "weaknesses": "The paper’s writing and structure could be improved for clarity. After reading, several conceptual issues remain ambiguous:\n\na. What is the motivation for jointly modeling group fairness and uncertainty? Are there concrete real-world applications that benefit from this combination?\n\nb. What are the main technical challenges in combining fairness and uncertainty? Have there been prior studies exploring this intersection? Why can’t existing fairness and uncertainty methods simply be combined?\n\nc. Why is the Bayesian inference framework particularly appropriate for this setting? What specific advantages does it offer compared to prior non-Bayesian approaches?\n\nThe work only considers DP as the fairness definition, which is too limited. Other widely used metrics, such as EO and EOdds, should also be discussed. It remains unclear whether the proposed theory and framework can generalize to these metrics.\n\nIn the experimental section, fairness evaluation is restricted to the Wasserstein distance under DP. The empirical validation could be strengthened by including additional fairness metrics to better demonstrate the generalizability and robustness of the proposed method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "z7cDTcXdqa", "reviewer": "Reviewer_vJDv", "summary": "The paper develops a Bayesian route to group fairness by defining fairness for posteriors and proposing a matched Gibbs posterior: a Gibbs posterior with a new matched deviation penalty that avoids adversarial IPM/MMD inner loops, comes with bounds relating it to Wasserstein/TV, and is trained via an HMC+MH sampler that jointly infers the predictor and a matching function $T$. Experiments on ADULT, DUTCH, CRIME, CELEBA, and CIVIL suggest improved utility–fairness and uncertainty–fairness trade-offs.", "questions": "Please see weakness.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Clear formulation of fairness for posteriors (average DP to strong DP via rejection) and a practical proxy—the matched Gibbs posterior—that circumvents adversarial critics and offers $O(n)$ updates with an explicit sampler.\n2. Sound theory + strong experiments. Bounds linking matched deviation to Wasserstein/TV give intuition, and the image/text/tabular results show stronger Pareto fronts.", "weaknesses": "1. Fairness target & positioning. The work focuses on demographic parity. Is there any justification for this choice against other metrics (Equal opportunity/equalized odds/calibration)?\n2. Fairness is measured on score distributions (W2(P_{f,0}, P_{f,1})); the relation to thresholded decisions (rate gaps) is unclear. Could you provide a more detailed analysis connecting score-level DP to rate-level DP, with sensitivity to thresholds or post-hoc calibration? \n3. The prior and MH proposal for T (swap k matches) are reasonable, but (i) what distance d is used for images/text (esp. CIVIL)? (ii) How does mixing/acceptance scale with n, class imbalance, and high-dimensional X? \n4. Complexity claims. The paper argues $O(n)$ per update vs. $O(n^2)$ MMD; it would be better if there were some running time/memory tables on large splits and report the cost of the HMC step (leap-frogs, step size) and MH over $T$.\n5. Concerns about the dataset. The empirical studies are very strong. However, the adult dataset has some known issues (https://arxiv.org/abs/2108.04884), it would be better to try out a more robust dataset or mention these caveats.\n6. Limitations. The current proposed method seems only applicable to binary sensitive attributes. Are there any discussions on how to extend to multi-categorical/continuous ones? Also, how could it be generalized to conditional fairness notions like equalized odds?\n\nI'm willing to raise the score if my concerns are discussed and addressed, thank you.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lrLXdCKPuW", "reviewer": "Reviewer_ZbGT", "summary": "The paper introduces a Bayesian framework for fair machine learning, addressing group fairness and uncertainty quantification  simultaneously. Traditional group-fair models ensure fairness by constraining optimization objectives but rarely account for predictive uncertainty, which is crucial for trustworthy AI. The authors propose a matched Gibbs posterior, derived from a novel fairness-aware penalty called the matched deviation, which upper-bounds the Wasserstein and total-variation fairness measures. This formulation avoids adversarial training and provides a computationally efficient approximation to the fair Bayesian posterior. They design an MCMC algorithm where both model parameters and the matching function 𝑇 are inferred jointly, ensuring fairness without heavy computation. Experiments on tabular, image, and text datasets show that the matched Gibbs posterior achieves better trade-offs between utility vs fairness and uncertainty vs fairness than baselines such as Reduction, GapReg, and Adversarial fairness methods, while also improving individual fairness.", "questions": "Impact of imperfect matching: How sensitive is fairness performance to suboptimal or noisy matching functions 𝑇? Can the authors quantify how deviation from optimal 𝑇 affects fairness bounds?\n\nScalability: How would the proposed MCMC perform on large neural models (e.g., transformers) or high-dimensional text embeddings?\n\nConnection to uncertainty geometry: Can the authors relate matched Gibbs fairness constraints to curvature of the posterior (e.g., Hessian eigen structure) or persistent-homology-based fairness landscapes? (This is not necessary for the paper, just something that came to mind during the review.)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "1. Well written and clearly structured: Despite heavy mathematics, the paper is well organized; intuitions precede theorems, notation is consistent, and experiments visually support claims.\n\n2. Novel combination of fairness and Bayesian inference: The paper is among the first to integrate group fairness constraints directly into Bayesian inference, explicitly addressing both uncertainty quantification and fairness.\n\n3. Avoids adversarial optimization: By replacing adversarial discriminators with a learnable matching function 𝑇, the approach sidesteps instability and 𝑂(𝑛^2) cost typical in IPM-based fairness.\n\n4. Improved fairness–utility trade-off: Across datasets, matched Gibbs posterior consistently outperforms prior baselines on accuracy, NLL, Brier, and calibration error (ECE).", "weaknesses": "1. Scalability concerns: Joint inference of 𝑓 and 𝑇 may become costly for high-dimensional or non-metric input spaces (e.g., text embeddings). No discussion on large-scale efficiency.\n\n2. Restricted to binary sensitive attributes: The method currently handles only 𝑆∈{0,1}; multi-group or intersectional fairness remains unexplored.\n\n3. Empirical scope and baselines: Experiments are thorough but limited to medium-sized datasets; modern large-scale deep architectures (e.g., BERT, ResNet-50) are absent.\n\n4. Unclear robustness under complex priors: The framework assumes tractable Gaussian priors; how matched Gibbs behaves with non-Gaussian or hierarchical priors is not tested.\n\nMy main concerns focus on scalability and applicability to modern large-scale architectures. Specifically, the paper lacks validation on high-capacity models such as BERT or ResNet-50, and it remains unclear whether the proposed matched Gibbs posterior remains computationally feasible in high-dimensional or non-metric spaces (e.g., text embeddings). In such settings, stability and sensitivity to noise could become significant issues.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)