Does Unsupervised Domain Adaptation Improve the Robustness of Amortized Bayesian Inference? A Systematic Evaluation
Authors: Lasse Elsemüller
Valentin Pratz
Mischa von Krause
Andreas Voss
Paul-Christian Bürkner
Stefan T. Radev
Venue: Accepted by TMLR ()

Abstract: Neural networks are fragile when confronted with data that significantly deviates from their training distribution. This is true in particular for simulation-based inference methods, such as neural amortized Bayesian inference (ABI), where models trained on simulated data are deployed on noisy real-world observations. Recent robust approaches employ unsupervised domain adaptation (UDA) to match the embedding spaces of simulated and observed data. However, the lack of comprehensive evaluations across different domain mismatches raises concerns about the reliability in high-stakes applications. We address this gap by systematically testing UDA approaches across a wide range of misspecification scenarios in silico and practice. We demonstrate that aligning summary spaces between domains effectively mitigates the impact of unmodeled phenomena or noise. However, the same alignment mechanism can lead to failures under prior misspecifications -- a critical finding with practical consequences. Our results underscore the need for careful consideration of misspecification types when using UDA to increase the robustness of ABI.

Published in Transactions on Machine Learning Research (09/2025)
Does Unsupervised Domain Adaptation Improve the Robust- ness of Amortized Bayesian Inference? A Systematic Eval- uation
Lasse Elsemüller Heidelbery University, Germany
lasse, elsseeesemeeller @gmail
Valentin Pratz
Heidelberg University, Germany Zuse School ELIZA, Germany
Mischa von Krause Heidelbery University, Germany
Andreas Voss
Heidelberg University, Germany
Paul-Christian Bürkner
TU Dortmand University, Germany
Stefan T. Radev Rensselaer Polytechnic Institute, USA
stefon.radevd38@maal.coo com
Reviewed on OpenReview: https: // openreview. net/ forum? id= eugLuvnEut
Abstract
Neural networks fragile when confronted with data that significantly deviates from their training distribution. This is true in particular for simulation- inference methods, such as neural amortized Bayesian inference (ABI), where models trained on simulated data are deployed on noisy real-world observations. Recent robust. approaches employ unsuper- vised domain adaptation (UDA) to match the embedding spaces of simulated and observed data. However, the lack of comprehensive evaluations across different domain mismatches raises concerns about. the reliability in high-stakes applications. We address this gap by systematically testing UDA approaches across wide range of misspecification scenarios in silico and practice. We demonstrate that aligning summary spaces between domains effectively mitigates the impact of unmodeled phenomena or noise. However, the same alignment mechanism can lead to failures under prior misspecifications a critical finding with practical consequences. Our results underscore the need for careful consideration of misspocification types when using UDA increase the robustness of ABI.
1 Introduction
Synthetic data can augment numerous real-world applications Savage 2023
including complex statistical
workflows Bürkner et al. 2025). In line with this perspective, amortized Bayesian inference (ABI; Gershman & Goodman 2014) redefines the classical sampling problem in Bayesian estimation by training generative
neural networks on simulations derived from computational models (Bürkner al. 2023 Cranmer et al.
2020). The trained neural networks are then deployed to efficiently solve inference tasks as diverse as inferring
evolutionary parameters (Avecilla et al. 2022
gravitational waves (Pacilio et al. 2024

---

Published in Transactions on Machine Learning Research (09/2025)
NPE-MMD
Backpropagate LNPE
Simulated data
Summary network
Summary statistics (sim)
Inference network
Posterior approximation
Summary statistics (obs)
Maximum Mean Discrepancy (MMD; simulated vs. observed)
Observed data
Additionally backpropagate À MMD
NPE-DANN
Backpropagate NNEE
Simulated data
Summary network
Summary statistics (sim)
Inference network
Posterior approximation
Summary statistics (obs)
Classifier network
Simulated vs. observed?
Observed data
Grodient reversal
Additionally backpropagate À DANN
Simulated data flow
Observed data flow
Backpropagation flow
Figure 1: Schematic overview of NPE-UDA methods that combine neural posterior estimation (NPE) with unsupervised domain adaptation (UDA). Standard NPE training optimizes posterior approximation a simulation-basse training loop. NPE-UDA approaches introduce observed data into the training procedure, targeting performance improvements in the (possibly shifted) observed domain via domain alignment in summary space. NPF-MMD (maximum mean discrepancy) directly minimizes the distance between distri- butions, whereas NPE-DANN (domain-adversaria) neural networks) uses adversarial competition between an auxiliary domain classifier and the summary network.
Evidently, the faithfulness of any simulation-base method rests on the critical assumption that statistical patterns learned from simulated data can be extrapolated to real observations. This assumption inevitably situates ABI in a domain-shift regime, exacerbated by the degree of potential mismatch between model simulations and reality. As such, robusiness to model misspecificaaiion has been identified as the primary challenge for amortized methods in different fields (Dingeldein et al. 2024 Rainforth al. 2024 Cannon
et al. 2022
Unsupervised domain adaptation (UDA) studies the transfer of knowledge from a labeled source domain to an unlabeled target domain. It aims to mitigate domain shifts by aligning the embedding spaces of the two domains. This property makes UDA a promising approach for addressing domain shifts in ABI, as the latter typically combines inference with embedding high-dimensional data into learned summary statistios (Radey ct al. 2020: Chan et al., 2018). Indoed, recent rescarch has underscored the critical role of in-distribution summary statistics for achieving robust simulation-based inference (Schmitt et al. 2024
Frazier et al. 2024
Huang al. 2023 Wehenkel al. 2024).
So far, only two pioncering studics Swiere et al. 2024 Huang et al. 2023 have explored the potential of UDA methods for robustifying simulation-based inference. Both approaches aligu the embedding spaces by minimizing the maximum mean discrepancy (MMD; Gretton et al.. 2012) between simulated and observed summary statistics (sce Figure 1). However, despite their promising results, several gaps remain. In particu- lar, Huang et al. (2023) did not make an explicit connection to UDA and explored non-amortized approach. While Swiere et al. (2024) acknowledged the connection to UDA, their work focused on specific gravita-
2

---

Published in Transactions on Machine Learning Research (09/2025)
tional lensing application. Both works mainly evaluated likelihood misspecification, leaving the behavior under prior shifts largely untapped. Finally, the utility of the widely used UDA method domain--auersarial neural networks (DANN; Ganin et al.) 2016) remains completely unexplored. To address these gaps, we make the following contributions:
1. We adapt domain-adversariil neural networks for neural posterior estimation (NPE; see Figure 1 and evaluate their utility for robust amortized Bayesian inference.
2. We theoretically categorize robust, methods by inference targets, explaining when and why UDA- based ABI methods succeed or fail under different sources of model misspecification.
3. We empirically evaluate the robustness of UDA-basod ABI mcthods across both prior and likelihood misspecification in multiple benchmarks, confirming the central role of the source of misspecification: whereas UDA improves performance under likelihood shifts, it is detrimental under prior shifts.
2
Background
2014 Ritchic al. 2016 Le et al. 2017] are a subset of the simulation-based inference (SBI; Cranmer et al. 2020) family. Their defining characteristic is the ability to perform zero-shot inference model parameters 0 by learning conditional distribution q() x) that requires no further training auxiliary algorithms for new data æ (see Appendix A for details). The amortized distribution ( x) is typically parameterized by an inference network, a generative neural network that can generate random samples 0 ~ q(0 x) akin to a standard Markov chain Monte Carlo (MCMC) sampler, but orders of magnitude faster. Often, the inference network is preceded by a summary network é that compressos raw observations to Icarned summary statistics Φ(x), leveraging probabilistic symmetries in the data (Radev al., 2020; Chen et al. 2021). Following a potentially expensive simulation-basee approximate the target distribution p(O [new). Initially dismissed as inefficient compared to sequential methods optimized for a specific data set Tobs (Papamakarios & Murray 2016), amortized methods have
Amortized Bayesian Inference (ABI) Amortized methods (Gershman & Goodman
training phase, the network can be queried with any new data Nnew to rapidly
since achieved notable successes across various domains (Bürkner et al. 2023 Zammit-Mangion et al. 2024)
Unsupervised Domain Adaptation (UDA) UDA is a subfield of transductive transfer learning where labeled data is only available for the source domain D3 = { ) distributed according to ps (z. y
but not for the target domain Dr = ) Nr distributed according to pr 3r) (Johansson et al. 2019
UDA methods are based on the seminal theoretical works of Ben-Daviid et al.] (2006) 2010), who introduced
generalization bounds for binary classification tasks that bound the risk in the target domain Ry of a hypothesis h H:
RT(h) S Rs(h) + dHAH(PS,PP) AH:
(1)
where (h) is the source domain risk, dzaH (Ps: PT) measures the divergence between the domain dis- tributions, and Xu is the minimum combined risk of the optimal hypothesis, Au = infnex|Rs(( RR(h)) + (h))
[Johansson et al. 2019). This suggests that domain adaptation from Dss to Dr can be facilitated by minimiz- ing the divergence between the marginal domain distributions. Although the domain distribution divergence cannot be reduced directly, the representation divergence d(o(xs), ((xx)) from a transformation o : Z can be readily minimized (Ben-David et al. 2006). The core idea of UDA thus twofold: (i) to mini- mize the source-domain error Rs(h) during training, and (ii) to align the domain representations d(xs) and )(æT) to achieve domainn-nnvrrian embeddings that generalize to the target domain. UDA methods include tiscrepancy-basee approaches, which minimize statistical divergences like the MMD between source and target embeddings (Tzeng et al.) (204)). and, most prominently, adversarial-basee approaches, such as domain-adversarti neural networks (DANN) (Ganin et al. 2016), which learn domain-invariant embeddings via a minimax game between a feature extractor and a domain classifier.
The vast majority of UDA rescarch, including its thooretical foundations, focuses on classification tasks Redko et al. 2022; Ben-David David et al. 2010 Liu et al.) (2022), with some works on regression tasks (Cortes) Mansour et. al. 2009) and only few on generative tasks (Uppaal et. al. 2024). More
& Mohri 2014
3

---

Published in Transactions on Machine Learning Research (09/2025)
Table 1: List of central acronyms, ordered by first appearance in text.
Acronym
Definition
ABI
Amortized Bayesian Inference
NPE
Neural Posterior Estimation
UDA
Unsupervised Domain Adaptation
MMD
Maximum Mean Discrepancy
DANN
Domain-AAlversarial Neural Networks
SBI
Simulation Caauee Inference
MCMC
Markov Chain Monte Carlo
NLL
Negative Log Likelihood
NRMSE
Normalized Root Mean Squared Error
ECE
Expected Calibration Error
PC
Posterior Contraction
PPD
Posterior Predictive Distance
SSDD
Summary Space Domain Distance
C2ST
Classifier Two-Sample Test
INLD
Inference Network Latent Distance
LAT
Implicit Association Test
recently. UDA methods have been successfully applied to address simulatioon-to-realiy (sim2real) problems (Ciprijanović et al., 2020; Swierc et al. 2023 Kong et al., 2023) which seek to generalize patterns learned in a simulated source domain to a real-world target domain. These problems seem pertinent to any simulation- based method relying on data generation from imperfect models.
From Simulated to Real Domains The preceding discussion makes the connection between UDA and ABI immediately apparent: When the distance between the data distribution p(Zobe) and the model- implied distribution p(æ) Eppc [p(æ 0)] non-zero, the risk of extrapolation error for atypical data Bobs may increase. Indeed, this behavior has been observed repeatedly in the context of SB1 (Ward)
et al. 2022 Schmitt et al. 2024 Huang et al. 2023 Frazier et al., 2024 Kelly et al. 2025 In partic- ular,
Frazier et al. (2024] noles that ABI is especially prone to "extrapolation bias" for observed sum- mary statistics ((x)) that are far in the tails of the model-implied (i.e., prior predictive) density p(æ)).
The scemario can be equivalently stated by invoking the notion of a typ-
ical set (Cover & Thomas, 2012), which denotes a subset of the support of p(æ) where most of the probability mass concentrates around the entropy (p)::
0.30
NRMSE
0.25
A. = (x X 2 (p)) < <e}.
0.20
Accordingly, for any problem-specific- e, observed data Tobs & A, may result in a biased posterior approximation g( Wohs). ). As further noted in the comprehensive theoretical exposition by Frazier et al. (2024).
0.0
0.2
0.4
matching summary statistics d(x.bx) to the model-implied distribution of ((x) can be a useful heuristic for reducing extrapolation bias. This observation harmonizes with the UDA literature as well (Ben-David et al. 2010). Pre-asymiptotically the success of such matching depends on multiple factors, including (i) the type and hyperparameters of the matching method (see Figure 2); (ii) the degree and nature of domain mismatch; (iii) the complexity of the learning problem; and (iv) even the choice of success metric. Thus, primary goal of this work is systematically examine the effects of these factors on variety of metrics that index patential robustness gains.
SSDD
NPE NNPE NPE-DANN NPE-MMD
Figure 2: Experiment 3. Sum- mary space domain distance (SSDD; MMD) normalized root mean squared error (NRMSF) for row deletions. We observe a sweet spot of domain alignment without losing important information.
4

---

Published in Transactions on Machine Learning Research (09/2025)
3 Methods
3.1
Unsupervised Domain Adaptation for Amortized Bayesian Inference
We start with the observation that model misspecification in ABI (Schmitt et al. 2024 and also more generally in neural SBI, can naturally be framed as an UDA problem: Ground-truth parameter values are only available for the simulated source domain == {(x', )' but not the observed target domain Dobs bbe Nob.. M In most machine learning applications, the collection of reliable ground-truth values is costly but leasible, whereas in SBI, collecting ground-truth parameter values Oobe of observed data is typically impossible. A general optimization objective for NPE-UDA methods can be formulated by extending the standard negative log pog-posterio NPE objective:
LNPE-UDA(q Ø) LNPE + A LUDA
-Bp(0xx)p(xaa log log ((x)) + X d(((ar), d((rab))))
(4)
where À controls the regularization weight of the UDA loss and d(-.-) is a divergence measure that attains its global minimum if and only if ((x)) ((xrbs))
LNPF-UDA incurs trade-off between approximation performance in the simulated domain and domain difference in the summary space, depending on the degree of domain mismatch. In the well-specified case, p(œ) = p(oobs), ENPE-UDA reduces to the standard NPE loss. In the misspecified case, p(æ) # p(zobs), the summary network & optimizes the summary statistics to both mazimize information extraction in the simulated domain and minimize domain shift in summary space. Thereby, the inference network q(O <bxr) needs to rely on domain-invarian information shared between the simulated and the observed domain.
The common UDA assumption that there exists low-error hypothesis for both domains (Redko et al. 2022 ef. D suggests X-dependent upper bound on the amount of domain shift that can be rectified by UPE-UDA methods. However, finding an optimal value for À despite missing target labels (i.c., fol...oounddrttt parameters in SBI) al lest. time is an open problem in UDA research (Zellinger et al. 2021 Musgrave el al., 2022h. Thus, we expect this problem to carry over to NPE-UDA methods as well. Next, we formulate two NPE-UDA variants based on popular UDA methods with strong performance on established benchmarks
Musgrave et al. 2021
3.2
NPE-MMD
The maximum mean discrepancy (MMD; Gretton et al. 2012 is a popular probability integral metric
SBI, since can be efficiently estimated from finite number of samples (Bischoff al. 2024 Schmitt al.
2024 For the same reason, it has been employed by various UDA works (Pan et al.
2010 Tzeng et al.
2014 Long et al. 2015) to measure the divergence between (transformed) samples from different domains.
We categorize the combination of NPE and UDA based on MMD, such as the variants of Huang et al. (0023)
and Swierc et al. (2024), as NPE-MMD. Choosing the MMD as LUDA, Eqq becomes
ENPE-MMD (g. Φ) := x)) log ((0 ((x))] + X MMD 2 [d(x) ) ((Eob=))
(5)
The most important hyperparameter of NPE-MMD is the choice of kernel in the sample-based MMD esti- mator. In our experiments, using a sum of inverse multiquadric kernels (Ardizzone et al.) 2018) led to the most stable training dynamics, but other choices have been explored in the context of robust ABI as well,
such as (sums of) Gaussian kernels (Schmitt et al. 2024 Huang et al. 2023
3.3 NPE-DANN
Domain-advvesaarial neural networks (DANN; Ganin et al.) 2016), which have not been considered for NPE to date, introduce a domain classifier (b() to reduce domain distance. Unlike typical adversarial training, which alternates between objectives, DANN achieves minimax optimization in single-step update via a gradient reversal layer (Ganin et al.) 2016 This layer flips the gradient sign from the classifier the feature extractor (e-g., summary network) Φ during backpropagation, encouraging the feature extractor to
5

---

Published in Transactions on Machine Learning Research (09/2025)
generate less domain-specifie summary statistics. Similarly to NPE-MMD, DANN can be integrated into Fq. 3 to achieve NPF-DANN:
ENPE-DANN( &, v) = Bp) log ((]]))) + A (D(..)).
(6)
The discriminator loss LD is given by:
(p(w.0) == -pp(e) log(p(((((x)))) - Ep(a..))) log(1 - pl p(((((((oba))))
(7)
where & is the domain classifier and the equation represents the binary cross-entropy loss on the domains, where a gradient reversal layer enables updating & and ai in opposing directions.
While DANN is a powerful and popular UDA method (Zhou ct al. 2022), it has two important drawbacks. First, the unstable training dynamics and convergence issues generally associated with adversarial learning also occur with DANN (Sener et al., 2016; Sun et al.) 2019). Second, adversarial training adds new hyperparamcters. including the domain classifier architecture, an optional weight for gradient reversal balance Ganin et al., 2016), and stabilization techniques like label smoothing (Zhang et al. 2023 Notably, although À is a shared hy typerparameter in NPE-MMD and NPE-DANN, its effect on training dynamics will vary across applications due to differing LUDA scales.
3.4 What Is the Target of Robustness?
To better understand the strengths and limitations of robust methods, including NPE-UDA, we suggest distinguishing between the following inference goals:
The analytic (true) posterior p(O Dohs)) p(zobs p(0) of the assumed probabilistic model given the observed data Tobs
Target 1:
Target 2: A posterior p(0 Fabs) x pl oooss adjusted data Tobs.
p(0) of the assumed probabilistic model given
Target 3: A posterior 7(0 Babs) x P(aos (0) ((0) from an adjusted prior p(0) given the observed data Fobs.
Target 1 is the most common target in Bayesian inference. Classical approximation methods such as MCMC almost always consider this larget (Carpenter el al.) 2017). Target 2, an explicit devialion from the true posterior, is often targeted by methods that seek to improve the robustness of Bayesian inference (see 4 Their goal is to reduce the influence of unmodeled phenomena in Lobs, such as additional noise or external contamination, by approximating a larget posterior p(0 Tabs) based on denoised or uncontaminated data Fabs- This can be achieved either explicitly, by transforming Pobs into Pobs: or implicitly, by using an adjusted (òmplicit) likelihood p(Zoba 0).
Since Target 2 implies ignoring parts of the data that are in disagreement with the assumed probabilistic model, we expect corresponding methods to perform worse under prior misspecification: When data- generating parameter 0* is impossible or highly unlikely under the assumed prior, igmoring conflicting information effectively reduces the amount of information available to counteract a poorly chosen prior. Generalized Bayes approaches (Bissiri al., 2016) also aim to reduce the influence of undesired parts of the data. They move away from the classical Bayes rule by replacing the likelihood with a loss function, which can be interpreted as an adjusted likelihood p(arata 0) according to Target 2. Lastly, Target 3 can directly reduce the impact of prior misspecification by adjusting the prior based on Tobe. However, compared to Target 2, it is more challenging to conceptualize the desired target priors p(0) and posteriors 5(( Eabs) under model misspecification.
Given this categorization, what is the target NPE-UDA? Unsurprisingly, the classic NPE loss CNPE aims at Target 1. In contrast, the additional LUDA loss governs the alignment of the summary space between simulated and observed data, effectively adjusting the observed data seen by the model. Thus, LUDA introduces shift towards Target 2, with À governing its relative importance compared to Target 1. As
6

---

Published in Transactions on Machine Learning Research (09/2025)
hypothesized above, methods aiming at Target 2 may not perform well under prior misspecifcation,, which is confirmed for the NPF-UDA methods throughout our experiments. While Huang et al. (2023) suggested that their NPE-MMD variant is robust to prior mean shift, this conclusion was based on a single tested Tobs and our comprchensive evaluation could not replicate the result.
In line with our hypothesis and empirical results, Huang et al. (0023) observed that increasing values of À encourage trading off the information content of æ to minimize the domain distance in summary space, leading the posterior to converge to the assumed prior p(0). Thus, the critical importance of the tunable hyperparameter À in UDA contexts (Zellinger et al. 2021] directly translates to ABI applications, where X controls a tradc-off between improving approximation under likelihood misspecification and degrading approximation under prior misspecification.
4 Related Work
Robust Neural SBI Robustness in neural SBI has become rapidly growing area of research, with most approaches en enhancing robustness for a single data set at the cost of amortization, e.g,, due to additional MCMC runs or post-hoc corrections. The majority of these approaches focuses on Target 2 by incorporating a misspecification model (Ward et al., 2022), shifting observed summary statisties with low support (Kelly et al.] 2023), reducing the influence of unmodeled data shifts via generalized SBI (Gao et al., 2023), or
Target 1. Siahkoohi et al. (2023) highlighted the role of the inference network's 's latent space in domain shifts and proposed latent space correction based on the observed data Tobs. Differently, Wang et al. (2024) focus
using the singlo-data-sct NPE-MMD variant previonsly discussed (Huang et al., 2023 Focusing
on Target 3 by using an upfront ABC run to filter the part of the parameter space causing the highest discrepancy between x and Tobs
Robust ABI In contrast, rescarch on robustifying inference while retaining amortization has boen sparse. Extending the scope of the training data via additive noise (Cranmer al.) 2020: Bernaerts et al.] 2023 such the spike-and-slab noise approach of noisy NPE (NNPE; Ward et al. 2022), can be seen as light modification to the simulator-implied likelihood of Target 2, but requires strong assumptions about the corruption process. Wehenkel et al. (2024) also approach Target 2 by framing domain shift as an optimal transport problem in summary space, but this requires observed "ground-truth" parameters O that are difficult to obtain in most ABI settings. Swierc et al. (2024) provided evidence for the potential of NPE-MMD for robust ABI but focused their evaluation solely on a gravitational lensing application with synthetically added noise. Finally, Glöckler al. (2023) proposed an efficient regularization technique that can increase robustness against adversarial attacks and attain more reliable performance under Target 1.
5 Experiments
In all experiments, we benchmark NPE-MMD and NPE-DANN against an NPE baseline as well as NNPE (Ward et al.) 2022 see Appendix B.1 for further details). We acknowledge that NNPE was not the primary focus of Ward et al.] (2022) and instead consider it an instantiation of a simple additive noise training modification (Cranmer et al. 2020 Bernaerts et al. 2023), the presumably most straightforward current approach for robustifying ABI in practice.
The two existing works on NPE-MMD approaches mainly evaluated performance against contamination (Huang et al. (002)), where a fraction of the sample is replaced with corrupted obscrvations (Huber 1981 or noise applied to all observations (Swiere et al., 2024).. Both of these scenarios are cases of likelihood misspecification where ignoring noise is desirable (Target 2). That is, the inference target is the posterior p(0) T) of the assumed probabilistic model given the uncontaminated or (implicitly) denoised data set ă.
Experiment 1 introduces a canonical contamination selting, in which we explore the sensitivity of NPE-UDA methods to hyperparameters using Bayesian optimization. Afterwards, we expand the scope by compre- hensively evaluating various aaaiiooddddtaa and prior misspecification scenarios to obtain clearer insights into the strengths and limitations of the robust methods. Experiment starts with a simple and con- trollable setting that allows for comparing the NPF-UDA methods not only against standard NPE and
7

---

Published in Transactions on Machine Learning Research (09/2025)
NNPE [Wad et al., 2022; but also to the analytic posterior under Target 1. Experiment 3 explores whether the result. patterns can be replicated in a challenging setting with high-dimensional parameter space. Lastly, since we are ultimately interested in the robustness of NPE in genuine scientific applications, Experiment 4 tests the methods on a massive real-world data set of human decision- makinn (von Krause
et al. 2022).
We evaluate a range of metrics to enable a holistic assessment of the compared methods:
Parameter space performance metrics: (i) Negative log likelihood (NLL.) as standard measure of posterior density estimation (ii); normalized root mcan squared error (NRMSE) to mcasure ap- proximation error; (iii) expected calibration error (ECE) to measure the fidelity of credible intervalst; (iv) posterior contraction (PC) to measure information gain from prior to posterior.
Data space performance metrics: (i) Posterior predictive distance (PPD) to the observed data, which is the standard approach but has the disadvantage that the observed data contains the noise that a robust method should ignore, and (ii) PPD to data resimulated from the ground-truth pa- rameters, a modification that allows PPD to represent the distance to well-specified (e.g., denoised) data. In Experiment 4, we approximate the denoised reference via intensive data pre-processing.
Network space metrics: (i) Summary space domain distance (SSDD) measuring domain align- ment via MMD, (ii) SSDD via the classifier two-sample test (C2ST), and (ii)) inference network latent distance (INLD) to the base distribution of the generative neural network (diagonal Gaussian in all our experiments). which has recently been highlighted as the central mediator of posterior errors Siahkoohi et al. 2023
Appendix B provides details on metries, experimental setups, network architectures, training and evaluation procedures, as well as additional results. Code for reproducing all results from this paper is available at http:///ggttubb.coobbaaeeee/f/ee
5.1 Experiment 1 Ricker: Hyperparameter Exploration
Setup Our first experiment explores the properties of amortized NPE-UDA methods in a classic contam- ination misspecification scenario. The inference task consists of inferring two parameters of the popular Ricker model of population dynamics (Wood] 2010 Ricker 1954). Using the same setting, previous work by Huang al. (2023) demonstrated the potential of non-amortized NPF-MMD, which specializes inference for single seen test data set.
Here, we evaluate both NPE-MMD and OPEN in an amortized setting on Nobe = 1 100 unseen observed (contaminated) test data sets. All methods are trained on a low budget of N = 5 000 uncontaminated training data sets. The NPE-UDA methods are trained with additional Nobs = 1 000 unlabeled data sets from the observed domain, non-overlapping with the validation and test set.
To achieve comprehensive evaluation of the trade-offs associated with the additional hyperparamoters of the NPE-UDA methods, we optimize for the NLL Nobs 1 1.000 contaminated validation data sets via Bayesian hyperparameter optimization (Akiba et al.] 2019) using 50 separate training runs per method. The fixed training run budget per method automatically accounts for the complexity of the hyperparameter space, since fewer hyperparamotors allow for a more thorough exploration of a method's hyperparameter space. We optimize the learning rate for all methods and the À hyperparameter controlling the alignment strength for NPE-MMD and NPE-DANN (see Table B.1 for the search ranges for all hyperparameters)
For NPE-DANN, we additionally optimize (i) the width and depth of the feodforward discriminator archi- tecture; (ii) the weight Agrt balancing the strength of the adversarial summary network updates relative to the discriminator network (with Agr! < diminishing and Agrt > 1 amplifying the reversed classification gradients passed to the summary network); and (iii) label smoothing, which has been found to be helpful for stabilizing the dynamics of domain- adversarial training (Zhang et al. 2023
We also explored scaling the gradient reversal weight Aget during training as suggested by Ganin et al. (0116), but found consistently worse results for the tested NPE setting. The same applies the kernel choice NPF.MDD, where the popular
8

---

Published in Transactions on Machine Learning Research (09/2025)
0.4
1.0
NRMSE
0.3
0.2
0.2
0.1
0.5
0.000
0.002
0.004
0.000
0.002
0.004
0.000
0.002
0.004
0.000
0.002
0. 0.004
Initial Learning Rate
Initial Learning Rate
Initial Learning Rate
Initial Learning Rate
0.4
1.0
NRMSE
0.3
0.2
CL 0.5
0.1
0.1
0.0
0.0
150
150
150
À
À
A
NPE NNPE
NPE-MMD
NPE-DANN
Figure 3: Experiment 1. Parameter space performance metrics resulting from 50 separate Bayesian hyper- parameter optimization runs per method. The solid trend lines represent the predictive mean of Gaussian process regression fitted to the individual run results, with the shaded areas representing 95% confidence intervals of the predictive distribution. If a parameter was not optimized, the method's average performance is depicted by a dashed horizontal line. Lower values indicate better performance for all metrics but PC, NLL = Negative Log Likelihood. NRMSE Normalized Root Mean Squared Error. ECE Expected Calibration Error. PC Posterior Contraction. Whereas learning rate oplimization is mostly ineffect ive for improving performance under contamination misspecification, the domain aligmment regularization parame- ter Ả controls a trade-off between error (NRMSE) vs. calibration (ECE) and contraction (PC) for NPE-UDA methods.
Results The parameter space performance metrics results are displayed in Figure 3 the data space per- formance metries results in Figure 4a and the network space metrics in Figure 4b) Overall, optimizing the learning rale does not improve approximation performance on contaminated data, with the only exception of NPE-MMD calibration improving at higher learning rates (see Figure 3). As expected, NNPE exhibits consistently robust performance as its data-generating process resembles the contamination misspecification scenario, whereas the success of the NPE-UDA methods depends on the domain aligument. regularization parameter X.
Concerning the parameter space performance metrics Figure 3). we observe a lower NLL for all robust methods compared to NPE. However, taking into account the other metrics unveils that robust methods tend to improve calibration, but not estimation error. For the NPE-UDA methods, increasing the domain alignment regularization parameter improves calibration at the cost of approximation error and posterior contraction. We also find NPE-MMD to be much more sensitive X, with inference breaking down very large values.
With regard to data space performance metrics Figure 4aj,, both approaches of obtaining the posterior predictive distance lead to similar results in this setting. While the PPD mostly mirrors the NRMSE in the parameter space, we find a surprising advantage of NNPE in the data space. Since we find a close correspondence between PPD and NRMSE in the other experiments, NNPE's unique advantage in the current setting could be caused by a peculiar sensitivity of the Ricker simulator to certain parameter constellations that NNPE is less likely to infer.
choice of using sums of Caussian kernels with different bandwidths (Maandet al. 2017 Schmitt al. 2024 destabilized training compared the sum of inverse multiquadratic kernels. Thus, we did include these hyperparameters the systematic hyperparameter assessment.
9

---

Published in Transactions on Machine Learning Research (09/2025)
PPD (obs)
75
PPD (resim)
75
1.0
SSDD (C2ST)
0.9
SSDD (MMD)
70
0.8
0.10
0.7
60-
0.6
0.000
0.002
0.004
0.000
0.002
0.004
0.000
0.002
0.004
0.000
0.002
0.004
0,000
0,002
0.004
Initial Learning Rate
Initial Learning Rate
Initial Learning
Initial Learning
Initial Learning
PPD (obs)
75
PPD (resim)
75
1.0
SSDD (C2ST)
SSDD (MMD)
0.15
0.8
0:30
65
0.5
0.7
60-
0.0
0.6
0.00
10
50
150
10
50
150
10
50
150
10
50
150
10
50
150
À
A
(a) Data space performance metrics.
(b) Network space metrics.
NPE
NNPE
NPE-MMD
NPE-DANN
Figure 4: Experiment 1. Further metrics resulting from 50 separate Bayesian hyperparameter optimization runs per method. The solid trend lines represent the predictive mean of a Gaussian process regression fitted to the individual run results, with the shaded areas representing 95% confidence intervals of the predictive distribution. If a parameter was not optimized, the method's average performance is depicted by dashed horizontal line. Lower values indicate better performance for all metrics but SSDD. PPD Postcrior Predictive Distance (RMSE). INLD Inference Network Latent Distance (MMD). SSDD Summary Space Domain Distance.
Considering network space metrics (Fiuure 4bj, we find that the deformation of the inference network's 's latent space, as measured by INLD, directly corresponds to density estimation quality as mcasured by NLL. This is unsurprising, given the change-of-variabl
mechanics of normalizing flows (Papamakarios et al. 2021) Further, observe overall lower summary space domain distance (SSDD) for both NPF-UDA methods in terms of C2ST but only for NPE-MMD in terms of MMD, and we do not observe the expected decrease in SSDD with increasing X. We suspect these patterns to be caused by two factors limiting SSDD variability: First, the overall domain distance being relatively small and second, the hyperparameter optimization search concentrating on higher values. Our next experiments inspect the lower range of X settings more closoly and reveal to be decisive for SSDD. Lastly, we find an overall little impact the additional NPE-DANN hyperparameters (see Figure BI).
5.2 Experiment 2 2D Gaussian Means: Simple and Controllable Benchmark
Setup Inspired by Schmitt et al. (2024), our next experiment tests the performance of NPE-UDA met against different types of misspecificatinn. Here, the simple approximation task of inferring the means of a 2-dimensional Gaussian model enables a comparison to an analytic posterior, which represents the optimal solution under Target 1. The well-specified setting uses a multivariate standard normal prior and an identity likelihood covariance matrix. We evaluate performance under increasing misspecification in two prior misspecification scenarios prior location oo and prior scale Eg 70 and two likelihood misspecification scenarios likelihood scale Σ rl2 and data contamination (see Table B.2). For the contamination misspecification, fraction e of the observations is replaced by negative and positive vectors of the constant = 1.5 to obtain atypical obscrvations without affecting overall location or scalc. Each simulated data set contains M = 100 exchangeable observations. All methods train on N = 48 000 well-specified data sets until convergence, with NPE-UDA methods additionally exposed to Nobs = 48 000 unlabeled data sets. All methods are evaluated on Nobs 000 observed data sets (unsoen by NPE-UDA methods).
Results Figure displays the results for all misspecification scenarios. We invert the meaning of the posterior contraction (PC) metric, such that lower means better for all metrics. Thus, the performance of
10

---

Published in Transactions on Machine Learning Research (09/2025)
Prior Location = 1-PC 0.15
Prior Scale
Likellhood Scale
Contamination Fraction =
To = 1 PC 0.03
T=1 1-PC 0.05
1-PC 0,02
ECE
NRMSE 0.84
ECE
NRMSE 0.13
ECE
NRMSE 0.27
ECE
NRMSE 0.08
PPD (resim) 5.72
SSDD (MMD)
PPD (resim)
S5DD (MMD)
(resim)
SSDD (MMD) 4.81
PPD (resim)
SSDD (MMD) 0.16
2.99
3.82
0.68
2.27
1.51
14o 2.5
To = 3
T= 10
E = 0.25
PC
1-PC
1-PC
1-PC
0.15
0.03
0.05
0.02
ECE
NRMSE 0.84
ECE 0.34
NRMSE 0.13
ECE 0.48
NRMSE 0.27
ECE
NRMSE 0.08
0.50
0.40
(resim)
S5DD (MMD)
PPD (resim)
S5DD (MMD)
(resim)
S5DD (MMD)
(resim)
S5DD (MMD)
2.99
0.68
2.27
4.81
0.16
Ho PC
To
I 1-PC
E 0.5
1CPC
0.15
0.03
0.05
0.02
ECE 0.50
NRMSE 0.84
ECE
NRMSE
ECE
NRMSE
ECE
NRMSE 0.08
0.34
0.13
0.48
0.27
0.40
PPD (resim)
SSDD
PPD (resim)
(MMD)
PPO (resim)
(MMD)
PPD (resim)
3.82
0.68
2.27
4.81
0.16
NPE-MMD 1.0)
APE-DANN 1.0)
Analytic
Figure 5: Experiment 2. Performance metrics and summary space domain distance (SSDD) of the methods in all misspecification scenarios (columns), aggregated via the median of 10 runs. The first shows the well-specified setting, with misspecification increasing from top to bottom within each column. Metric values are centered at 0 and normalized by each column's/ "aaumn's/scenari''s maximum value, which is displayed below the metric name at the border of each radar plot. Lower values indicate better performance for metrics but SSDD. 1- PC = 1- Postcrior Contraction. NRMSE = Normalized Root Mcan Squared Error. SSDD (MMD) Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) Posterior Predictive Distance measured via the RMSE resimulated data. ECE Expected Calibration Error. NPE-UDA methods fail under prior misspecification but can be advantagcous under contamination.
method can mostly be inferred from its area. All methods perform well in the well-specified case (first row), whereas we observe distinct but consistent patterns for the different methods under increasing mismatch.
In the prior misspecification scenarios, NPE and NNPE perform well compared the analytic posterior under location shift, but fall off under scale shift, with increasing error (NRMSE) and drastically increased miscalibration (ECE). The two NPE-UDA methods, on the other hand, perform poorly for both prior location and scale shift (see Section 3.4 for the complementary prior scale shift scenario of too broadly chosen priors).
In the likelihood scale misspecification scenario, the NPE methods are generally less sensitive to the misspec- ification than the analytic posterior. NPE-MMD successfully aligns the summary space between domains, but the practical effects of this alignment are limited slightly improved PC. NPE-DANN, on the other hand, shows the unstable training dynamics described in Section 3.3 Tt. fails to align the summary space
11

---

Published in Transactions on Machine Learning Research (09/2025)
for both likelihood scale misspecification levels (as well as the Ho = 5 prior location shift scenario), which translates to poor performance. Crucially, this drastic failure in the observed domain is not detectable in the simalated domain, where all methods, even NPE-DANN in the T = 20 scenario, perform well according to all metrics and only SSDD signals irregularities (see Figure B3
In the data contamination scenario, deviating from the true posterior via. Target enables NPE-MMD and NPE-DANN to excel, achieving much lower NRMSE and ECE than NPE, NNPE, and even the analytic posterior. Across all misspecification scenarios, NNPE performs similarly to NPE, with NNPE's noisier training process typically resulting in slightly lower contraction and calibration. This applies even to the contamination scenario, where we expected an advantage for NNPE due to its similarity to the corruption process.
With regard to differences between metrics, PPD reliably detects NPE-UDA failures under prior misspecifi cation, but is less sensitive under likelihood scale shifts and insensitive under contamination. We suspect that this lessened informativeness compared to Experiment is caused by the simple data structure of the Gaus- sian mean model. Further, the deformation of the inference network's latent space, measured via INLD, is again strongly associated with approximation quality, with rank correlations of .79 with NRMSE, = .95 with ECE, and r= .74 with PPD (ro-simulated).
Furthermore, in this low-dimensional example, we observed two sources of instability for NPE-UDA methods. First, despite the good performance across a wide range of À settings in Experiment 1, we observed a high sensitivity the regularization hyperparameter. For X = 0.1, the domain aligmment in the contamination scenario is too weak, eliminating the advantage of NPF-UDA methods (see F Figure B5p. Differently, setting x = 10 leads to drastic failures due to overly aggressive domain alignment of NPE-MMD and increases the training instabilities of NPE-DANN (see Figure B6). Second, these extreme failures necessitated aggregation of the training run results via the median instead of the mean, since single extreme results (not for À = 0.1, occasionally for x = 1, frequently for A = 10) rendered visualization of the results impossible.
5.3 Experiment 3 Bayesian Denoising: Righ-Dimensional Benchmark
Setup This experiment tests the generalization of our results on a high-dimensional (in the context of SBI) benchmark simulating a noisy camera model (Ramesh et al. 2022). The parameter vector 0 E 250 represents a crisp image, whereas the observation æ E 2226 is a blurred version of the original image generated by the noisy camera. The training data set consists of N = 50 000 images from the MNIST data set (Lecun et al. 1998), downscaled to 16 16 pixels for compatibility with the USPS data set (Hull 1994].
We test four different misspecification scenarios (see Table 3 for examples). In the prior misspecification scenario, we keep the settings of the noisy camera model constant but use images from the USPS data set (Hull 1994). While both data sets contain digits, the USPS data set features smaller margins, giving the priors different support. In the likelihood scale scenario, we increase the amount of blur. In the noise contamination scenario, we replace 10% of the pixels with calt-and-pepper noise (i.e., set them to black or white). In the row contamination scenario, we randomly set two rows (12.5% of the pixels) of each observation to black. The NPE-UDA methods are trained with Nots = 1 000 observed training data sets. We evaluate the performance on further Note = 000 observed test data sets (unseen by the NPE-UDA methods during training).
Results Table displays an overview of the metrics in all scenarios. Table 3 shows samples from the best run (lowest NRMSE) for each scenario and method. We observe worse approximations for all robust methods compared to NPE in the prior misspecification scenario, even though the summary space domain distance (SSDD) is strongly diminished for NPE-DANN and NPE-MMD, This is somewhat expected, performance improvements would also require an adaptation of the inference network, which cannot be induced by the methods lested here. NNPE is beneficial in the two contamination scenarios, whereas NPE-DANN and NPE-MMD improve performance in all three likelihood misspecification scenarios. The results highlight the differences between the robust methods: While NNPE mainly excels in the noise contamination scenario, where its misspecification model matches the domain shift, NPE-UDA methods effectively adapt to different likelihood shifts. Figure 6 shows a two-dimensional t-SNE, (Van der Maaten & Hinton 2008) representation
12

---

Published in Transactions on Machine Learning Research (09/2025)
Prior (MNIST USPS)
Likelihood Scale
Contamination (Noise)
Contamination (Rows)
Method
NRMSE 4 PPD
SSDD
NRMSE PPD 4 SSDD NRMSE & PPD SSDD
NRMSE PPD & SSDD
NPE
0.259
0.131
0.256
0.165
0.036
0.101
0.312
0.117
0.463
0.315
0.112
0.386
NNPE
0.274
0.137
0.206
0.173
0.041
0.088
0.154
0.035
0.019
0.214
0.064
0.071
NPE-DANN 0.01
0.329
0.193
0.027
0.130
0.031
0.027
0.217
0.065
0.017
0.180
0.056
0.016
NPE-DANN 0.10
0.326
0.193
0.029
0.134
0,031
0.016
0.211
0.057
0.015
0.178
0.045
0.014
NPE-DANN 1.00
0.352
0.205
0.038
0.147
0.032
0.014
0.240
0.068
0.015
0.201
0.051
0.014
NPE-MMD 0.01
0.303
0.184
0.029
0.145
0.027
0.022
0.268
0.085
0.016
0.262
0.085
0.016
NPF-MMD 0.10
0.312
0.189
0.018
0.189
0.059
0.009
0.245
0.071
0.013
0.181
0.047
0.012
NPE-MMD 1.00
0.374
0.225
0.004
0.322
0.129
0.000
0.321
0.129
0.000
0.322
0.129
0.000
Table 2: Experiment 3. Metrics of the methods in all misspecification scenarios, averaged across 3 runs.
NRMSE: Normalized Root, Mean Squared Error (lower is better). PPD = Posterior Predictive Distance (RMSE) to resimulated data (lower better). SSDD = Summary Space Domain Distance (MMD). Lower values indicate better summary space aligument, but too much alignment (i.e., vanishing SSDD) can lead to an uninformative summary space (e.g., NPE-MMD with 1.00).
Train
Prior (MNIST USPS)]
Likelihood Scale
Contamination (Noise)
Contamination (Rows)
3
0
7
Z
7
z
2
Parameters 0
Observations X
6
6
NPE
4
2
NNPE
NPE-DANN
9
4
2
L
NPE-MMD
Table 3: Experiment 3. Parameters, observations, and samples from the run with the lowest NRMSE for each scenario and method. Train shows sample from the parameters O of the training distribution and the corresponding observations Labs- The observations are identical for NPE, NPF-DANN, and NPE-MMD, whereas spike-and-slab noise added for NNPE. The similarity to the observations in the Contamination (Noise) scenario explains the good performance of NNPE in that scenario.
of the summary spaces produced by the different methods. The observed overlap corresponds well to the SSDD values reported in Table
wide range of À values. In contrast, NPE-MMD is prone to overregularizing the summary space, leading to complete loss of information in the summary space (sce also Figure 2). This is indicated by a huge drop in performance and vanishing SSDD. We found NPE-MMD highly sensitive to the chosen batch size, which we had to increase from 32 to 128 to achieve acceptable results. Thus, increasing the batch size and reducing À can counteract excessive regularization in higher-dimensional problems. Finally, the close correspondence between the NRMSE and PPD metrics confirms our hypothesis that the limited diagnostic power of PPD in the likelihood misspecification scenarios of Experiment was caused by the limited informativeness of data simulated from simple Gaussian model.
Overall, NPE-DANN achieves good performance
13

---

Published in Transactions on Machine Learning Research (09/2025)
X1
&1
81
&1
Xabs
Xoba
Xoos
Xass
(a) NPE
(b) NNPE
(c) NPE-DANN
(d) NPE-MMD
Figure 6: Experiment 3. Contamination (Rows): t-SNE representation the summary spaces from the best run (lowest NRMSE) of each method. The t-SNE map calculated jointly using summary statistics of data from both the simulated and the observed domain. For NPE, the two domains are clearly separated. With increasing alignment in the summary space, the overlap between the domains increases: The domain embeddings overlap partially for NNPE (medium SSDD in Table 2) and fully for NPE-DANN and NPE- MMD (low SSDD in Table 2). Since t-SNE can introduce artificial clustering, the overlap and not the specific shape is relevant.
5.4 Experiment 4 Decision Making: Large Human Behavior Data Set
Setup Finally, we evaluate the effectiveness of NPE-UDA methods using real-world example from cog- nitive science, based on large-scale empirical data set of binary decision-making task. The data set comprises response time data from millions of participants who completed the Implicit Association Test (IAT), widely used test in social psychology /Greenwald et al. 1998 2003 The data are publicly available through Project Implicit [Xu et al. 2014).
To gain insights into the underlying cognitive processes, response time data are often analyzed using ev- idence accumulation models (Evans & Wagenmakers, 2020 Ratcliff et al.) (2066). However, fitting these models within Bayesian framework is computationally intensive, making them prime further challenge with online response time data is the presence of substantial noise, stemming from limited experimental control (Gong & Huskey) 2023).. For example, participants may guess, experience attentiona lapses, or behave inconsistently. This issue can be framed as a form of likelihood misspecification, where the obscrved data are partially generated by processos not captured by the model.
for ABI. A
Traditional approaches to address this include data cleaning procedures or extending the model with ad- ditional parameters (e.g., to capture guessing behavior, see Ratcliff & Tuerlinckx 2002). Here, we explore whether NPE-UDA methods model data with those of the noisy empirical data. This alignment should allow for improved parameter estimation despite the noise. Thus, we aim for Target 2: approximating the posterior distributions of cognitive model parameters while implicitly filtering out the noise components in the empirical data. All methods are trained on = 32 000 simulated response time data sets to approximate the parameters of an evidence accumulation model adapted to IAT task data. The NPE-UDA approaches additionally train on Nobs = 32 000 unprocessod empirical data sets.
offer alternative by aligning the summary statistics of simulated (clean)
For evaluation, construct several empirical test data sets using held-out observed data sets along two dimensions (see Appendix B.6 for details): (i) data either unprocessed cleaned using gold standard pre-processing procedures from the cognitive modeling literature, and (ii) data are categorized as cither well- specified where the cognilive model is expected to be valid - or misspecified, representing cases likely affected by processes not captured by the model (i.e., likelihood misspecification) based on their atypicality in NPE summary space. The misspecified test set consists of data from Nobr 730 participants, while the well- specified set consists of Nobs 10000 randomly selected participants for whose data the probabilistic model classified well-specified.
14

---

Published in Transactions on Machine Learning Research (09/2025)
Misspecified
Well specified
Misspocified
Well specified
0.250
0.550
0.300
0.275
0.070
0.500
0.200
550D
0.250
0.450
0.065
0.150
0.225
0.200
0.060
0.350
0.100
0.175
0,300
0.600
Age Cor Baundary Sep.
0.320
0.400
0.400
0.315
0.550
0.395
0.350
0.310
RMSE
0.500
0.385
0,250
0.450
0.200
0.380
Age Car Non Diersiion Time
0.375
0.400
0.230
0.048
0.350
Accuracy
0.220
0.325
0.390
0.210
0.047
0.300
0.380
0.200
0.048
0.190
0.370
0.180
0.045
0.225
0.01
0.10
1.00
0.01
0.10
1.00
0.01
0.10
1.00
0.01
0.10
1.00
A
A
NPE
NPE-MMD
NPE
NNPE
NPE-DANN
MMD
(a) Summary space domain distances (SSDDs; MMD) and external validity metrics for empirical data. The empirical
(b) Posterior predictive distances (PPDs; RMSE) to cleaned empirical respouse time (RT) data. The first row shows the PPD for response times (in seconds) correct trials, averaged across posterior samples, partici- pants, response time quantiles, and experimental condi- tions. The second row shows the PPDs for error response times, while the third row shows the PPDs for accuracy rates (in %).
first shows the SSDDs of simulated data. Rows two and three show (across-person) age cor- relations for posterior medians in two cognitive model parameters: boundary separation (in the congruent ex- perimental condition) and non-decision times (for cor- recl trials).
Figure 7: Experiment 4. Metric results for all methods and different À weights for the NPE-UDA methods. All runs display the averaged results across three runs per method, with across-run standard deviations shown as shaded areas (for NPE and NNPE) or error bars (for NPE-DANN and NPE-MMD). Please note that y-axis scales differ between the misspecified and the well-specified settings. X UDA regularization weight, The left column shows results for misspecified data sets, while the right column shows results for the (much larger) well-specifted data sct. While NNPE improves performance misspecified data sets, the adaptation of NPE-UDA methods to the general observed domain does not cover the minority of misspecified data sets.
To evaluate the networks' performance, we first compare the methods by assessing the summary space domain distances between simulated and empirical data scts. Next, because certain cognitive paramcters are known to covary with participant age (von Krause al. 2020 Ratcliff et al., 2010 Theisen et al. 2021), then examine how well each method captures this relationship by comparing age correlations for two key parameters across estimation approaches. Finally, we use the PPD to assess the methods based on their ability to predict unseen empirical data. Similar to before, we use a denoised version of the data as PPD reference, which we approximate for this real-world application via the intensive pre-processing procedures.
We train the NPE-UDA methods with weights of 0.01, 0.1, 1, and 10. As before, À 10 frequently caused training instabilities (particularly for the MMD approach) leading extreme results. To retain readable visualizations, leave out the À 10 failure setting in the following.
15

---

Published in Transactions on Machine Learning Research (09/2025)
Results Figure presents presents the main results from our decision modeling experiment. Concerning Figure 7a the NPE-UDA methods show the expected reduction in SSDD in the well-specified data sets, with domain alignment becoming more pronounced as the UDA weight parameter À increases. However, this domain adaptation effect does not extend to the minority of misspocified test sets, where NNPE shows the smallest distance between domains in summary space. With respect to age correlations for the two cognitive pa- rameters, NNPE leads to on average slightly higher correlations on misspecified data compared to the other approaches.
Figure 7b presents posterior retrodictive RMSE values between posterior resimulations and the cleaned test data; corresponding results for the uncleaned data are provided in the appendix (Figure B14). Consistent with the SSDD and correlation results, we do not observe clear advantages for any method on the well- specified data sets but an advantage of NNPE on the misspecified data sets for response time fits. All methods perform comparably concerning the accuracy fits. The NPE-UDA approaches perform similarly to NPE (cf. Figure 7b). with higher À values leading to increased instability as indicated by RMSE variability across runs for misspecified data. Overall, while additive training noise via the NNPE method proved beneficial for misspecified data sets, the general domain adaptation induced by the NPE-UDA methods did not carry over from the majority of well-specified data sets to the minority of misspecified data sets.
6 Discussion
NPE-UDA Methods In this paper, we argued that introducing UDA to NPE methods shifts the inference target from the standard analytic posterior p(0) zobs) to a "denoised posterior" p(@ Tobs) based on adjusted data Tos- This shift implies potential robustness gains under likelihood misspecification, where implicitly ignoring unmodeled phenomena in the observed data can be desirable. However, also roduces the amount of information available to counteract prior misspecification. We consistently found these patterns Chroughout our systematic evaluations for both the existing NPE-MMD and our new NPE-DANN method. Consequently, prior predictive or pushforward checks (Bürkner et al. 2025 Box 1980) become an even more essential part of the Bayesian workflow (Gelman et al. 2020) for NPE-UD UDA methods.
Our results suggest that while posterior accuracy is related to domain distance in the summary space, the relationship is not straightforward, pointing to more subtle effects of domain alignment. Compared to simpler methods, such as NNPE (Ward et al.) 2022), the flexibility of NPE-UDA methods to automatically adapt to various types of likelihood misspecification comes at the cost of reduced transparency during inference. Thus, closer examination of UDA mechanisms is necessary to determine the nature of domain adaptation and how exactly these adaptations affect the interpretation of the resulting posterior p(0 @obe). Employing interpretability methods or decoders to track summary space adaptations back to the data space seems a particularly promising avenue for future rescarch.
Role of the Regularization Weight NPE-UDA methods, and especially NPE-DANN, introduce a num- ber of additional hyperparameters that could potentially influence inferential results. While we found limited effects of most hyperparameters in our hyperparameter exploration experiment, we confirmed the existence of an application-speeiff optimal amount of UDA regularization (Zellinger el al. 2021) in the NPE con- text across all experiments. Both NPE-UDA methods exhibited substantial instabilities for higher À values: We observed the typical unstable training dynamics associated with adversarial training for NPE-DANN, whereas NPE-MMD exhibited overly aggressive domain alignment, suppressing all information contained in the data. Notably, which À values qualify as high varied substantially between the experiments: while a broad range of À values was tolerable in Experiment 1, we found severe inference breakdowns already for x = 10 in all other experiments.
Mcasuring Robustness in the Real World The critical influence of the À hyperparameter directly lcads to the next question: How can we measure robustness gains in the real world, where ground-Iruth parameter values are unavailable, and find an optimal value for X? Posterior predictive measures that compare posterior resimulations the observed data are usually the tool of choice. For measuring the success of robust mothods, however, their standard application flawed, since using the observed data directly as reference implics that successfully ignoring noise is punished by increased posterior predictive distance to noisy outliers. Our
16

---

Published in Transactions on Machine Learning Research (09/2025)
evaluation metrics sought to account for this by constructing "denoised" reference data set. Although this approach is useful for benchmarking robust methods on real-world data, creating application-specinn reference data solely for hyperparameter tuning would negate the benefit of automatic domain adaptation. Alternative approaches to measuring the robustness of NPE in the real world are also challenging to realize for NPE-UDA methods: The summary space is actively reshaped by NPE-UDA methods and therefore loses its validity as proxy for detecting model misspecification (Schmitt et al. 2024). In contrast, the predictive variability of a deep ensemble (Lakshminarayanan et al.) 2017] on the observed data, which directly measures the predictive fragility resulting from model misspecification in NPE (Elsemüller et al,, 2024), remains a meaningful diagnostic for NPF-UDA. However, using deep ensemble variability to guide lyperparameter search is problematic: Ensemble variability can straighttorwardly be minimized by collapsing the posterior to the prior, thus rewarding excessive regularization during hyperpararntte search. While such collapses could be partially addressed by jointly optimizing hyperparameter search for posterior contraction, doing so introduces the additional challenge of balancing these two scarch objectives in label-free real-world settings. Furthermore, embedding deep ensembles into hyperparameter searches multiplies the already large computational cost of the latter and consequently remains impractical. Thus, finding generally reliable robustness measures for real-world settings remains an unsolved issue, embedded into the overarching open UDA problem of guiding hyperparameter optimization despite missing target labels (Zelinger et al. 2021 Musgrave et al. 2022
Future Avenues Based on the results of our experiments, we believe two further pathways to be especially interesting for future research. First, we focused on UDA approaches that target the summary space to retain the modularity of summary/inferonn
network NPE architectures. This directly enables extensions to different downstream tasks, such as amortized Bayesian model comparison (Radev et al. 2021 Elsemüller et al.) 2023 Jeffrey & Wandelt 2024). The latter reframes model comparison as a classification task and is thus situated in the most extensively studied UDA task setting. Second, in contrast to the non-amortized NPE-MMD approach by Huang et al. (2023), which specializes inference for a single observed data set, we evaluated amortized approximation performance on data sets unseen by the NPF-UDA methods. While we observed good performance with low amounts of observed training data, would be valuable to systematically assess the minimum amount of data necessary for effective adaptation to the observed domain. Here, generalization gaps of NPE-UDA methods could be measured via the difference in performance on dala sets seen and unseen during training.
Conclusion Our results reveal a rather complex story of NPE-UDA shaped by the interplay of factors such as problem dimensionality, misspecification type, and UDA model choice. They also emphasize the need for pairing stylized theoretical investigations with thorough empirical evaluation. In sum, UDA offers a straightforward way to incorporate real data into simulation-basse various types of likelihood misspecification. However, our systematic evaluation uncovered major obstacles and unanswered questions that hinder the direct application of NPE-UDA methods in critical settings.
training and shows promise in handling
Acknowledgments
The authors are grateful for the constructive feedback received during reviews and live discussions of an early version of this work at the 'Frontiers in Probabilistic Inference: Learning meets Sampling' workshop at ICLR 2025. L.F. and V.P. are supported by the state of Waade---tttteberrggg through bwHPC. Additionally, L.E. is supported by a grant from the Deutsche Porschungsgemeensschatt (DFG, German Research Foundation; GRK 2277: project number 310365261) to the research training group Statistical Modeling in Psychology (SMiP).
V.P. is supported by the Konrad Zuse School of Excellence in Learning and Intelligent Systems (ELIZA) through the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research, and the German Research Foundation (DFG) through grant INST 35/1597- FUGG. M.V.K. acknowledges funding by the German Research Foundation (DFG) through grant KR 6065/1-1. P.B. acknowledges support of DFG Project 528702768 and DFG Collaborative Rescarch Center 391 (Spatio-Temporal Statistics for the Transition of Energy and Transport) 520388526. S.T.R. is supported by NSF under Grant No. 2448380.
17

---

Published in Transactions on Machine Learning Research (09/2025)
Author Contributions
L.E. and S.T.R. conceived the initial idea, L.E., P.C.B., and S.T.R. developed the methodology, and A.V. and S.T.R. supervised the project. L.B., V.P., and M.V.K. implemented the experiments and wrote the experiment sections. L.E., P.C.B., and S.T.R. wrote the rest of the initial manuscript draft, which all authors reviewed and refined.
References
Takuya Akiba, Shotaro Sano, Toshihiko Yanase, Takeru Ohta, and Masanori Koyama. Optuna: A next- generation hyperparameter optimization framework. In Proceedings of the 25th ACM SIGKDD Interna- tional Conference on Knowledge Discovery and Data Mining, 2019.
Lynton Ardizzone, Jakob Kruse, Sebastian Wirkert, Daniel Rahner, Eric W Pellegrini, Ralf S Klessen, Lena Maier-Hein, Carsten Rother, and Ullrich Köthe. Analyzing inverse problems with invertible neural networks. ar preprint arXic:1808, 04730, 2018.
Lynton Ardizzone, Jakob Kruse, Carsten Lüth, Niels Bracher, Carsten Rother, and Ullrich Köthe. Con- ditional invertible neural networks for diverse image-to-image translation. In Pattern Recognition: 42nd DAGM German Conference, DAGM GCPR 2020, Tübingen, Germany, September 28 Oclober 1, 2020, Proceedings 42, pp. 373 387. Springer, 2021.
Grace Avecilla, Julie N Chuong, Fangfei Li, Gavin Sherlock, David Gresham, and Yoav Ram. Neural net- works enable efficient and accurate Aimulation-baseed inference of evolutionary parameters from adaptation dynamics. PLoS Biology, 20(5))c3,,1633, 2022.
Shai Ben-David, John Blitzer, Koby Crammer, and Fernando Pereira. Analysis of representations for domain adaptation. Advances in neural information processing systems, 19, 2006.
Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine learning, 79:151 175, 2010.
Yves Bernaerts, Michael Deistler, Pedro J Gonçalves, Jonas Beck, Marcel Stimberg, Federico Scala, Andreas S Tolias, Jakob Macke, Dmitry Kobak, and Philipp Berens. Combined mechanisiiieca modeling links ion channel genes to physiology of cortical neuron types. bioRxie, pp. 2023 03, 2023.
Michael Betancourt. Calibrating model-based inferences and decisions. arXie preprinl or X 1111188008999 2018.
Scbastian Bischoff, Alana Darcher, Michael Deistlor, Richard Gao, Franziska Gerken, Manuel Gloeckler, Lisa Haxel, Jaivardhan Kapoor, Janne K Lappalainen, Jakob Macke, et al. A practical guide to statistical distances for evaluating generative models in science. arXiv preprint arXiv:2408. 12636, 2024.
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):1103- 1130, 2016.
George EP Box. Sampling and Bayes' inference in scientific modelling and robustness. Journal of the Royal Statistical Society Series A: Statistics in Society, (43(())383-41))) 404, 1980.
Paul-Christian Bürkner, Maximilian Scholz, and Stefan T Radev. Some models useful, but how do we know which ones? Towards unified Bayesian model taxonomy. Statistic Surveys, 17:216 310, 2023.
Paul-Christian Bürkner, Marvin Schmitt, and Stefan T Radev. Simulations in statistical workflows. arv preprint AXft:2503.24011 2025.
Patrick Cannon, Daniel Ward, and Sebastian M Schmon. Investigating the impact of model misspecification in neural simulation-based inference. arXiv preprint orXin:2209.01845, 2022.
18

---

Published in Transactions on Machine Learning Research (09/2025)
Bob Carpenter, Andrew Gelman, Matthew D Hoffman, Daniel Lee, Ben Goodrich, Michael Betancourt, Marcus Brubaker, Jiqiang Guo, Peter Li, and Allen Riddell. Stan: A probabilistic programming language. Journal of statistical software, 76(1), 2017.
Jeffrey Chan, Valerio Perrone, Jeffrey Spence, Paul Jenkins, Sara Mathieson, and Yun Song. A likelihood- free inference framework for population genetic data using exchangeable neural networks. Advances in neural informalion processing systems, 31, 2018.
Yanzhi Chen, Dinghuai Zhang, Michael U. Gutmann, Aaron Courville, and Zhanxing Zhu. Neural approx- imate sufficient statistics for implicit models. In International Conference on Learning Represenlations, 2021. URL htttps://opperreeveer wttppperrreeeeeeeeetorrrrrrn wet.:/ooroun.ddSnoosstttt
A Ciprijanovié, Diana Kafkes, S Jenkins, K Downey, Gabriel N Perdue, Sandeep Madireddy, T Johnston, and Brian Nord. Domain adaptation tochniques for improved cross-domain study of galaxy mergers. arXiv preprinl ar (tf::2011.03590) 2020.
Corinna Cortes and Mehryar Mohri. Domain adaptation and sample bias correction theory and algorithm for regression. Theoretical Computer Science, 519:108- 126,, Jamuary 2014. ISSN 0304-3975. doi: 10.1016/ (rces.0013.9.c0m
Thomas M Cover and Joy A Thomas. Elements of information theory. John Wiley & Sons, 2012.
Kyle Cranmer, Johann Brehmer, and Gilles Louppe. The frontier of simulation baase inference. Proceedings of the National Academy of Sciences, 117(48) 2020.
Lars Dingeldein, Pilar Cossio, and Roberto Covino. Simulation-based inference of single-molecule experi- ments, 2024. URL
Lasse Elsemüller, Martin Schnuerch, Paul-Christian Bürkner, and Stefan T Radev. A deep learning method for comparing Bayesian hicrarchical models. arXiv preprint arXim:2301.11873, 2023.
Lasse Elsemüller, Hans Olischläger, Marvin Schmitt, Radev. Sensitivity-aware amortized Bayesian inference. Transactions Machine Learning Research (TMLR), 2024.
Bürkner, Ullrich Köthe, and Stefan T
Nathan J Evans and Eric-Jan Wagenmakers. Evidence accurulaatin models: current limitations and future directions. Quantitative Methods for Psychology, 16(2):73 90, 2020.
David T. Frazier, Ryan Kelly, Christopher Drovandi, and David J. Warne. The statistical accuracy of neural postorior and likelihood ostimation, 2024. URL
12068
Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario March, and Victor Lempitsky. Domain-adversarial training of neural networks. Journal of machine learning research, 17(3)):1-35, 2016.
Richard Gao, Michael Deistler, and Jakob H Macke. Generalized Bayesian inference for scientific simulators via amortized cost estimation. Advances in Newral Information Processing Systems, 36:80191-80219 2023.
Andrew Gelman, Aki Vehtari, Daniel Simpson, Charles C Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, and Martin Modrák. Bayesian workflow. arXiv preprint arXiv:2011.01808, 2020.
Samuel Gershman and Noah Goodman. Amortized inference in probabilistic reasoning. In Proceedings of the annual meeting of the cognitive science society, volume 36, 2014.
Manuel Glöckler, Michael Deistler, and Jakob H. Macke. Variational methods for simulation-based inference. arXiv preprint arXiv:220s.00 0467 2022.
Manuel Glöckler, Michael Deistler, and Jakob H Macke. Adversarial robustness of amortized Bayesian inference. Xiv preprint arXi::2305.11 144942 2023.
19

---

Published in Transactions on Machine Learning Research (09/2025)
Manuel Gloeckler, Michael Deistler, Christian Weilbach, Frank Wood, and Jakob H Macke. All-in-one simulation-based inference. aix preprint arXi::2444.09636, 2024.
Xuanjun Gong and Richard Huskey. Moving behavioral experimentation online: A tutorial and some rocommendations for drift diffusion modeling. American Behavioral Scientist, pp. 000276422331200773, 2023. ISSN 0002-7642. doi: 10.1177 01177/000266222310700000 000276422312070733 Publisher: SAGE Publications Inc.
URL https://doi. fttps://doi.orr/10..11 org/10.117/
Anthony G. Greenwald, Debbie E. McGhoe, and Jordan L. K. Schwartz. Mcasuring individual differences in implicit cognition: The implicit association lest. Journal of Personality and Social Psychology, 74(6): 1464-1480) 1998. ISSN 1939- 1315, 0022-3514. doi: 10.1037 001700002-331.744606
URL http:://oi. apa.
00000030002200000 3514. 6. 7464.1464
Anthony G. Greenwald, Brian A. Nosek, and Mahzarin R. Banaji. Understanding and using the implicit association test: I. an improved scoring algorithm. Journal of Personality and Social Psychology, 85( 197-216, 2003. ISSN 1939-1315, 0022-3514. doi: 10. 10.1037700022-3514.852222 org/ ctt.iddii...mmmii11110..1 1000000.00000022235111111 00..54445...1..... 2.17.997
URL http:///ol: apa.
A Gretton, K. Borgwardt, Malte Rasch, Bernhard Schölkopf, and AJ Smola. A kernel two-sample test. The Journal of Machine Learning Research, 13:723-772 03 2012.
Joeri Hermans, Volodimir Begy, and Gilles Louppe. Likelihood frre meme with amortized approximate ratio ostimators. In International conference on machine learning, pp. 4239-4248. PMLR, 2020.
Daolang Huang, Ayush Bharti, Amauri Souza, Luigi Acerbi, and Samuel Kaski. Learning robust statistics for simulation based inference under model misspecification. Advances in Neural Information Processing Systems, 36, 2023.
Peter J Huber. Robust statistics. John Wiley & Sons, 1981.
J.J. Hull. database for handwritten text recognition research. IEEE Transactions on Pallern Analysis and Machine Intelligence, 16(5):550-58) 554, 1994. doi: 10.1109/34.291440.
Niall Jeffrey and Benjamin D Wandelt. Evidence networks: simple losses for fast, amortized, neural Bayesian model comparison. Machine Learning: Science and Technology, 5(1):015008, 2024.
Fredrik Johansson, David Sontag, and Rajesh Ranganath. Support and invertibility in domain-invarien invariant representations. In The 22nd International Conference on Artificial Intelligence and Statistics, pp. 527- 536. PMLR, 2019.
Ryan P Kelly, David J Nott, David T Frazier, David J Warne, and Chris Drovandi. Misspecffcattion-rouust sequential neural likelihood. arXiv preprint ar Xin:23301 13366, 2023.
Ryan P Kelly, David J Warne, David T Frazier, David J Nott, Michael U Gutmann, and Christo- pher Drovandi. Simulation-basck Bayesian inference under model misspeciffcation. arXio preprint arxiv:2003. 12315, 2025.
Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1xl convolutions. Advances in neural information processing systems, 31, 2018.
Karl Christoph Klaner, Andreas Voss, Florian Schmitz, and Sarah Teige-Mocigembaee Process components the implicit association test: a diffusion-monn analysis. Journal of Personality and Social Psychology, 93(3):353, 2007.
Xianghao Kong, Wentao Jiang, Jinrang Jia, Yifeng Shi, Runsheng Xu, and Si Liu. Dusa: Decoupled unsupervised sim2real adaptation for vehiclettoeeerrything collaborative perception. In Proceedings of the 31st ACM International Conference Mullimedia, pp. 1943 1954, 2023.
20

---

Published in Transactions on Machine Learning Research (09/2025)
Peter D Kvam, Louis H Irving, Konstantina Sokratous, and Colin Tucker Smith. Improving the reliability and validity of the iat with a dynamic model driven by similarity. Behavior Research Methods, 56(3): 2158 2193, 2024.
Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncer- tainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017.
Tuan Anh Le, Atilim Gunes Baydin, and Frank Wood. Inference compilation and universal probabilistic programming. In Artificial Intelligence and Statistics, pp. 1338-1348. PMLR, 2017.
Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-bass learning applied to document recognition. Proceedings of the IEEE, 86(11):2278 2324, 1998. doi: 10.1109/5.72677
Yaron Lipman, Ricky T. Q. Chen, Heli Ben- Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. In The Eleventh International Conference on Learning Regresentations, 2023. URL https:://penreevie net/
net/forum? ftrum?ormmm/m Pa-rrunndddPPWW
Xiaofeng Liu, Chaehwa Yoo, Fangxu Xing, Hyejin Oh, Georges El Fakhri, Je- Lon Kang, Jonghye Woo, ct al. Deep unsupervised domain adaptation: A review of recent advances and perspoctives. APSIPA Transactions on Signal and Information Processing, 11(1), 2022.
Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan. Learning transferable features with deep adaptation networks. In International conference on machine learning, pp. 97-105. PMLR, 2015.
Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Domain adaptation: Learning bounds and algorithms. Xiv preprint orXiv:0902.343) 2009.
Krikamol Muandet, Kenji Fukumizu, Bharath Sriperumbudur, Bernhard Schölkopf, et al. Kernel mean embedding of distributions: A review and beyond. Foundations and Trends0 in Machine Learning, 10 (1-2):1-141, 2017.
Kevin Musgrave, Serge Belongie, and Ser-Nam Lim. Unsupervised domain adaptation: A reality check. arXiv preprint arXic:2111.15673 2021.
Kevin Musgrave, Serge Belongic, and Ser-Nam Lim. Three new validators and a large-scale benchmark ranking for unsupervised domain adaptation. arXiv preprinl arxiv:200. 07360, 2022.
Costantino Pacilio, Swetha Bhagwat, and Roberto Cotesta. Simulation- inference of black hole ring- downs in the time domain. Physical Review D, 110(8):083010, 2024.
Sinno Jialin Pan, Ivor W Tsang, James T Kwok, and Qiang Yang. Domain adaptation via transfer component analysis. IEEE transactions neural networks, 22(2):199 210, 2010.
George Papamakarios and lain Murray. Fast s-free inference simulation models with Bayesian conditional density estimation. Advances in neural information processing systems, 29, 2016.
George Papamakarios, David Sterratt, and lain Murray. Sequential neural likelihood: Fast likelihood-free inference with autoregressive flows. In The 22nd international conference on artificial intelligence and statistics, pp. 837-848. PMLR, 2019.
George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshmi- narayanan, Normalizing flows for probabilistic modeling and inference. The Journal of Machine Learning Research, 22(1):20 (11):2617-22800 2021.
Christian S Perone, Pedro Ballester, Rodrigo C Barros, and Julien Cohen-Adad. Unsupervised domain adaptation for medical imaging segmentation with self-ensembling. Neurolmage, 194:1 11, 2019.
Stefan T Radev, Ulf K Mertens, Andreas Voss, Lynton Ardizzone, and Ullrich Köthe. Bayesflow: Learning complex stochastic models with invertible neural networks. IEEE transactions on neural networks and learning systems, 2020.
21

---

Published in Transactions on Machine Learning Research (09/2025)
Stefan T Radev, Marco D'Alessandro, Ulf K Mertens, Andreas Voss, Ullrich Köthe, and Bürkner. Amortized Bayesian model comparison with evidential deep learning. IEEE Transactions on Newral Networks and Learning Systems, 2021.
Stefan T Radev, Marvin Schmitt, Lukas Schumacher, Lasse Flsemüller, Valentin Pratz, Yannik Schälte, Ull- rich Köthe, and Paul-Christian Bürkner. Bayesflow: Amortized Bayesian workflows with neural networks. arviv preprinl ar Xic:23006 16015, 2023.
Tom Rainforth, Adam Foster, Desi R Ivanova, and Freddic Bickford Smith. Modern Bayesian experimental design. Stalistical Science, 11:000 114, 2024.
Poornima Ramesh, Jan-Matthis Lueckmann, Jan Boelts, Álvaro Tejero-Cantero, David S. Greenberg, Pe- dro J. Goncalves, and Jakob H. Macke. GATSBI: Generative adversarial training for simulation--ase inference. In International Conference on Learning Representations, 2022. URL https::/openrevvie.. het/fforunn1d=xRR1nc655-33
Roger Rateliff and Francis Tuerlinckx. Estimating parameters of the diffusion model: Approaches to dealing with contaminant reaction times and parameter variability. Psychonomic Bulletin & Review, 9(3):438-45 2002. ISSN 1531-5320.
doi: 10.3758/BBB /BF0319602. URL hataps://8o..org/// http:///..0.0./000.007/////11130
Roger Ratcliff, Anjali Thapar, and Gail McKoon. Individual differences, aging, and IQ in two-choice tasks. Cognitive psychology, 60(3):127 157, 2010. ISSN 1095-5623.
doi:
200..0.001.
Roger Ratcliff, Philip L. Smith, Scott D. Brown, and Gail McKoon. Diffusion decision model: Current issues and history. Trends in cognâtive sciences, 20(4):260-281 2016. doi: http:///.tics.20..0...00
levgen Redko, Emilie Morvant, Amaury Habrard, Mare Sebban, and Younès Bennani. A survey on domain adaptation theory: Learning bounds and theoretical guarantees, July 2022.
William Edwin Ricker. Stock and recruitment. Journal of the Fisheries Board of Canada, 11(5):559-623, 1954.
Daniel Ritchie, Paul Horsfall, and Noah D Goodman. Deep amortized inference for probabilistic programs. arXiv preprint Xre: 16xxi:1610.. 1111661..5550 2016.
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Medical image computing and computer-assistedr interventions -MCCAAL 2015: 18th inter- national conference, Munich, Germany, October 5-9, 2015, proceedings, part III 18, pp. 234-241. Springer, 2015.
Neil Savage. Synthetic data could be better than real data. Nature, 2023.
Marvin Schmitt, Paul-Christian Bürkner, and Ulirich Köthe. Detecting model misspecification in amortized Bayesian inference with neural networks. Proceedings of the German Conference on Pattern Recognition (GCPR), 2024.
Ozan Sener, Hyun Oh Song, Ashutosh Saxena, and Silvio Savarese. Learning transferrable representations for unsupervised domain adaptation. Advances in neural information processing systems, 29, 2016.
Ali Siahkoohi, Gabrio Rizzuti, Rafael Orozco, and Felix J Herrmann. Reliable amortized variational inference with physies-based latent distribution correction. Geophysics, 88(3):R297 R322, 2023.
Yu Sun, Eric Tzeng, Trevor Darroll, and Alexei A Efros. Unsupervised domain adaptation through self-
supervision. arXie preprint arXiv: 11825, 2019.
Paxson Swierc, Megan Zhao, Aleksandra Čiprijanović, and Brian Nord. Domain adaptation for measurements of strong gravitational lenses. arXiv preprint arXiv:2311.12288, 2023.
Paxson Swierc, Marcos Hamargo-Arizment Aleksandra Ćiprijanović, and Brian D Nord. Domain-adaptive neural posterior estimation for strong gravitational lens analysis. arXiv preprint arXiv:24 16347, 2024.
22

---

Published in Transactions on Machine Learning Research (09/2025)
Maximilian Theisen, Veronika Lerche, Mischa von Krause, and Andreas Voss. Age differences in diffusion model parameters: A meta-analysis. Psychological Research, 55:2012-2024 2021. Publisher: Springer.
Eric Tzeng, Judy Hoffman, Ning Zhang, Kate Saenko, and Trevor Darrell. Deep domain confusion: Maxi- mizing for domain invariance. arxiv preprint arXiv:1422.3474 2014.
Rheeya Uppaal, Yixuan Li, and Junjie Hu. How useful is continued pre-training for generative unsupervisod domain adaptation? In Proceedings of the 9th Workshop on Representation Learning for NLP (RepL4NLP. 2024). pp. 99-117, 2024.
Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008.
Stéfan van der Walt, Johannes L. Schönberger, Juan Nunez-Iglesias, François Boulogne, Joshua D. Warner, Neil Yager, Emmanuelle Gouillart, Tony Yu, and the scikit-image contributors. scikit-image: image processing in Python. PeerJ, 2:0453, 6 2014. ISSN 2167-8359. doi: 10.7717/pcorj.45. URL https:
//doi. wtt..or///..77//pee....... 711.777er7/...4.. .453
Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Ev- geni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew J. Nelson, Eric Jones, Robert Kern, Eric Larson, C Carey, Îlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul Mulbregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17:261-272, 2020. doi: (..1038/$415922 --00000000222-0
Mischa von Krause and Stefan Radev. A big data analysis of the associations between cognitive parameters and socioeconomic outcomes. OSF preprint http:///ootom///133111/////////
ort.103311110////////// /o... /88,, 2025.
Mischa von Krause, Veronika Lerche, Anna-Lena Schubert, and Andreas Voss. Do non-decision times mediate the association between age and intelligence across different content and process domains? Journal of Intelligence, 8(3):33, 2020. ISSN 2079-3200. doi: (0.3330/jucclligaccc00333333 URL httpss:/wwww..... mdpi. com/2079-3200/883/32
Mischa von Krause, Stefan T. Radev, and Andreas Voss. Mental speed is high until age 60 as revealed by analysis of over a million participants. Nature human behaviour, 6(5):700-700 2022. Publisher: Nature Publishing Group UK London.
Xiaoyu Wang, Ryan P. Kelly, David J Warne, and Christopher Drovandi. Preconditioned noural postcrior estimation for likelihood-free inference. Transactions on Machine Learning Research, 2024. ISSN 2835- 8856. URL htttps://operrevieee http:///oopgaeeeeeeeeee/e/rraaaa het:://foorrmmm//sgg//AAAA
Daniel Ward, Patrick Cannon, Mark Beaumont, Matteo Fasiolo, and Sebastian Schmon. Robust neural posterior estimation and statistical model criticism. Advances in Neural Information Processing Systems, 35:33845-33533 2022.
Antoine Wehenkel, Juan I, Gamella, Ozan Sener, Jens Behrmann, Guillermo Sapiro, Marco Cuturi, and Jörn-Henrik Jacobsen. Addressing misspecification in simulation-based inference through data-driven calibration. Xiv preprint arXiv:2405.08710 2024.
Jonas Bernhard Wildberger, Maximilian Dax, Simon Buchholz, Stephen R Green, Jakob H. Macke, and Bernhard Schölkopf. Flow matching for scalable simulation- based inference. In Thirty Conference on Neural Information Processing Systems, 2023. URL http::///pppppooieeee.eetrorrrrs
Simon N Wood. Statistical inference for noisy nonlinear ecological dynamic systems. Nature, 466(7310): 1102-1104, 2010.
23

---

Published in Transactions on Machine Learning Research (09/2025)
Kaiyuan Xu, Brian Nosek, and Anthony Greenwald. Psychology data from the race implicit association test on the project implicit demo website. Journal of Open Psychology Dala, 2(1);a3, 2014. ISSN 2050-9863. doi: 10.5334/jopd.ac. URL http://wwwsaaoollrrrr........oor http:////.......///........
Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R. Salakhutdinov, and Alexan- der Smola. Deep sets. Advances in neural information processing systems, 30, 2017.
Andrew Hammit-Manghan Matthew Sainsbury-Dake and Raphaël Huser. Neural methods for amortized inference. Annual Review of Statistics and Its Application, 12, 2024.
Werner Zellinger, Natalia Shepeleva, Marius- Coniss-Conttnninn Dinu, Hamid Eghbal-zadeh, Hoan Duc Nguyen, Bernhard Nessler, Sergei Pereverzyev, and Bernhard A Moser. The balancing principle for parameter choice in distance-regulariieed domain adaptation. Advances in Neural Information Processing Systems, 34:20798-208111 -0071, 2021.
Yi Zhang and Lars Mikelsons. Solving stochastic inverse problems with stochastic Bayesflow. In 2023 IEEE/ASMI International Conference on Advanced Intelligent Mechatronics (AIM), pp. 966 972. IEEE, 2023.
YiFan Zhang, Xue Wang, Jian Liang, Zhang Zhang, Liang Wang, Rong Jin, and Tieniu Tan. Free lunch for domain adversarial training: Environment label smoothing. arXiv preprint arXiv:2302.0019,, 2023.
Kaiyang Zhou, Ziwei Liu, Yu Qiao, Tao Xiang, and Chen Change Loy. Domain generalization: A survey. IEEE Transactions on Pattern Analysis and Machine Intelligence, 4444):4396-4415, 2022.
24

---

Published in Transactions on Machine Learning Research (09/2025)
APPENDIX
A Theoretical Details
Defining Amortized Bayesian Inference The term "amortized" has been used inconsistently through- out the literature, often denoting different generalization scopes. To clarify this concept for the discussion within this work, we offer the following definition:
Definition 1. Let A denote a learner, denote target variables, a represent input data, and c denote contert variables. A learner y ~ A(x, e) is an amortized Bayesian approximator of a largel quantity y with respect to a joint distribution p(a, y, C) if it can directly approzimate p'y x, c) for any (x,c) ~ p(a, c) without requiring further training or additional opprozimation algorithms.
By this definition, sequential methods that necessitate further training for new data (Paparakaaiiss & Murray 2016 Glöckler et al., 2022) are not considered amortized. Similarly, neural likelihood estimation (NLE: Papamakarios et al. (2019) and neural ratio estimation (NRE) (Hermans et al. 2020), which depend MCMC algorithms, do not qualify as amortized. In contrast, recent transformer-basee (Gloeckler et al.
2024) or context-aware methods [Flsemüller et al. 2024 clearly fall within the scope of amortized neural
posterior estimation (NPE).
B
Experimental Details
Since the analytic posterior is only obtainable in Experiment 2, we measure performance relative to the data- generating parameters l to enable a direct comparison between the experiments. For likelihood misspeci- fication settings, O is closely related to the posterior p(0 @ots) based on adjusted (e-g., decontaminated) data Tobs (Target 2). Thus, the NPE-UDA posterior approximations being closer 0" than the analytic posterior p(0 æ:) in the contamination scenario of Experiment indicates that NPE-UDA methods indeed focus Target 2.
In all experiments, we build upon the BayesFlow Python library for amortized Bayesian workflows using generative neural networks (Radev et al. 2023
B.1 Method Details
NNPE We implemented NNPE following the original implementation of Ward el al. (022) at https:
//gi thub. t/g.thbbccommaaaeee hataal/..cmmmmmleaaaddd////aa who used a spike scale of = 0.01 and a slab scale of t 0.25 for all experiments. To remain consistent with the original implementation of Ward al. (0022), , we applied NNPE to standardized data in all experiments (in Experiment 3, we applied equivalent scaling instead, see Appendix B.5 Whether spike (standard normal) or slab (standard Cauchy) noise is applied to simulated data point is determined by sampling from a Bernoulli distribution with = 0.5.
Sensitivities of NPE-UDA In both experiments, we found the typical UDA phenomenon of sensitivity to higher learning rates (Perone et al. 2019) in the form of unstable learning dynamics such exploding gradients. We also found sensitivity to short training times, suggesting that finding a stable optimum for the two-component NPE-UDA loss in Eq. 3 requires more gradient updates than usual.
Computational Cost of NPE-UDA NPE-UDA methods only modify the training process of standard NPE and therefore do not influence the computational cost of inference. Since NPE-UDA methods operate in the compressed summary space, the runtime increase during training is minimal compared to NPE. Experiment 3 features the largest (32-dimensional) summary space our experiments and thus constitutes an upper bound mcasurement of the time increase due to integrating UDA: Here, NPE and NPE-MMD took 12s/epoch (amounting a total wall clock time of 4 minutes) and NPE-DANN 13s 13s/epoch (amounting total wall clock time of ~ 4 minutes and 20 seconds) during GPU training on cluster.
25

---

Published in Transactions on Machine Learning Research (09/2025)
B.2 Metrics
We compute multiple metrics that measure the performance based on the approximation performance of J getta-geeraatng parameters {0]) via S posterior samples (we forego the obs notation where possible for brovity here). Depending on the metric, results are averaged across the J paramcters and/or N observed data sets.
B.2.1 Parameter Space Performance Metrics
Negative log likelihood (NLL):
NLL 1 N N
Σ ==1
log (00) d(x,)),
(8)
where we utilize the fact that normalizing flows allow us to easily compute approxinate (log) densities. Normalized root mean squared error (NRMSE):
N N
E0= (0) 0/
NRMSE
(9)
J
max min(07)
==1
Expected calibration error (ECE) via the fraction of ground-truth inliers for R linearly spaced confidence intervals in [0.005, 0.995] (Ardizzone et al. 2018 Radev et al. 2020
(NN I(e (6
N
( ar or).
ECE = J Σ median R
S 87 Q
(10)
1=1
where median, R represents the median fraction of inlicrs across the R 20 credible intervals and Q (x)) represents the k-th quantile of the posterior samples for the n-th data set. We estimate the ECE all test data sets via the median calibration error of R = 20 linearly spaced credible intervals, averaged across J model parameters.
Posterior contraction (PC) relative to the prior distribution
Betancourt 2018
Var(ō(+)) J.n Var(0 J.n
1
PC =
(11)
N
J
R=1
==1
B.2.2 Data Space Performance Metrics
Posterior predictive distance (PPD):
Σ d ona 2(.0)
PPD
(12)
where ((0) represents a re-simulation based on a posterior sample of all estimated parameters, ((0), and we use RMSE for d(.,-). To keep computation times reasonable, we limit the number of re-simulations a(s) by using random subset of all S posterior samples in the experiments with large number of posterior samples (Experiment 1: 100 samples; Experiment 2: 100 samples). We investigate two different PPD variants: (i) the default calculation defined in Equation 12 with E,, as the reference data set, (ii) and a calculation where vn is replaced with a "denoised" reference data set. T, that is obtained by re-simulating from the ground- truth parameter the synthetic experiments and intensive data processcss in the real-world experiment. One limitation of the PPD metric is that any parameter values that would break the simulator have to be excluded before re-simulating, which can reduce the sensitivity of the metric for detecting approximation failures.
26

---

Published in Transactions on Machine Learning Research (09/2025)
B.2.3 Network Space Metrics
Summary space domain distance (SSDD): SSDD, which does not quantify approximation performance but the degree of summary space alignment, is measured with two variants. The first variant is based on the biased sample-based MMD estimalor (Gretton et al. 2012):
MMD 2 [(o(=x))]]Ho((a")))
SSDDMMD
(13)
N
where {0(xx)) and {ō(2))))) are sets summary statistics over which the expoctations are approximated. The second variant. SSDDc2sT based on the classifier two-sample test (C2ST) and represents the accuracy of an MLP classifier trained to distinguish the sets of summary statistics {d(x,)) and (o(*)))) (Bischoff et al. 2024).
Inference network latent distance (INLD): Following (Siahkoohi et al. 2023
consider distortions in the inference network's latent space proxy for approximation quality, which is a direct consequence of maximum likelihood training. To measure general distortions (beyond location and scale), we again consider
the biased basple-based MMD estimator Gretton et al. 2012):
N Σ MMD [I
1
INLD
(14)
where f(0n; denotes the forward direction of the conditional invertible network realizing the normalizing flow.
B.3 Experiment 1 Ricker
Probabilistic Model We follow the model specification of Radev et al. (2020), including their prior specifications for the growth rate parameter and the scaling parameter p (see Radev et al. (2020) for details). The only deviation from Radev et al. (2020) is the specification of the parameter a governing the standard deviation of Gaussian noise, where we follow Huang el. al. (2023) and fix a 0.3 to allow for an easy visual inspection of the resulting 2D posterior landscape during method development.
Network Architecture We use an LSTNet architecture as described in Zhang & Mikelsons (023) for the summary network é, compressing the input to (-dimensional summary statistics. For the generative inference network g, we use an affine coupling flow architecture (Ardizzone et al.) 2021 Kingma & Dhariwal 2018) with 3 coupling layers. See Table B.1 for the optimized hyperparameters (regarding architectural well as training choices) per method and their respective search range.
Table B.1: Hyperparameter search ranges for all methods.
Hyperparameter
Range
Method(s)
Initial learning rate (a)
1 x 10 4.5 x 10
NPE, NNPE, NPE-MMD, NPE-DANN NPE-MMD, NPE-DANN
NPE-UDA alignment weight
0.01-150.0
Discriminator depth
2-4
NPE-DANN
Discriminator width
128 1024
NPE-DANN
Gradient reversal weight Agri
0.5 15.0
NPE-DANN
Label smoothing
0.0-0.3
NPE-DANN
Training and Evaluation Details We use an AdamW optimizer with an initial learning rate set by the hyperparameter search algorithm and cosine decay. We further use batch size of 32 and train for 20 epochs. For the evaluation, generate S 000 posterior samples per method and test data set.
27

---

Published in Transactions on Machine Learning Research (09/2025)
Additional Results Figure B1 contains all performance metries for the additional NPE-DANN hyper-
parameters, showing an overall little effect of the additional hyperparameters. We therefore favor simple settings of these hyperparameters in the following experiments.
NAMSE
PRO
Decdimanttor Depth
Dantiminator Depm
Dixrr mitaor Depth
Dicrimiratsr Depth
Berrmbsarr Depth
Diec morerr Beath
Iicrrrr Death
DEST in roter Dephh
Diaariinee Disaimmntter Witth
Disc minaatr waah
Disommattr Wath
Disciminator Motts
Discimmator Weth
Daccmmitor With
Dictmminator with
Diccimintoo With
PHD ceeteo
NOMSE
PPO Dreamn
WHISE
22
2
SED0
Label Smiothing
LobdSmoocio
Labe Smouthing
Labd Smoothing
Label Smoothhin
Label Smouthing
Labe Smoothing
Label Smoothing
NPE
NNPE
NPE-MMD
NPE-DANN
Figure B1: Experiment 1. Performance metrics for the additional NPE-DANN hyperparametors resulting from 50 separate Bayesian hyperparameter optimization runs per method. The solid trend lines represent the predictive mean of a Gaussian process regression fitted to the individual run results, with the shaded areas representing 95% confidence intervals of the predictive distribution. If parameter the method's average performance is depicted by dashed horizontal line. Lower values indicate better performance for all metrics but PC. NLI. Negative Log Likelihood. NRMSE Normalized Root Mean Squared Error. ECE Expected Calibration Error. PC Posterior Contraction.
not optimized,
B.4 Experiment 2D Gaussian Means
Misspecification Setting
Prior
Likellhood
Well-specified
p N/O, 1z)
N(,, 12)
Prior location misspecification
# ~ N(Ho, 12) ~N(,, 12)
Prior scale misspecification
# N/O, mol2)
æa ((,,
Likelihoud scale misspecification
p N/O, Iz) ax 71,
Contamination misspecification
p I2)
f 6(x c) -N(,
Table B.2: Experiment 2: Overview of the model specifications in the different misspecification settings.
Table B.2 provides overview of the well-specified setting and the different misspecification scenarios inspired by Schmitt et al. (2024).
Network Architecture We use deep set architecture Zaaheer et al. 2017 for the summary network
Φ, compressing the input to 4-dimensional summary statistics. For the generative inference network q, we
use an affine coupling flow architecture (Ardizzone et al. 2021 Kingma & Dhariwal 2018) with 3 coupling layers.
For the domain classifier vb in NPE-DANN, we use standard foedforward network with 2 hidden layers of width 256. We do use label smoothing weight the gradient reversal balance.
28

---

Published in Transactions on Machine Learning Research (09/2025)
Training and Evaluation Details To rule out any overfitting effects on the results, we online training approach where new data from the simulated and the observed domain is simulated at each training step, resulting in overall simulation budgets of N = 48 000 and Nobs = 49 000. Since we use a batch size of 32, also for the observed data in NPE-UDA methods, online training amounts to 1 500 mini-batches and thus gradient updates. We use an Adam optimizer with an initial learning rate of 5 10-4 and cosine decay. We generate = 5 000 posterior samples per method and test data set.
We provide additional results iterating over (i) performance in the simulated vs. the observed domain and (ii) À 10
Additional Results F Figure B2 Figure B3 and Figure B4 show the performance in the simulated domain for varying À settings. Despite notable performance differences in the observed domain, all methods perform well in the simulated domain for the vast majority of settings, with the only exception being failures of NPE-UDA methods for high regularization weights in Figure B4] NNPE performs worse in the simulated (noiseless) domain since it was optimized based on noisy training data. Besides the clear NPE- UDA failures in À = 10 settings, we mostly do not observe trade-off of the summary space alignment of the NPE-UDA methods in the simulated domain.
F Figure B5 and Figure B6 show the performance in the observed domain for varying À settings. The results confirm our finding of an application- and also method-specific À optimum: Whereas the difference of the NPE-UDA methods to NPE is often small for = 0.1, À = 10 renders the NPE-UDA methods highly unstable under misspecification, especially when large domain shifts are present.
Figure B7 directly contrasts the effects of prior misspecification due to too narrowly chosen priors (see "Prior Scale" results in Figure 5) with the complimentary scenario of too broadly chosen priors. While too narrow priors that exclude relevant parts of the prior space constitute clear misspecification, the case is less clear for too broad priors, to which practitioners commonly lean when in doubt. In Figure B7] we indeed find that, for this experimental setting, too broad priors are less detrimental for NPE than too narrow priors. The symmetric summary space domain distance (SSDD) of NPE (partially overlapping with NNPE in the plot) when deviating from the well-specified setting suggests that both regimes are equally atypical from the perspective of the trained network. However, while we observe drastic differences between the analytic posterior and the NPE methods in terms of parameter recovery error (NRMSE), posterior predictive data space distance (PPD), and calibration (ECE) in the too-nartow-priors regime (right side), these errors are much less pronounced in the roo-broad-priors regime (left side). Therefore, the differences between the various NPE mothods in the loo-broad-priors regime are mainly constrained to different contraction (1 PC) vs. calibration (ECE) trade-offs. Here, the contraction patterns mirror the general trend of UDA posteriors across our experiment to reduce the influence unmodeled information in the observed data and thus stay closer to the prior than NPE (i.e., less contraction equals broader NPE-UDA posteriors in the too- broad-priors regime and vice versa). We hypothesize that the dimensionality of the inference problem plays crucial role: In this low-dimensional (2D) inference task, the too-broad- -aroors regime is much less clear misspecification/ wwaattiionnuttoo-----i oot--o--distribueeeeeee the test setting is still rell-represented in the typical set of the simulated training data. As dimensionality increases, we expect this to become less and less the case due to the curse of dimensionality. Thus, for high-dimennsionll inference problems, we expect the choice of overly broad priors to become as detrimental as overly narrow priors. Comprehensive results for the too- too-broad--oo broad-priors regime for all settings are displayed in Figure B8 for the simulated (well-specified) domain and Figure B9 for the observed domain.
setting than the too-narrow-prion regime, since in the former regime,
29

---

Published in Transactions on Machine Learning Research (09/2025)
Metrics in the Simulated Domain
Prior Location Ho
Prior Scale
Likelihood Scale
Contamination Fraction
To 1
PC
0.12
0.17
0.02
0.02
ECE
ECE 0.09
0.15
0.10
0.08
0.41
0.12
0.09
PPD (resim)
SSDD (MMD)
PPD (resim) 1.53
SSDD (MMD)
PPD (resim) 1.66
SSDD (MMD)
PPD (resim)
SSDD (MMD)
1.59
1.43
Mo 2.5
3
10
0.25
PC
PC
NRMSE
ECE
NRMSE
NRMSE
NRMSE
PPD (resim) 1.59
SSDD (MMD)
PPO (resim) 1.53
SSDD (MMD)
PPO (resim)
SSDD (MMD)
PPD (resim)
SSDD (MMD) 0.16
2.64
0.68
1.66
1.11
1.43
= 5
To = 5
T = 20
E = 0.5
1 PC 0.12
PC 0.17
1 PC 0.02
--PC PC 0.02
ECE
NRMSE 0.10
ECE
NRMSE 0.0B
ECE
NRMSE 0.12
ECE
NRMSE 0.02
(resim)
SSDD (MMD)
(resim)
$500 (MMD)
(resim)
SSDD (MMD)
(resim)
SSDD (MMD)
1.59
2.64
1.53
0.68
1.66
1.11
1.43
0.16
NPE
NNPE
MME (A = 0.1)
DANN (A 0.1)
Analytic Posterior
Figure B2: Experiment 2. Performance metrics and summary space domain distance (SSDD) of the methods in all misspecification scenarios (columns) on simulated (i.e., well-specified) data for X = 0.1 in NPE-MMD and NPE-DANN, aggregated via the median of 10 runs. Lower values indicate better performance for all metries but SSDD. 1- PC 1- Posterior Contraction. NRMSE Normalized Root Mean Squared Error. SSDD (MMD) Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) Posterior Predictive Distance measured via the RMSE to resimulated data. ECE Expected Calibration Error.
30

---

Published in Transactions on Machine Learning Research (09/2025)
Metrics in the Simulated Domain
Prior Location Ho
Prior Scale
Likelihood Scale
Contamination Fraction
To 1
PC
0.12
0.17
0.02
0.02
ECE
ECE 0.09
0.15
0.10
0.08
0.41
0.12
0.09
PPD (resim)
SSDD (MMD)
PPD (resim) 1.53
SSDD (MMD)
PPD (resim) 1.66
SSDD (MMD)
PPD (resim)
SSDD (MMD)
1.59
1.43
Mo 2.5
3
10
0.25
PC
PC
NRMSE
ECE
NRMSE
NRMSE
NRMSE
PPD (resim) 1.59
SSDD (MMD)
PPO (resim)
SSDD (MMD)
PPO (resim) 1.66
SSDD (MMD)
PPD (resim)
SSDD (MMD) 0.16
2.99
1.53
0.68
4.81
1.43
= 5
To = 5
T = 20 1 PC 0.02
E = 0.5
1 PC 0.12
1 PC 0.17
--PC PC 0.02
ECE
NRMSE 0.10
ECE
NRMSE 0.0B
ECE
NRMSE 0.12
ECE
NRMSE 0.02
(resim)
SSDD (MMD)
(mesim)
$500 (MMD)
(resim)
SSDD (MMD)
(resim)
SSDD (MMD)
1.59
2.99
1.53
0.68
1.66
4.81
1.43
0.16
NPE
NNPE
MMD (A 1.0)
DANN (A 1.0)
Analytic Posterior
Figure B3: Experiment 2. Performance metrics and summary space domain distance (SSDD) of the methods in all misspecification scenarios (columns) on simulated (i.e., well-specified) data for À = 1 in NPE-MMD and NPE-DANN, aggregated via the median of 10 runs. Lower values indicate better performance for all metries but SSDD. 1- PC 1- Posterior Contraction. NRMSE Normalized Root Mean Squared Error. SSDD (MMD) Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) Posterior Predictive Distance measured via the RMSE to resimulated data. ECE Expected Calibration Error.
31

---

Published in Transactions on Machine Learning Research (09/2025)
Metrics in the Simulated Domain
Prior Location Ho
Prior Scale
Likelihood Scale
Contamination Fraction
To 1
PC
93964310027.38
0.95
0.99
0.93
ECE
ECE 0.12
0.15
35376.87
0.18
0.41
0.18
0.10
0.17
PPD (resim)
SSDD (MMD)
PPD (resim) 1.93
SSDD (MMD)
PPD (resim) 1.95
SSDD (MMD)
PPD (resim)
SSDD (MMD)
1.95
1.91
Mo 2.5
3
10
0.25
PC
PC
93964310027.38
NRMSE 5376.88
ECE 0.12
NRMSE
NRMSE
NRMSE
PPD Iresim) 1.95
SSDD (MMD)
PPO (resim) 1.93
SSDD (MMD)
PPO tresim)
SSDD (MMD)
PPD (resim)
SSDD (MMD) 1.25
6.38
1.05
1.95
5.03
1.91
= 5
To = 5
T = 20
E = 0.5
1 PC
1 PC 0.95
PC
1-PC PC 0.93
93964310027.38
0.99
ECE
NRMSE
ECE
NRMSE 0.18
ECE
NRMSE 0.18
ECE
NRMSE 0.17
5376.887
(resim)
SSDD (MMD)
(resim) 1.93
$500 (MMD)
(resim)
SSDD (MMD)
(resim) 1.91
SSDD (MMD)
1.95
6.38
1.05
1.95
5.03
1.25
NPE
NNPE
NPE-MMD MMD 10.0)
NPE DANN A 10.0)
Analytic Posterior
Figure B4: Experiment 2. Performance metrics and summary space domain distance (SSDD) of the methods in all misspecification scenarios (columns) on simulated (i.e., well-specified) data for x= 10 in NPE-MMD and NPE-DANN, aggregated via the median of 10 runs. Lower values indicate better performance for all metries but SSDD. 1- PC 1- Posterior Contraction. NRMSE Normalized Root Mean Squared Error. SSDD (MMD) Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) = Posterior Predictive Distance measured via the RMSE to resimulated data. ECE Expected Calibration Error.
32

---

Published in Transactions on Machine Learning Research (09/202))
Prior Location Ho 0
Prior Scale 1
Likelihood Scale
Contamination Fraction
T=1 1
PC
PC
1-PC PC
NRMSE
ECE 0.33
NRMSE
NRMSE
NRMSE
0.40
PPD (resim)
SSDD (MMD) 2.64
PPD (resim) 3.50
SSDD (MMD)
PPO (resim) 2.27
SSDD (MMD) 1.11
PPD (resim) 1.51
SSDD (MMD) 0.16
5.05
0.68
14o
2.5
To = 3
T= 10
= 0.25
1 PC 0.15
PC 0.03
1-PC 0.05
1CPC 0.02
ECE
NRMSE 0.75
ECE
NRMSE 0.12
ECE
NRMSE 0.27
ECE
NRMSE 0.08
PPD (resim)
S5DD (MMD)
PPD (resim)
$500 (MMD)
(resim)
SSDD (MMD) 1.11
(resim)
SSDD (MMD)
5.05
2.64
3.50
0.68
2.27
1.51
0.16
Mo = 5
To = 5
T= 20
E = 0.5
1 - PC 0.15
- PC 0.03
1CPC
1CPC
0.05
0.02
ECE
NRMSE 0.75
ECE 0.33
NRMSE 0.12
ECE 0.47
NRMSE 0.27
ECE 0.40
NRMSE
0.50
0.08
(resim)
S5DD (MMD) 2.64
(resim)
S5DD (MMD)
(resim)
S5DD (MMD)
S5DD (MMD)
5.05
3.50
0.68
2.27
1.11
1.51
0.16
NPE
NNPE
0.1)
NPE-DANN 0.1)
Analytic Posterior
Figure B5: Experiment Performance metrics and summary space domain distance (SSDD) of the meth- ods in all misspecification scenarios (columns) for λ 0.1 in NPE-MMD and NPE-DANN, aggregated via the median of 10 runs. Lower values indicate better performance for all metrics but SSDD. 1- PC 1- Posterior Contraction. NRMSE Normalized Root Mean Squared Error. SSDD (MMD) Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) Posterior Predictive Distance measured via the RMSE to resimulated data. ECE Expected Calibration Error.
33

---

Published in Transactions on Machine Learning Research (09/202))
Prior Location Ho
Prior Scale 1
Likelihood Scale =
Contamination Fraction
PC
NRMSE
ECE 0.39
NRMSE
NRMSE
NRMSE
PPD (resim)
SSDD (MMD)
PPD (resim) 4.66
SSDD (MMD)
PPO (resim) 2.27
SSDD (MMD) 5.03
PPD (resim)
SSDD (MMD) 1.25
5.25
6.38
1.05
1.93
I4 2.5
To = 3
T= 10
E = 0.25
PC
PC 0.04
1-PC
1 PC 0.89
3730 76306668
1.08
ECE
NRMSE 959 85
ECE
NRMSE 0.15
ECE
NRMSE 0.27
ECE
NRMSE 0.21
PPD (resim)
SSDD (MMD)
PPD (resim)
$500 (MMD)
(resim) 2.27
SSDD (MMD) 5.03
PPD (resim)
SSDD (MMD)
5.25
6.38
4.66
1.05
1.93
1.25
Ho = 5 1 - PC 37 18.888888
To = 5
T= 20
E = 0.5
- PC 0.04
1CPC
1 1
1.08
0.89
ECE 0.50
NRMSE 1959.6 85
ECE
NRMSE 0.15
ECE 0.47
NRMSE 0.27
ECE
NRMSE 0.21
0.39
0.40
(resim)
S5DD (MMD)
(resim)
55DD (MMD)
(resim)
S5DD (MMD)
S5DD (MMD)
5.25
6.38
4.66
1.05
2.27
5.03
1.93
1.25
NPE
NNPE
10.0)
NPE-DANN
Analytic Posterior
Figure B6: Experiment Performance metrics and summary space domain distance (SSDD) of the meth- ods in all misspccification scenarios (columns) for X = 10 NPE-MMD and NPE-DANN, aggregated via the median of 10 runs. Lower values indicate better performance for all metrics but SSDD. 1- PC 1- Posterior Contraction. NRMSE Normalized Root Mean Squared Error. SSDD (MMD) Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) Posterior Predictive Distance measured via the RMSE to resimulated data. ECE Expected Calibration Error.
34

---

Published in Transactions on Machine Learning Research (09/202))
0.10
NRMSE
0.05
0.00
0.4
0.3
0.2
0.1
0.0
0.3
ECE
0.2
0.0
PPD (resim)
2
0.6
SSDD (MMD)
0.2
0.0
0.2
0.33
1
too broad prior
too narrow prior
To (log scale)
NPE
NPE-DANN (Λ = 1.0)
NNPE
Analytic Posterior Well-specified
NPE-MMD (A= 1.0)
Figure B7: Experiment 2. Performance metrics and summary space domain distance (SSDD) of the methods in all misspecification scenarios (columns) for prior scale misspecification with too broad vs. too narrow priors (A = 1), aggregated via the median of 10 runs. Lower values indicate better performance for all metrics but SSDD. NRMSE = Normalized Root Mean Squared Error. 1- PC = 1- Posterior Contraction. ECE Expected Calibration Error. PPD (resim) = Posterior Predictive Distance measured via the RMSE to resimulated data. SSDD (MMD) = Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior).
35

---

Published in Transactions on Machine Learning Research (09/202))
Metrics in the Simulated Domain
Metrics in the Simulated Domain
Metrics in the Simulated Domain
Prior Scale
Prior Scale
Prior Scale
To PC
To
To P-PCC 0.59
0.02
0.02
0.05
0.05
0.5R
0.18
MO [reim]
S5DD (MND)
PPD (tesim)
SSD0 (MMD)
PIO Iresim)
S5DD (MMD) 1.30
0.63
1.46
0.63
= 0.33
= 0.33
To = 0.33
0.02
0.07
0.99
ECE
NAMSE
ECE
NRMISE
ECE
NAMST
PRO fresim] 1.45
PPD (resim) 1.48
PO (resim) 1.98
0.65
0.63
1.30
0.2
0.2
0.2
PPC
PCPC
NAMSE
NRMSE
NAMSE
0.38
0.05
0.38
0.38
(NND)
(MMD)
(MMD)
1.45
NPE
NPE MMD 0.1)
NPE: HHD (A - 1.01
NIEE MIND 10.00 0)
NPP -AYE-DANN lA 0.11
NPE-DANN CA 1.0)
NFE-DAAN LA 10.0
(a) 0.1.
(b) ==
(c) A = 10.
Figure 18: Experiment 2. Performance metrics and summary space domain distance (SSDD) of the mothods in all misspecification scenarios (columns) on simulated (i.c., well-speeified) data for prior scale misspecification with too broad priors, aggregated via the median of 10 runs. Lower values indicate better performance for all metrics but SSDD. 1- PC = 1 Posterior Contraction. NRMSE = Normalized Root Mean Squared Error. SSDD (MMD) = Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) Posterior Predictive Distance measured via the RMSE to resimulated data. ECF Expected Calibration Frror.
36

---

Published in Transactions on Machine Learning Research (09/2025)
Prior Scale To = 1 1CPC 0.34
Prior Scale To 1 PCPC 0.38
Prior Scale To 1 1-PC 23.82
ECE 0.12
NRMSE 0.11
ECE 0.12
NRMSE 0.12
ECE 0.38
NRMSE 0.75
PPD (resim) 1.42
SSDD (MMD)
PPD (resim) 1.42
SSDD (MMD) 0.63
PPD (resim) 1.74
SSDD (MMD) 1.30
0.65
To = 0.33
To = 0.33
To
0.33
PC 0.38
PC 23.82
0.34
ECE 0.12
NRMSE 0.11
ECE 0.12
NRMSE 0.12
ECE
NRMSE 0.75
0.38
(resim) 1.42
SSDD (MMD)
(resim) 1.42
SSDD (MMD)
(resim)
SSDD (MMD) 1.30
0.65
0.63
1.74
To 0.2
To 0.2
To 0.2
PC
1 PC 0.38
PC 23.82
0.34
ECE 0.12
ECE 0.12
ECE 0.38
0.11
0.12
0.75
PPD (resim) 1.42
(MMD)
PPD (resim) 1.42
(MMD) 0.63
PPD (resim) 1.74
SSDD (MMD)
0.65
1.30
NPE
NPE
NNPE
NNPE
NNPE
NPE-MMD (Λ = 0.1)
NPE-MMD (A 1.0)
NPE-MMD (A = 10.0)
NPE-DANN 0.1)
NPE-DANN (A 1.0)
NPE-DANN (A = 10.0)
Analytic Posterior
Analytic Posterior
Analytic Posterior
(a) À = 0.1.
(b) A ==1.
(c) = 10.
Experiment Performance metrics and summaay space domain distance (SSDD) of the methods in all misspecification scenarios (columns) for prior scale misspecification with too broad priors, aggregated via the median of 10 runs. Lower values indicate better performance for all metrics but SSDD. 1- PC 1- Posterior Contraction. NRMSE Normalized Root Mean Squared Error. SSDD (MMD) = Summary Space Domain Distance measured via MMD (not applicable for Analytic Posterior). PPD (resim) = Posterior Predictive Distance measured via the RMSE to resimulated data. ECE Expected Calibration Error.
Figure B9:
37

---

Published in Transactions on Machine Learning Research (09/2025)
B.5 Experiment 3
Probabilistic Model We adopt noisy camera model similar to the one presented in Ramesh el. al. [2022). First, the imput image is clipped to the range [-1,1]. Next, we use scikit-image (van der Walt et al. 2014) to add Poisson noise to the image, then filter using a Gaussian filter from SciPy (Virtanen ot al., 2020) with a standard deviation a for the Gaussian kernel. The result a blurred image with identical size
as the input image.
Data Preparation For each data set, we normalize the images to the range [-1, The MNIST (Lecun et al. 1998) images are rescaled from 28 x 28 to 16 x 16 with anti-aliasing enabled. To produce the training data H, the images are processed by the simulator, with do 1.4. We adapt NNPE, which is originally defined for standardized data, by scaling the noise scales by the standard deviation dæ of the training data.
While the training data remains constant across scenarios, the observed data is generated in different ways. For the prior misspecification scenario, we use the USPS data set (Hull 1994) instead of MNIST, but the parameters of the simulator remain identical (i.e., = oo). For the likelihood scale scenario, we use ā 1.25 oo leading to an increased blur. For the noise contamination scenario, we randomly set 10% of the pixels of cach observation to black white. For the contamination scenario, randomly set 2 rows of each observation (i.e., 12.5% of the pixels) to black. Refer to Table 3 for samples from each scenario.
Network Architecture For the summary network, we use a 4-layer convolutional neural network, which oulputs 32 learned summary variables.
For the inference network, we use flow matching (Lipman et al. multivariate Gaussian distribution to the approximate postcrior distribution. We use U-Net architecture RRnneberger al. 2015) to learn the flow field conditional on the summary variables.
2023 Wildberger et al. 2023) to convert a
For NPE-DANN, we domain classifier 1b consisting of a standard feedforward network with 3 hidden layers of width 256, gradient reversal layer (GRL) weight of 1, and no label smoothing.
Training and Evaluation Details We use an AdamW optimizer with an initial learning rate of 5 10 and cosine decay. We use a batch size of 32 and train for 20 epochs, except for NPE-MMD, which required increasing the batch size 128. To keep the number of gradient updates constant, we also increased the number of epochs 80 for NPE-MMD, The training budget 000 training images, and 1000 observed images. Training one neural network takes approximately 10 minutes on GPU.
We use moderate number of posterior samples of S = 100 per method and test data set to limit the computational cost of the experiment, allowing for a broader exploration of hyperparameters and the variance between multiple runs.
Additional Results Table B.3 displays the performance on a held-out in-distribution data set, to assess the influence on the loss on the in-domain observations. We consistently observe similar performance for NPE and the NPE-UDA methods for low X and the expected performance loss of the noisy NNPE training on in- distribution data. Interestingly, NPE-MMD with low À shows slight but consistent improvement over NPE in this setting, The reason for this is unclear the additional observed data might serve as an extra. training signal that improves the learned summary statistics, even for in-distribution data. Additionally, we can verify that for NPE-MMD with A = 1.0 with vanishing SSDD, the summary space does not contain information, as also the in-distribution performance drops to a value we expect of unconditional generation. Table B.4 displays the same data as Table 2 but with uncertainty indicators (standard deviation). Figure B10 shows the plots corresponding
Figure 2 for the remaining three scenarios.
38

---

Published in Transactions on Machine Learning Research (09/2025)
0.30
0.30
0.35
NRMSE
NRMSE
0.25
NRMSE
0.25
0.20
0.30
0.20
0.15
0.0
0.1
0.2
0.00
0.05
0.10
0.0
0.2
0.4
SSDD
SSDD
SSDD
NPE-DANN NPE-MMD
NPE NNPE NPE-DANN NPE-MMD
NPE-DANN NPE-MMD
(a) Prior (MNIST + USPS)
(b) Likelihood Scale
(c) Contamination (Noise)
Figure B10: Experiment 3. Relationship of summary space domain distance (SSDD; MMD) and normalized root mcan squared error (NRMSE, lower is better). For a), we see that despite the reduced SSDD, there is no gain in performance. For b) and c), we observe a sweet spot at a low SSDD value, before performance drops again when approaching zero. Refer to Table 2 for numerical values.
X1
X
81
81
Xabs
Xatx
Matx
Xaza
(a) NPE
(b) NNPE
(c) NPE-DANN
(d) NPE-MMD
Figure B11: Experiment 3. Prior: t-SNE representation of the summary spaces from the best run (lowest NRMSE) of each method. Please refer to Figure G for detailed caption.
x
X1
8
81
Mazs
(a) NPE
(b) NNPE
(c) NPE-DANN
(d) NPE-MMD
Figure B12: Experiment 3. Likelihood (Scale): t-SNE representation of the summary spaces from the best run (lowest NRMSE) of each method. Please refer to F Figure 6 for detailed caption.
39

---

Published in Transactions on Machine Learning Research (09/2025)
Prior (MNIST + USPS)
Likelihood Scale
Method
A
NRMSE +
PPD +
NRMSE +
PPD +
NPE
0.104 (1.6e-03)
0.020 (3.0e-04)
0.103 (9.9e-04)
0.020 (2.0e-04)
NNPE
0.140 (8.9c-04)
0.030 (2.1e-04)
0.141 (1.7e-04)
0.031 (1.6c-04)
NPE-DANN
0.01
0.116
(2.0c-03)
0.024 (4.9c-04)
0.110 (8.2c-04)
0.023 (2.6c-04)
NPE-DANN
0.10
0.163
(2.0e-02)
0.037 (5.5e-03)
0.122 (1.3e-03)
0.026 (2.9e-04)
NPE-DANN
1.00
0.267
(7.6e-03)
0.080 (4.6e-03)
0.140 (9.3e-04)
0.030 (2.9e-04)
NPE-MMD 0.01
0.092
(9.0e-04)
0.019 (2. (2.7e-04)
0.091
(3.2e-04)
0.018 (2.4e-04)
NPE-MMD
0.10
0.092
(3)-03)
0.018 (2.7e-04)
0.169 (1.1e-01)
0.055 (5.1e-02)
NPE-MMD
1.00
0.298
(3. (e22e-02
0.111 (2.3c-02)
0.320 (4.2o-04)
0.127 (1.8c-04)
Contamination (Noise)
Contamination (Rows)
Method
NRMSE +
PPD +
NRMSE +
PPD +
NPE
0.104 (1.8e-03)
0.021 (2.7e-04)
0.104 (1.3e-03)
0.020 (2.7e-04)
NNPE
0.141 (1.1e-03)
0.031 (1.7e-04)
0.139 (6.5e-04)
0.030 (1.6e-04)
NPE-DANN 0.01
0.114 (3.1e-04)
0.023 (7.6e-05)
0.115 (4.4e-03)
0.023 (7.2e-04)
NPE-DANN
0.10
0.139
(1.2e-03)
0.029 (4.8e-04)
0.136 (5.8e-03)
0.028 (2.0e-03)
NPE-DANN
1.00
0.192
(1.0e-02)
0.045 (3.6o-03)
0.179 (1.8c-02)
0.040 (6.2o-03)
NPE-MMD
0.01
0.091
(1.2c-03)
0.018 (2.3e-04)
0.091 (5.9c-04)
0.018 (1.1e-04)
NPE-MMD
0.10
0.092
(1.7e-03)
0.018 (4.1e-04)
0.093 (6.8e-04)
0.018 (9.4e-05)
NPE-MMD 1.00
0.319
(2.6e-04)
0.128 (2.5e-04)
0.320 (5.6e-04)
0.128 (2.2e-04)
Table B.3: Experiment 3. Overview of the metrics on a held-out validation data set from the training distribution (mean and standard deviation of three runs). NRMSE: Normalized Root Mean Squared Error (lower better). PPD Posterior Predictive Distance (RMSE) resimulated data (lower is better). For NNPE and NPE-DANN, that for successful runs, the performance on the training distribution improves. For settings with vanishing SSDD (compare Table 2) the performance drops massively, for both training distribution and observed distribution. This supports the notion that no meaningful information is learned in the summary space.
reduced performance
the training distribution. For NPE-MMD, we see
X1
X1
X1
81
Xans
Mazs
(a) NPE
(b) NNPE
(c) NPE-DANN
NPE-MMD
Experiment 3. Contamination (Noise): t-SNE representation of the summary spaces from the Figure B13:
best run (lowest NRMSE) of each method. Please refer to Figure 6 for detailed caption.
40

---

Published in Transactions on Machine Learning Research (09/202))
Prior (MNIST
USPS)
Likellhond Scale
Method
NRMSE 4
PPD 4
SSDD
NRMSE 1
PPD J
SSDD
NPE
0.259 (2.5e-03)
0.131 (1.7e-03)
0.258
(4.4e-03)
0.165 03.5--33)
0.036 (1.0e-03)
0.101 (3.4e-03)
NNPR
0.274
(2.3o-03)
0.137 (1.6o-03)
0.206 (1.2x-02)
0.173 (1.5-03)
0.041 (1.3e-03)
0.088 (5.3e-03)
NPE-DANN
0.01
0.329
(4. lc-03)
0.193
(2.6n-03)
0.027
(1.1e-03)
0.130
(2.8c-03)
0.031
(1.8e-03)
0.027 (1.4c-03)
NPE-DANN
0.10
0.326
(3.7e-03)
(1.4e-03)
(2.8e-03)
0.134 (8.5e-04)
(1.0e-03)
(1.2e-03)
NPENDANN
1.00
0.352
(4.1e-03)
0.205
(4.3e-03)
0.038
(..2--33
0.147 (1.0e-33)
0.032 (2.7e-04)
0.014 (8.6o-04)
MPE
0.303
(1.6c.03)
(7.4o.04)
(9.9s-04)
(8.3e.05)
(2.9c-04)
(6.1e-04)
NPE-MMD
0.10
0.312
(1. 1e-03)
0.189
(5.7e-04)
0.018
(4.6e-04)
0. (9.4e-02)
0.059 (4.9e-02)
0.009 (6.5e-03)
NPE-MMD
1.00
0.374
(1.3e-02)
0.225 (1.1e-02)
0.004 (5.9e-03)
0.322 (2.1e-04)
0.129 (2.7e-01)
0.000 (2 (2-0))
Contamination (Noise)
Contarnination (Rows)
Method
NRMSE
PPD
SSDD
NRMSE
PPD 4
SSDD
NPE
0.312 (6.8c-03)
0.117 (4.2e-03)
(9.2c-03)
0.315 (7.3e-03)
(4.0e-03)
(4.0e-02)
NNPE
0.154 (6.1e-04)
0.035 (2.4c-04)
0.019 (1.7e.03)
0:114 (7.4e-03)
0.064 (4.2c-03)
0.071 (1.5e-02)
NPE-DANN
0.01
(3.2c-03)
0.065 (8.Te-04)
(1.8c-03)
(7.96-04)
0.056 (1 6e-033
(1.2c-03)
NPE-DANN
0.10
0.211
(4.4e-03)
0.057 (2.2e-03)
0.015 (9.6c-04)
0.178
(1.7e-02)
0.045 (5. (5-1c-03)
0.014 (6. le-04)
NPE-DANN
1.00
0.240
(9.0e-03)
0.068 (3.6e-03)
0.015 (4.3e-04)
0.201 (2.0e-02)
0e-03)3)
(2.4e-04)
NPE-MMD
0.01
0.268
(4.26-03)
0.085 (3. 1e-03)
0.016 (6.46-04)
0.262 (1-16-03
0.085 (6.4e-04)
0.016 (3.9e-04)
NPE-MMD
0.10
0.245
(6.3e-03)
0.071 (3.6e-03)
0.013 (1.5c-04)
0.181 (9.7c-08)
0.047 (3.8)-03)
0.012 (3.0e-04)
NPE-MMD
1.00
0.321 (4.2o-04)
0.129 (3.4e-04)
0.000 (3.1o-06)
0.322 (5.3o-04)
0.129 (2.9o-04)
-0.000 (2.7o-06)
Table B.4: Experiment 3. Overview of the metrics in the different misspecification sccnarios (mean and standard deviation of three runs). Please refer to Table for a detailed description. Note that each standard deviation given for a constant set of hyperparameters, so it only covers the computational uncertainty for given setting. As shown by the performance changes when changing X, hyperparameters have large influence the results, and different hyperparameter choices might lead to qualitative changes in the results.
41

---

Published in Transactions on Machine Learning Research (09/2025)
B.6 Experiment 4 Decision Making
Real-World Data We utilize empirical response time data from Implicit. Association Test (TAT) ex- periments conducted in online settings and provided by Project Implicit (Xu et al. 2014]. In the 1AT, participants categorize words and images into two target categories as quickly and accurately as possible, aiming to respond swiftly while minimizing errors. We use a subsample from the standard Race IAT data set, collected between late 2014 and early 2015, with an initial sample size of N 1666 283. For each person, we use 120 experimental trials, 60 from each of the main experimental conditions in the IAT.
Probabilistic Model As our cognitive model, we employ the diffusion decision model (DDM; Ratcliff 2016). The DDM models the observed choices and reaction times as outcomes of a noisy evidence accumulation process and has been successfully applied to this and similar data sets (Klauer et al., 2007 Kvam
et al.
it al.
2024). We base our modeling approach on previous work with the samne data set 4yon Krause et al, 2022 von Krause & Radev 2025, see these papers for the exact model formulation). To capture differences between experimental conditions tondiiiis-typicalllyl labeled congruent. and incongruent we specify separate drift introduce distinct non-
rates and boundary separation parameters for each condition. Additionally, decision time parameters for correct and error responses, to account for how error response times are recorded in the Project Implicit data set (i.e., the error response time is not saved directly, but only after the initially erroncous answer has been corrected). We adopt broad yet informative priors based on previous modeling studies utilizing the same data (von Krause & Radev 2025), to maximize the closeness of our approach to real-world applications.
Network Architecture For the summary network é, we utilize deep set architecture (Zaheer et al. 2017) that reduces the imput (response time, accuracy, and experimental condition of 120 trials per person) to a 12-dimensional representation. We use a dropout rate of 0.05 to avoid overfitting due to offline training. The generative inference network q is based on an affine coupling flow model (Ardizzone et al. 2021: Kingma & Dhariwal, 2018), consisting of six invertible layers. As before, the domain classifier & used in NPE-DANN is conventional feedforward neural network with three hidden layers, each comprising 256 units. We omit both label smoothing and any reweighting of the gradient reversal component. We use an AdanW optimizer with an initial learning rate of 1- 10 and cosine decay.
Training and Evaluation Details All neural networks are trained offline using fixed simulation budget of 32, 000 data sets. Each model is trained for 100 epochs with batch size of 32, resulting in 1, 000 iterations per epoch. For validation during training, we generate an additional 1, 000 simulated data sets. In the case of NPE-UDA methods, we also use 32, 000 empirical data sets for training and another 1, 000 for validation. We use 5 = 100 posterior samples per method and test data set to keep computation times reasonable.
To evaluate model performance, we construct four distinct test sets from the empirical data. We begin by categorizing the remaining 133, 283 examples after removing training and validation data into well- specified and misspecified subsets. Well-specified data sets closely resemble the simulated data, whereas misspecified ones are more likely to reflect model mismatch. To define this distinction, we use out-of- distribution detection approach in summary space inspired by SShmiit et al. (2024): We train three standard NPE networks with equal settings and extract 12-dimensional summarr embeddings for 10, 000 simulated test sets. We then compute the 90th percentile of Mahalanobis distances within the embedding space for each network. Averaging the resulting quantiles across the three runs yields a threshold, which we use to classify empirical data sets: Those with embeddings within the threshold are labeled well-specified, while those exceeding it are considered misspecified. We calculate the mean Mahalanobis distance for the embeddings of each empirical test data set- vet-using the covariance matrix derived from simulated data- averaged across the three networks. Based on these distances, we categorize each data set accordingly. For subsequent analyses, we use as test data all 730 data sets labeled as misspecified, along with a separate random subsample of 10, 000 well-specified examples.
Both well-specified and misspecified test sets undergo a data cleaning procodure, resulting in two versions of each: a raw (uncleaned) and a cleaned data set. Cleaning involves removing response times below 200 ms above 10 seconds, well as intra-individual outlier trials. These outliers are identified as trials with
42

---

Published in Transactions on Machine Learning Research (09/2025)
log-transnn transformedd response times falling outside 1.5 interquartile ranges (IQRs) from the 25% 75% quantiles, calculated separately for each participant, condition, and trial correctness.
For the NNPE method, we again apply the spike-and-slab procedure to introduce noise into the input data- in this case, the response times. To prevent any sign reversal due to contamination, we enforce a lower bound by setting any negative response time values to 0.01 seconds.
Metrics As our first evaluation metric, we compute the MMD between the embedding spaces of simulated test data and uncleaned empirical test data. We hypothesized that UDA methods would yield better align- ment i.., lower MMD values compared to standard NPE, reflecting improved generalization to empirical data.
Our second metric is concerned with external validity. We examine correlations between individual par- ticipants' posterior median coguitive parameters and their age.Prior research has shown robust age-related effects, particularly on the boundary separation parameter and non-decision time (for correct trials). We therefore compute these correlations across all networks and test sets to evaluate the extent to which these known effects are recovered.
Our third metric focuses on posterior predictive distances. For each participant in the empirical test sel, we use our trained networks to approximate 100 samples from the joint posterior distribution over cognitive model paramcters. Using these samples, we re-simulate data and compare them to the original empirical data using root mean square error (RMSE) on a set of sumunary statistics: mean accuracy and response time quantiles (10%, 30,, 50%, 70,, and 90%), split by condition and further separated into correct and error trials. RMSEs are averaged across posterior samples, participants, quantiles, and experimental conditions. This yields one RMSE value each for accuracy, correct response times, and error response times per estima- tion method and per test data set (i.e., well-specified//cean misspecified/clen well-speciffed//ume /unccean, and patpeeiffeed/nnleaann
Additional Results Our last metric again uses MMD, but this time to quantify the discrepancy wetween the prior distributions of cognitive parameters and their corresponding posterior distributions after inference on empirical data. We utilize this divergence between prior and posterior distributions of cognitive param- eters to test the hypothesis of Huang al. (2023) that increasing À values lead to a convergence of the posterior to the prior. Figure B15 shows the results, with little evidence for a convergence to the prior up to x = 1.0. However, for x == 10 (not shown), suggesting that the model learns less from the data at this level of UDA influence. Similarly, while the lowest summary space distances are observed at À = 10, the PPD reveals that, at this setting, the estimated parameters fail to adequately reproduce the empirical data.
observe sharp decrease in the distance between prior and posterior distributions
Together, these findings highlight the importance of carefully balancing the UDA component in training: while stronger alignment can improve summary-level similarity, excessive emphasis may drastically impair parameter recovery and reduce the informativeness of the posterior. As parameter estimation effectively fails for the x = 10 networks, we exclude thes from further analyses.
Finally, we also compared results for shorter training time (50 instead of 100 epochs) and uniform instead of informative priors for the DDM parameters. In both cases, the alternative settings lead to worse performance across metrics and methods, so we omit these results for reasons of brevity.
43

---

Published in Transactions on Machine Learning Research (09/2025)
Misspecified
Well-specifed
RMSE Correct RT (Undlcar)
0.260
0.072
0.240
0.070
0.220
0.068
0.200
0.180
0.064
0.160
RMSE Error RT [Unclean)
0.860
0.840
0.425
0.820
0.420
0.780
0.415
0.760
RMSE Accuracy (Undlean)
0.009
0.110
0.009
0.100
0.009
0.090
0.008
0.080
0.01
0.10
1.00
0.01
0.10
1.00
A
NPE
NNPE
NPE-DANN
NPE-MMD
Figure B14: Experiment 4. Posterior predictive distance (RMSE) to uncleaned empirical response time data, averaged across three runs per method. Across run standard deviations are shown as shaded areas (for NPE and NNPE) or error bars (for NPE-DANN and NPE-MMD). The left column shows results for misspecisied data sets, while the right column shows results for the (much larger) well-specified data set. The first row shows the network's prediction error for response times (in seconds) on correct trials, averaged across posterior samples, participants, response time quantiles, and experimental conditions. The second row shows the same metric for error response times, while the third row shows results for accuracy rates (in %). Please note that y-axis scales differ across subplots. À UDA weight.
Misspecified
Well-specified
PriorPPstertor Distance
1.300
0.925
1.200
0.875
1.100
2
NPE
NNPE
NPE DANN
NPE MMD
Figure B15: Experiment 4. MMD of prior posterior distributions in the parameter space, averaged across three runs per method. Across run standard deviations are shown as shaded areas (for NPE and NNPE) or error bars (for NPF.DANN and NPF-MMD). The left. column shows results for misspecified data sets, while the right column shows results for the (much larger) well-specified data set. Please note that y-axis scales differ across subplots. A = UDA weight.
44


Reviews:
[{"review_id": "ThCUyJI6aL", "reviewer": "Reviewer_UKEB", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "$\\textbf{Strengths}$\n\n1.The paper introduces and systematically evaluates a novel amortized version of NPE-DANN, expanding the applicability of domain-adversarial training to amortized Bayesian inference for the first time.\n\n2.The proposed framework is conceptually simple and easy to implement, making it practical for integration into existing ABI pipelines.\n\n3.The empirical evaluation is comprehensive, covering a diverse range of synthetic and real-world scenarios, and provides valuable insights into when domain adaptation improves or harms inference.\n\n\n$\\textbf{Weaknesses}$\n\n\n1.The identified fragility of ABI under model misspecification is well-known and not specific to ABI alone. Similarly, the application of domain adaptation is a generic remedy. The paper does not offer novel methodological contributions to either ABI or UDA individually; rather, it combines existing tools without clear innovation beyond integration.\n\n2.The paper lacks theoretical contributions—for instance, it does not establish tighter generalization bounds or characterize how the proposed method narrows known robustness gaps.\n\n3.The approach is highly sensitive to the choice of the regularization weight, and no principled method is provided for tuning it in real-world, label-free settings—limiting practical usability.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper identifies a key shortcoming in existing applications of amortized Bayesian inference (ABI) under model misspecification and introduces a regularization-based approach to enhance robustness. The proposed model structure is simple yet effective in addressing this issue. The evaluation is thorough, covering both synthetic and real-world datasets, and highlights the trade-offs and practical limitations of applying unsupervised domain adaptation (UDA) to ABI.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "It would strengthen the paper if the authors could address some of the weaknesses outlined above. In particular, demonstrating a more substantial methodological or theoretical advancement—beyond combining existing ABI and UDA techniques—would significantly improve the paper’s contribution. Without such progress, the innovation may be seen as limited.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "kCLmqsrWjJ", "reviewer": "Reviewer_o89H", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- Clear and well-written.\n- First adaptation to domain-adversarial neural networks to amortised NPE. NPE-DANN convincingly shown to be capable of decent performance for the goal of robust inference.\n- Thoughtful updates to NPE-MMD. Nice to see useful updates (amortised, looked into hyperparameters) and comments on this, as well as connections to UDA.\n- Experimentation well-done, interesting and insightful comments.\n- Recognise and explicitly confront potential issues with prior misspecification. \n\n\nWeaknesses:\n- Hyperparameters and neural architecture choice. Both DANN and MMD introduce several extra hyper-parameters, increasing the tuning effort and computational cost. Although the paper notes this limitation, it still poses a hurdle to practical adoption. In addition, the learning summary approach requires custom summary networks and, for DANN, an accompanying discriminator tailored to each new domain, further expanding the set of design decisions the practitioner must manage.\n- Adversarial stability - acknowledged in the paper, but training stability might pose an obstacle for practitioner use. I am unsure on any specifics for DANN, wonder if any mitigations/techniques were found to be needed in experimentation for stable training.\n- Computational cost. Only briefly mentioned time per epoch in experiment 3. While it makes sense that NPE-UDA and NPE-MMD would have minimal impact on runtime, I wouldn't mind seeing a touch more concrete results on this across factors such as: training time, inference time, if we might expect more simulations to be required, if hyperparameter search is expected and would lead to increased overall computational cost.\n- Model building. The paper does not really cover the case where the simulator is inadequate (fails to reproduce key features of observed data that the modeller cares about). It remains a bit unclear at which stage NPE-DANN (or a more general workflow) would flag such misspecification and what the modeller should do next (e.g., modify the simulator, redesign summary network, or abandon UDA).\n- Unresolved handling of prior misspecification. The authors clearly show that UDA can break down when the assumed prior is wrong. Perhaps more guidance could be given on how users should remedy such a mismatch. Perhaps out of scope for paper, but if the authors have any useful comments on what could be done (modeller should consult more general Bayesian literature on prior misspecification, not use UDA, switch (or combine) with alternative robust SBI approaches?).\n- Novelty. The methodology is not highly novel, given NPE-MMD (Huang, 2023) and NPE-UDA (Swierc, 2024). That said not an issue for TMLR, especially given the useful insights and updates from the extensive experimentation here.\n- On robust methods in experimentation. I wonder why the NNPE from Ward's 2022 paper was used here, rather than the RNPE from the same paper? Also wonder if amortised cost estimation would be worth comparing against.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper investigates unsupervised domain adaptation (UDA) applied to amortised simulation-based inference, specifically by making neural posterior estimation (NPE) more robust to model misspecification.\nIt contributes a novel robust approach for amortised Bayesian inference, NPE-DANN, that augments the standard NPE loss with a domain-adversarial classifier.\nThe paper also re-examines NPE-MMD, extending it to be amortised and to include alternative hyperparameter choices.\nThe authors explain and empirically verify that UDA implicitly learns a denoised posterior, which helps under likelihood misspecification, yet can worsen performance under prior misspecification.\nFinally, experimentation was done on four examples that show NPE-DANN and NPE-MMD achieve good performance in well-specified and misspecified (likelihood) settings, supported by a thoughtful selection of evaluation metrics and clear justification. Open questions and obstacles to using NPE-DANN are addressed.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall, this is an enjoyable and well-written paper. While some open questions and obstacles remain for the use of NPE-DANN, the authors acknowledge these limitations and address them well. The manuscript is interesting and well-executed, and in my view is appropriate for TMLR as is.\n\nSome optional changes following weaknesses:\n- Some more results or details on computational costs - while it makes sense that runtime increases minimally, it might still be good to see some more empirical results or justification. \n- Regarding \"a closer examination of UDA mechanisms is necessary to determine the nature of domain adaptation\", while interesting future work given, is there anything people can do right now for this (e.g., posterior predictive checks)?\n- Any remaining points in weaknesses not already clear in paper.\n\nSpelling and grammar:\n- p7. \"intervalst\"\n- p8. Experiment 5.1 \"uncontamined\"", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bONggXMivl", "reviewer": "Reviewer_XfNk", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well written although 1) very heavy on acronyms (I'm not sure this is avoidable but a table of them might help) and 2) somewhat on the long side- there isn't content I would obviously suggest moving to an appendix but I would encourage the authors to do an editing sweep for brevity/clarity. Figure 1 is a nice exposition of NPE-MMD and NPE-DANN. Maybe I missed it but was NNPE not really described? A couple of sentences might be worth it to make the manuscript more self-contained. \n\nI don't have any substantial technical criticisms of the manuscript. I would have been interested to see an application in population genetics since ABI is being used quite widely in that context. One could imagine training on a simpler demographic model (e.g., one that doesn't include migration) and asking if UDA improves performance estimating parameters like TMRCA for data simulated under a more complex model (e.g., including migration). However, given the length of the paper already this would probably be too much.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors empirically assess approaches to ameliorating the effects of the distribution shift inherent in Amortized Bayesian Inference (ABI), where simulated data is used to train a (neural network) map from data to the posterior over parameters. They primarily consider 3 approaches: noisy neural posterior estimation (NNPE), NPE-MMD and NPE-DANN. NPE-MMD includes a loss term penalizing distribution shift between embeddings of the true and simulated data. DANN uses an adversial approach vs a classifier that tries to distinguish true vs real data embeddings. Their key finding is that UDA can help with a mis-specified likelihood (e.g. accounting for unmodeled noise processes) but can actually hurt performance under a mis-specified prior, since the weakened likelihood has less signal to counteract the bad prior. They also explore/discuss the utility of different metrics for assessment.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I'd be interested to understand whether there is different behavior for prior mis-specification that is 1) too high variance vs 2) too low variance. Intuitively one would expect that using relatively broad/high variance/uninformative priors would be \"safer\", but presumably there is some trade-off.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)