Uncertainty-Aware Surrogate-based Amortized Bayesian Inference for Computationally Expensive Models
Authors: Stefania Scheurer
Philipp Reiser
Tim Brünnette
Wolfgang Nowak
Anneli Guthke
Paul-Christian Bürkner
Venue: Accepted by TMLR ()

Abstract: Bayesian inference typically relies on a large number of model evaluations to estimate posterior distributions. Established methods like Markov Chain Monte Carlo (MCMC) and Amortized Bayesian Inference (ABI) can become computationally challenging. While ABI enables fast inference $\text{\emph{after}}$ training, generating sufficient training data still requires thousands of model simulations, which is infeasible for expensive models. Surrogate models offer a solution by providing $\text{\emph{approximate}}$ simulations at a lower computational cost, allowing the generation of large data sets for training. However, the introduced approximation errors and uncertainties can lead to overconfident posterior estimates. To address this, we propose Uncertainty-Aware Surrogate-based Amortized Bayesian Inference (UA-SABI) -- a framework that combines surrogate modeling and ABI while explicitly quantifying and propagating surrogate uncertainties through the inference pipeline. Our experiments show that this approach enables reliable, fast, and repeated Bayesian inference for computationally expensive models, even under tight time constraints.

Published in Transactions on Machine Learning Research (01/2026)
Uncertainty-Aware Surrogate-based Amortized Bayesian In- ference for Computationally Expensive Models
Stefania Scheurer"
stefaniia.chheurrekkuuss.uc wtw.://iwisuuiiitttgtrr......
Department of Stochastic Simalation and Safety Research for Hydrosystems
Cluster of Excellence Sim Tech
University of Staligart
Philipp Reiser
Cluster of Excellence Sim Tech University of Stuttgert
Tim Brünnette
tim, bwwennetteeWess wni-ssullgra..rd.ee
Depariment of Stochastic Sinvalation and Sufely Research for Hydrosystems University of Stuligart
Wolfgang Nowak
wol.gangg.nounkkBuuss.. wtw.::/wwwwoouu.ttttttttrr......
Deportment of Stochastic Simulation and Safety Research for Hydrosystems Cluster of Excellence Tech University of Stuligart
Anneli Guthke
Chuster of Excellence Sim Tech University of Stutigert
Paul-Christian Bürkner
www.bbucrkuer@@to-oorrmmaaal de
Depariment of Statistics TU Dortmund University
Reviewed on OpenReview: https: // openreview. net/ forum? ad=aVSoQxtfyl
Abstract
Bayesian inference typically relies large number of model evaluations to estimate pos- terior distributions. Established methods like Markov Chain Monte Carlo (MCMC) and Amortized Bayesian Inference (ABI) can become computationally challenging. While ABI enables fast inference after training, generating sufficient training data still requires thou- sands of model simulations, which is infeasible for expensive models. Surrogate models offer lower computational cost, allowing
solution by providing approrimate simulations the generation of large datasets for training. However, the introduced approximation lead to overconfident posterior estimates. To address this,
rors and uncertainties propose Uncertainty-Awaro Surrogatc-based Amortized Bayesian Inference (UA-SABI) framework that combines surrogate modeling and ABI while explicitly quantifying and prop- agating surrogate uncertainties through the inference pipeline, Our experiments show that this approach enables reliable, fast, and repeated Bayesian inference for computationally expensive models, even under tight time constraints.
"These authors contributed equally this work.

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Published in Transactions on Machine Learning Research (01/2026)
1 Introduction
Mathematical models are essential for simulating real-world processes, typically mapping parameters to observable data. Estimating these parameters from real-world observations, ie., the inverse problem, is fundamental across scientific disciplines (e-g. Etz & Vandekerckhove, 2018; Von Toussaint, 2011; Hülsenbeck et al., 2001; Ellison, 2004). However, because models never perfectly capture reality and observed data are often sparse and improcise, parameter estimation inherently involves uncertainty. Bayesian inference offers a systematic framework for estimating parameters while incorporating uncertainty in a statistically grounded manner (e.g. Gelman et al., 1995).
Markov Chain Monte Carlo (MCMC) methods are widely used for Bayesian inference to generate high- quality samples from the posterior distribution given fixed observations (Gilks et al., 1995). However, MCMC is computationally expensive and slow, rendering it impractical in scenarios where near-instant inference is required (Robert et al., 2018). Near-instant inference is, for example, necessary in adaptive robotic control or closed-loop medical devices, where parameters such as object mass or patient sensitivity must be estimated immediately to allow accurate prediction and safe action (e.g. Marlier, 2024; Tasoujian et al., 2020; Malagutti et al., 2023). Furthermore, cach new set of obscrvations requires restarting the entire process, further increasing costs when inference is needed for multiple datasets. Multiple inference runs may be needed for ongoing adaptation conditions evolve or more data become available. Tracking the spread of discase such as influenza or COVID-19 is one example of this. New case data continuously arrive, and separate datasets exist for different regions, requiring repeated Bayesian updates to infer transmission rates or reproduction numbers for each location (e.g. Radev et. al., 2021; Yang et al., 2015). MCMC also relies on a known likelihood function that can be evaluated either analytically or numerically.
Amortized Bayesian Inference (ABI), a dcep-learning-basee based approach originating from simulation- based in- ference (SBI), addresses these linitations (Crammer et al., 2020; Radev et al., 2020; Lückmann et al., 2021). SBI methods are typically employed when evaluating the likelihood is infeasible, but simulations from the model are possible, allowing learning mapping from observed data to the posterior of the model parameters via simulated data. ABT in particular leverages generative neural net to learn this mapping. Training data is generated through multiple model evaluations. Once trained, ABI enables near-instant posterior in- ference for new datasets, as the computational cost is incurred during training. Being likelihood-free makes ABI well-suited for complex problems involving noisy or nigh-dimensional data, where evaluating the like- lihood is infeasible unreliable. However, the learned posterior only approximates the true posterior, and high accuracy requires well-designed network architectures and extensive training data.
Both MCMC and ABI face limitations when the simulation model is computationally expensive, since both require a large number of model evaluations. MCMC requires many likelihood evaluations per generated pos- terior sample; ABI requires extensive model simulations to generate sufficient amounts of training data. As a result, if computational time is limited, oblaining a good posterior becomes infeasible for both approaches. Note that such computational effort can arise for different reasons for instance, from a high-dimensional parameter space or from the complexity of the governing equations. In Earth sciences, for example, low- dimensional models can be computationally very expensive, particularly if they involve (coupled systems of) non-linear partial differential equations or long simulated time periods (e.g. Mohammadi et al., 2018; Hommel et al., 2015), resulting in runtimes up to multiple days for a single model run.
Our goal is to enable ABI for computationally expensive models to benefit from the aforementioned ad- vantages that ABI offers. To this end, surrogate models are the crucial tool, allowing us to reduce the computational cost of expensive simulations for training data generation. Surrogate models (Sudret, 2008; Rasmussen & Williams, 2006), however, are only approximations (i.e., imperfect representations) of the ref- erence simulation, and therefore not necessarily reliable. It is crucial to quantify the uncertainties associated with surrogate modeling, such as thosc arising from limited training data or any inherent inflexibility of the surrogale. These uncertainties then need to be propagated through the inference pipeline, ensuring consis- tent posterior estimation. Previous methods have integrated these uncertainties into MCMC methods with surrogate models (o.g. Lückmann et al., 2018; Zhang et al., 2020; Reiser et al., 2025). However, this requires many additional MCMC runs to incorporate the surrogate uncertainty, largely eliminating the computational advantages of surrogates.
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We introduce -ncertainty-Awwre Surrogate-based Amortized Bayesian Inference (UA-SABI), which enables ABT for computationally expensive models while accounting for the uncertainty inherent in surrogate approx- imations. By leveraging a surrogate, we reduce the training error that arises from insufficient data during ABI training, as simulating from the surrogate easy and fast. This comes at the cost introducing a surrogate approximation error. However, unlike the ABI training error, the surrogate approximation error can be quantified and propagated, enabling reliable ABI for computationally expensive models.
For this purpose, we adapt and further develop the uncertainty estimation and propagation method for surrogate modeling proposed in Reiser et al. (2025). Our main novelty is to replace the MCMC with ABI for inference. This novelty and its benefits can be viewed from two perspectives: 1) Replacing MCMC with ABI substantially improves sampling efficiency, as parallel chains for draws from the surrogate posterior are no longer required. 2) UA-SABI enables ABI training with surrogate data while accounting for the uncertainty inherent in the surrogate approximation. This significantly reduces the required runs from the expensive simulation model while retaining the amortization property and, therefore, enabling ABI for computationally expensive models.
An illustration of the workflow, including training phases and inference, is given in Fig, 1. We demonstrate the workflow and substantiate the claimed advantages both theoretically and practically with toy example and two real-world case studies.
Simulation
Sparse
train
Uncertainty- Aware Surrogate Model
generate
ABI Training Data
train
UA-SABI Model
infer
Model
Model Data
Parameter
Posteriors
Surregate Training Phase
ABI Training Phase
Double Amortization through Repeated Inference
Figure 1: Illustrative workflow of UA-SABI training and inference.
2 Methods
2.1
Amortized Bayesian Inference (ABI)
This work focuses on using ABI to infer the parameters of computationally expensive simulation models. The typical ABI procedure (Radev et al., 2020) can be divided into training phase and an inference phase. Training includes simulating data and then training a neural posterior estimator (NPE) on that data (Papamakarios & Murray, 2016; Durkan et al., 2019; Greenberg et al., 2019b).
ABI Training Data Generation Training data is generated using simulation model M = M w), typically a realistic first-principles model. M depends on inputs x and parameters w and yields an output y. Additionally, it can contain stochastic parts omitted in the following to simplify the nolation. To learn the mapping from observations to posterior draws, a training set DB = {xx(1, w(()) (()) of NB samples must first be generated. This can be performed by simulating NB pairs of parameters, inputs, and outputs from the simulation model:
y( M(x((),wl) with (x(9) w))) p(x,w).
(1)
Neural Posterior Estimation After creating the training dataset, the NPE, including summary net- work Se(x, y) and an inference network I (s), must be trained. The goal of 5x(x,y) y) is to embed (stacked) observations, i.e., model inputs and outputs yx of polentially variable length (e.g., a measurement series) in a fixed-length vector s that serves as input to (s). Then, (s) generates samples from the approximate posterior 96 (w s). Here, and &p are Iearnable coefficients Sg(x, y) and (s). In our case studies, we use coupling flows (Papamakarios et al., 2021) as inference networks. Coupling flows have been empirically and theoretically shown to be expressive and allow for fast evaluation and sampling (Draxler et al., 2025).
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However, ABI is generally flexible in its choice for the inference network and works with any generative neural network (e.g. Ardizzone al., 2019; Padmanabha & Zabaras, 2021; Denker et al., 2021; Wildberger et al., 2023; Schmitt et al., 2024b). Thus, the summary and inference networks are modular and can be independently replaced or adapted to suit the specific problem at hand.
The optimal parameters of the summary and inference network (0", (0":"") are obtained jointly by minimizing the Kullback-Leibler (KL) divergence between the true posterior p(w x, and the approximate posterior 9p (w s) (Radev et al., 2020):
(0" ,4') argmin Bp(x,y) [KL(p(w x, |
( |s]]
(2)
0.4
argmin E (pykyyy
s)]l
(3)
0.
/|| p(x,y. log 1o S)( dx dw.
argmin
(4)
0.
According Radev et al. (2020), the expectations can be approximated with the Monte Carlo estimates, utilizing simulations {(x(1) w(),, (()) No from the (potentially expensive) simulation model (1). This results
in the following loss function:
NB
L(4,
Σ
log (w)
(0(x()),y()))
(5)
2.2 Uncertainty-Aware Surrogate Modeling
Uncertainty- surogate modeling is introduced in and adapted from Reiser et al. (2025). A surrogate model M aims to approximate the behavior of a simulation model M with negligible computational cost. The surrogate model is typically parametrized by learnable surrogate coefficients, denoted as c. The simulation model output can then be approximated by
y M(x, w) M)(x =
(6)
However, due to limited simulation data from M used for training Mc. the surrogate coefficients only be estimated with substantial epistemic uncertainty. For the same reason, the expressibility of Me must be restricted, ie., the complexity of Me must be kept relatively low. This introduces an approximation error, denoted as c, which captures the misspecification of the surrogate. In practice, one usually has to expect a non-negligible approximation error due to limitations in model capacity, optimization challenges, or regularization constraints. We assume the approximation error 6 p(e a) to follow a distribution parametrized by o, leading to the error-adjusted surrogate output
yc = f(),c) with E p(c ())
which is essentially a perturbed version of y. This setup defines the surrogate likelihood
p y, 0) = p(ve xx c,
(8)
that approximates the (usually unknown) likelihood of the simulation model.
Using sparse training data (Ny < NB) obtained from the simulation model Dy = ((x(0) (.)(())))))))))))))))) )
with y+)) = (x(0), w)) and (x(6), (+)) sampled from a given prior, we infer the surrogale model coefficients and a via Bayesian inference. This enables capturing both the surrogate model's epistemic uncertainty and the irreducible approximation error. For this purpose, consider the likelihood equation 8 and the prior p(c, The joint posterior over the surrogate parameters and the approximation noise, given training data Dr from the simulation model M, is then
p Dr)
(() x()) (), c.),c.o)pcc...
(9)
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assuming an i..d. sampled approximation error. This assumption serves simple stand-in, since surrogate construction is not the focus of this work. It suffices to remonstrate that even a basic error model outperforms making no error assumption at all. In general, CA-SABI is not limited to the i.i.d. case; heteroscedastic surrogate error models (e.g. Kohlhaas et al., 2023) are fully compatible with the method.
The conditions under which M is considered optimal are implicd by the assumed distributions of x and w in Dr and the specified error model p(e | a). Sampling-based algorithms such as MCMC can be used to estimate the surrogate posterior equation 9. For the considered surrogates, this is tractable since their likelihoods are fast to evaluate, and the surrogate posterior compared to model parameter posteriors across datascts estimated only once. While this formulation fully captures surrogate uncertainty, we now temporarily simplify the setup by omitting this uncertainly to introduce an uncertainly-unauuer baseline method - Surrogate-basee ABI. This allows us to isolate and demonstrate the benefit of propagating surrogate uncertainty through to inference.
2.3
Surrogate-based ABI (SABI)
The quality of the NPE depends sufficient training budget Dg to ensure that p(w x, is well approximated via (p(w] (x,)) However, for the expensive models considered here, generating sufficient ABI training data becomes infeasible. To mitigate this, we propose using surrogate model of the expensive simulator to esficiently generate sufficient training data. Surrogate-based ABI (SABI) thus differs from standard ABI only in the generation of training data. Instead of using the expensive model, a (point) surrogate model with a point estimate e used to generate the training data. DB = ((x(1, (000),))))))
y M) (!) with (x(1),w(() p(x,w).
(10)
For the point surrogate, the median of the posterior in Eq. (9) can be used it is robust to outliers and invariant under monotonic transformations, but also the mean is a valid option.
However, in general, Me is only an approximation of the simulation model M. Therefore, the SABI posterior trained with surrogate data p(w] x, x) will be systematically different from the true posterior p(w x, y); as no surrogate uncortainties or errors are considered in the ABI training process. Specifically, neither the epistemic uncertainty of Mc. nor its irreducible approximation error propagated. As we show in our experiments, this lead to highly inaccurate posterior approximations.
2.4 Uncertainty-Awar Surrogate-bar based ABI (UA-SABI)
To ensure reliable inference, the uncertainty of the surrogate model needs be propagated through the ABI training and inference procedure, illustrated in Fig. 1. This results in our proposed Uncertainty-Aware Surrogate-based Amortized Bayesian Inference (UA-SABI) approach.
Instead of generating training data with the point surrogate model, UA-SABI utilizes the uncertainty-aware surrogate model:
y p(Se (0) (() c(1) ,(0)((())) with 1x(0,w()) ~ p(x, o ~ p(c, a Dr).
(11)
To propagate uncertainty to the ABT training, now additionally draw the surrogate coefficients e() and the approximation error parameter g( from the surrogate posterior equation 9. These coefficients then used to evaluate the surrogate model and oblain (00, The pair (((()) is subsequently used to generate a sample y(9) from the distribution of the error-adjusted surrogate output. We now approximate the KI. divergence in Eq. (5) as:
No
L( 0) = Σ
log uo (w | (x(x)),)))))
(12)
Figure gives a detailed graphical overview. Additionally, pseudocode summarizing the UA-SABI training procedure can be found in Appendix A.
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Stmulate Simulation Model (M) Inputs x and Parameters a² Dy (xx00 w y )
Simulate Surrogate Model (M,i) 2n)) Data from Inputs x and Parameters a)
Correct Surrogate Output 8 p...18.0 {(x (1 0 50 )
Sample Surrogate
Calculate Summary Statistics
Choose Prior on Surrogate Parameters (C., d)
[[e()) (0)) Ma 1i= from Posterior
Parameters
Choose Likelihood for Model Surrr//Suroggtt p c,
Compute Loss
No
Sample Inputs and Parameters
((p,, 0) (- 2 (c00)/((0))
((x(0, (1) )
with ap (ts ( (1)) implied (p(5(1)
Train via Bayesian Inference (MCMC)
Update (g.0)
Surrogate Training
AB] Training
Epochs
Choose Likellhood on Surrogate Approximation Error 9,- p(.18.
Uncertainty-Awwe Awwrre Surrogate Model M, with Posterior for Surrogate: Parameters Wr. (( w.. c.. o o((.o)
Choose Prior on Inputs Parameters p(X, d))
UA SABI Model
p(c,a|Dy)
Figure 2: Detailed graphical overview of UA-SABI training as implemented in Algorithm 1. First phase: training of uncertainty-awarr surrogate model using sparse simulation data, yielding posterior over surro- gate parameters. Second phase: ABI training using sarogate--genrrate data, where full surrogate uncer- tainty is propagated through sampled surrogate outputs.
3 Related Work
Multi-Fidelity Approaches in SBI Multi-fidelity methods leverage models or simulations of varying accuracy and cost to accelcrate inference, combining cheap, approximate simulations with a smaller num- ber of expensive, high-fidelity runs. In Approximate Bayesian Computation (ABC), such strategies reduce computational burden: Warne al. (2018) apply multi-level Monte Carlo variance reduction to rejection sampling in ABC, improving efficiency. Prescott & Baker (2020) introduce multi-fidelity ABC approach that combines early acceptance and rejection of parameter samples and selectively runs high-fidelity simula- tions based on low-fidelity outputs, balancing computational cost and accuracy. Building on this, Prescott & Baker (2021) integrate the multi-fidelity strategy with sequential Monte Carlo sampling, further enhancing efficiency by more effectively exploring the parameter space for MF-ABC proposals. More recent work by Warne et al. (2022) integrates low- and high-fidelity simulations within a multilevel Monte Carlo framework to accelerate SBT for partially observed stochastic processes, using inexpensive approximations preselect parameters and high-fidelity simulations refine posterior estimates, while Prescott et al. (2024) dynamically select between low- and high-fidelity models based predictive accuracy, optimizing resource allocation. In multi-fidelity, simulation- based Inference (MF-SBI) (Krouglova et al., 2025) neural posterior estimator is first pre-trained on large amounts of low-fidelity data and then finc-tuned with smaller set of high-fidelity simulations. While sharing similar goals, none of the aforementioned methods explicitly take the uncertainly of the cow-fideitt simulations into account. UA-SABI's incertainty-awareenn in regions with scarce data- protection which appears not be in place for the low-fidelity training re- gions in MF-SBI, for example. Further, unlike many of the -ideiiy methods, UA-SABI does not require domain expertise for the construction of the surrogate.
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Sequential Neural Posterior Estimation (S-NPE) S-NPE iteratively refines posterior estimates by adaptively choosing simulations that are most informative, improving efficiency. Papamakarios & Murray (2016) use initial simulations to fit a Bayesian conditional density estimator for the posterior and then sample parameters from the current posterior estimate to guide the next round of simulations. Similarly, Toni et al. (2009) introduced Sequential ABC (ABC-SMC), which iteratively refines posterior distributions by progressively tightening acceptance thresholds. Kulkarni & Moritz (2023) build on ABC-SMC, leveraging massively parallel architectures, such as GPUs, to more efficiently explore the parameter space. Lueckmann et al. (2017) sequentially train Bayesian mixture-density network. Greenberg et al. (2019a) developed Automatic Posterior Transformation, which employs flexible flow-hased density estimators and dynamically updated proposal distributions. Deistler et al. (2022) introduce truncated S-NPE, restricting simulations to truncaled regions around high-probability areas. However, UA-SABI and S-NPE pursue related but distinct goals: UA-SABI focuses on fully amortized inference, enabling fast posterior estimation for new datasets through one-time training, whereas S-NPE sequentially refines the postcrior on the same datasct to improve accuracy and sample efficiency across rounds. Additionally, the referenced S-NPE procedures implement an active learning strategy for simulation selection, assuming data are generated iteratively, while UA-SABI can also operate pre-existing simulation datasets (offline training).
Gaussian Process Surrogates in SBI Several methods employ Gaussian Process (GP) surrogates to improve sampling efficiency in likelihood-free inference. Meeds & Welling (2014) construct GP to emulate the expensive simulator and use its predictive mean and uncertainty to estimate the acceptance probability in ABC, avoiding simulator calls when the surrogate is sufficiently confident. Wilkinson (2014) fit GP to the ABC log-likelihood to prune implausible parameter regions before sampling, greatly reducing simulation cost. Bayesian Optimization likelihood-free inference (Gutmann & Corander, 2016) fits a GP to the discrep- ancy between simulated and observed data and uses Bayesian optimization to adaptively choose simulator parameters, concentrating effort on informative regions and enabling accurate posterior estimation with few simulations. As in UA-SABI, these approaches apply surrogates, focusing specifically on GPs. However, they employ the surrogate to intelligently select simulation parameters for subsequent likelihood-free inference, whereas UA-SABI uses the surrogate itself to directly generate training data and is agnostic to the surrogate model class.
Structure-Aware SBI Methods Several computational strategies exploit the structure of simulators or information about the inference problem to improve efficiency in SBI. Compositional simulation- inference for time series (Gloeckler et al., 2025) breaks long, high-dinensionall time series into local transi- tions, performing inference on each step and composing the results to recover global posterior. Cost-aware sinulation- inference (Bharti et al., 2025) tackles heterogencous simulation costs by prioritizing cheaper and more informative simulations using a combination of rejection and importance sampling, introducing a cost function to guide allocation of simulation resources. In contrast to UA-SABI, structure- aware methods exploit knowledge about the simulator, whereas UA-SABI treats the simulator as a black box and requires no such knowledge.
Neural Likelihood Estimation Neural likelihood estimation is a likelihood-free inference approach that models the likelihood function directly. Sequential neural likelihood (Papamakarios et al., 2019) uses auto- regressive neural flows and focuses simulations on high-posterior regions for efficient inference. Bayesian synthetic likelihood (Price et al., 2018) approximates the likelihood of summary statistics with a multivari- ate normal, incorporating their uncertainty to provide robust and accurate inference. Jointly Amortized Neural Approximation (JANA) (Radev et al., 2023a) combines neural likelihood and posterior estimation. In addition to learning neural posterior, it simultaneously learns neural likelihood to enable marginal likelihood estimation, which is essential for tasks such as Bayesian model selection. However, its objective differs slightly, as it aims to learn both the likelihood, serving as surrogate model, and the posterior jointly. Such a highly parameterized neural likelihood requires a sufficient amount of training data (Frazier et al., 2024). While UA-SABI also requires simulation data to train a surrogate, it achieves effective inference with far fewer simulations compared to neural likelihood estimation, making it more practical for very expensive models.
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Model Type
Point
UA-
True
Surrogate
Surrogate
Inference Method
MCMC
Infeasible
Point
E-Post
Foll/Low Budget
ABI
SABI
UA-SABI
Figure 3: Context of UA-SABI and its alternatives under tight computational constraints.
Low-Budget Simulation-based Inference with Bayesian Neural Networks Delaunoy et al. (2024) also locus on the challenge of limited training data in ABI. They employ Bayesian Neural Networks (BNNs), which model uncertainty via distributions over network parameters. However, selecting an appropriate distribution and defining a mcaningful prior for weights and biases remains speculative and can undermine posterior reliability, as noted in (Delaunoy et al., 2024). Moreover, BNNs still demand substantial training data and compute resources (-25,000 GPU hours in Delaunoy al. (2024)), thus only shifting computational resources from evaluating expensive simulation models to training BNNs.
Uncertainty-Aware Surrogate-Based Inference Reiser et al. (2025) consider the setting, re- placing a computationally expensive model with uncertainty-aware surrogate. However, their inference approach differs: they run MCMC separately for each observation set and for every surrogate sample. This results in a computational cost that scales with both the number of observations and the number of surrogate samples, making their method significantly more expensive. Their proposed method, E-Post, marginalizes over the surrogate coefficient posterior p(c,a a Dr) to obtain the posterior of the parameters of interest w:
p(w y) ff plw v,c,o) dedo.
(13)
In practice, this integral is approximated via Monte Carlo integration with sufficient number of samples from the surrogate posterior p(c, a Dr).
During inference with UA-SABI, this explicit marginalization is not required, as the NPE is trained over the entire posterior space of both the outputs and the surrogate parameters. This results in a form of double amortization of the surrogate and ABI training costs through repeated inference. Since UA-SABI and E-Post share the same overall objective and framework, E-Post with MCMC inference serves as benchmark for UA-SABI. A formal proof of asymptotic equivalence is given in Proposition 1 in Appendix B.
4
Case Studies
We evaluate and benchmark UA-SABI against its uncertainty-unawer counterpart SABI, the corresponding MCMC-based methods, and (if possible) standard ABI using a toy example and two real-world case studies. Code is publicly available on GitHub'.
4.1 Objectives
We aim to validate the parameter posterior obtained with UA-SABI by assessing its quality and justifying the computational effort required to train the surrogate by reducing the effort to train the ABI model. Our central rescarch question is "Can we train ABI with surrogate data, do we have to, and, if so, how should we account for the uncertainty introduced by the surrogate?". To keep focus on that, we deliberately remove unrelated complexities. Specifically, in our real-world case studies, we use computationally expensive but low- dimensional problems. High-dimensional models present significant challenges for surrogate training, as they
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typically require substantially more data to adequately capture the model's behavior. Additionally, inference on high-dimensional problems becomes more complex due to multi-modality and non-identifiability issues that complicate posterior estimation. This would make result interpretation less transparent and potentially obscure insights about the proposed approach itself.
We want to show the advantage of using surrogate data instead of scarce simulation data, and demonstrate the importance of quantifying and propagating the uncertainty of the surrogate model. To do so, we compare UA-SABI, SABI, and standard ABI for the true model trained with full and low budget (shown in blue in Fig. 3). To validate the posteriors produced by these methods, we also compare them with those obtained using the corresponding MCMC-based approach (shown in purple in Fig. 3). Further, to highlight the efficiency of our method, we compare the runtimes of UA-SABI and E-Post (shown in red in Fig. 3).
To assess posterior calibration for our first two case studies, we generate multiple syuthetic ground truth parameters and perform inference on the simulated datasets. For cach datasct, we compute the rank of the ground truth within the posterior samples and summarize the results using empirical cumulative distribution function (ECDF) difference plots procedure known simulation- calibration (SBC) checking (Talts et al., 2020; Säilynoja et al., 2022; Modrák et al., 2023). SBC assesses whether the inference method is calibrated by verifying that the rank of the true parameter among posterior samples follows a uniform distribution across repeated simulations. To compare runtimes, we estimated how many observation sets are needed before the accumulated computational cost of repeated E-Post inference runs exceeds the total cost of UA-SABI, including both training and its repeated (quasi-instant) inference.
General Setup For all case studies, we use polynomial chaos expansion (PCE) for the surrogate model (Wiener, 1938; Sudret, 2008; Oladyshkin & Nowak, 2012), as it is fast-to-ovaluate and hast-to-train surrogate and a flexible black-box model which can be applied to (almost) arbitrary simulation models. In principle, PCE is just one example among many possible choices any uncertainty-aware surrogate could be used and replaced in a modular fashion depending on the application domain. A deterministic PCE constructs the surrogate through a spectral projection onto orthogonal (w.r.t. p(x,w)) polynomial basis functions, expressed as
Mc(x, w) Σ oj W)( ==0
(14)
with L {w,) being the multivariate orthogonal polynomial basis and {cy15mm the corresponding coefficients. The number of expansion terms J is computed via the slandard (runcalion scheme (Sudret, 2008). For our case studies, we choose relatively low polynomial degree, as we operate in a data-scarce regime and aim to avoid overfitting. In the real-world case studies, the underlying models are highly compu- tationally expensive, so a higher polynomial degree would require additional model runs that could exceed the available computational budget. In general, selecting the appropriate surrogate complexity is important but challenging to determine a priori, as it depends on the specific application.
To construct Bayesian PCE (Shao et al., 2017; Bürkner et al., 2023), we define prior distribution over the surrogate coefficients p(c) and the approximation error parameter p(a). We choose normal likelihood for the simulator output y = (x)
ply x,w =
Me(x, o).).
(15)
Employing Hamiltonian Monte Carlo (Hoffman & Gelman, 2014) yields samples from the surrogate posterior p(c, Dr) (Bürkner et al., 2023). The resulting Bayesian PCEs serve our uncortainty-aware surrogate.
4.2 Case Study 1: LogSin Model
We first evaluate our surrogato- basod ABI approaches in a simple synthotic scenario: simulation model with one parameter and one-dimensional input x::
y Mxx w log(x) + (0.00)) 0.01x + 1.
(16)
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4.2.1 Setup
We generate surrogate training data by evaluating the simulation model M at the first 16 points of a two-dimensional Sobol sequence (Sobol', 1967), scaled to [00] for x and [0.6, 1.4] for w. The resulting input-output pairs are used to train a Bayesian PCE. This introduces significant surrogate uncertainty and approximation error; as a perfect match to the simulation model is unaltainable. The surrogate's posterior is sampled using HMC via Stan (Carpenter et al., 2017; Stan Development Team, 2024). We employ 4 chains, each with 1,000 warm-up and 250 sampling iterations, yielding 1,000 surrogate posterior samples in total to be propagated.
To perform inference on observation sets, we sample 200 ground truth parameters w from prior = = (11, 0.2) and generate four y) observations for each. We compare the inference results of our pro- posed surrogate- based ABI to surrogate-base summary network (Zaheer et al., 2017), given p(y x) being ii.d.,, and a coupling flow inference network. We employ online training, where newly sampled surrogate outputs are used at each iteration. All ABI models were trained for 100 epochs, with a batch size of 64 and 128 batches per epoch, using BayesPlow (Radev et al., 2023b). For given observations, ABI generates 4,000 posterior samples via the inference network. Surrogate-based MCMC (Point) draws 4,000 samples using 4 chains (1,000 warm-up and 1,000 sampling iterations). For E-Post, we run MCMC separately for each surrogate posterior draw (1,000 warm-up, 4 sam- pling iterations per run), a process that is embarrassingly parallelizable but still computationally expensive. Further computational details are given in Appendix C.
MCMC methods. For ABI, we used an equivariant DeepSet
4.2.2 Impact of Uncertainty Propagation
First, we compare the performance of our two surrogate-basod methods, SABI and UA-SABI, against cach other. Then we compare them against low-budget ABI, trained with the same simulation data as used for surrogate training, and a full-budget ABI trained on sufficient simulation data. Figure 4a shows corre- sponding recovery plots. They show the posterior median (circles) and median deviation (vertical lines) for any given ground truth value w.
Figure 4a shows that a low-budget ABT model unable to recover the ground truth values, particularly towards the boundaries of the parameter domain. Using a surrogate as part of SABI or UA-SABI, it becomes possible to generate sufficient ABI training data. Assigning the inter- and extrapolation tasks to the surrogate, specifically designed for this purpose, produces better outcomes than relying on the NPE to implicitly interpolate the posterior from scarce data. However, without propagating surrogate uncertainty (i.e., using SABI), estimated posterior uncertainty remains minimal and does not cover the true values, thus producing overconfident posteriors. Conversely, incorporating surrogate uncertainty via UA-SABI leads to an incrcase in postcrior uncertainty, now better covering the true values. Indcod, the ECDF difference plots showing calibration in Fig. 4b confirm that both low-budget ABI and SABI produce severely miscalibrated posteriors, while UA-SABI and full-budget ABT yield well-calibrated posteriors. However, achieving this calibration with full-budget ABI requires substantially more training data, highlighting the efficiency of UA-SABI under limited simulation budgets.
4.2.3 Validation of Parameter Posterior
We validate the correctness of the surrogate-based ABT approaches and benchmark their performance against a reference solution. To this end, we compare the two surrogate-based ABI methods, SABI and UA-SABI, against the corresponding surrogate-based based MCMC methods, Point and E-Post. We plot the posterior medians (circles) and their median deviations (lines) for the ABI methods along with the corresponding MCMC results, as well as the ECDF differences for all the methods in Fig. 5.
When comparing the ABI posteriors to the MCMC full reference solution, we observe that for the uncertainty- unaware methods, the SABI results align closely with the MCMC counterpart. Comparing the respective ECDF differences it shows that both Point and SABI produce similarly overconfident posteriors. For the uncertainty-aware methods, E-Post and UA-SABI, we observe a slight difference in the recovery plots and the ECDF differences. To investigate this discrepancy, we performed convergence analysis for the MCMC
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Figure 4: LogSin recovery plots (top) and ECDF difference plots (bottom) for full-budger ABI, low-budget ABI, SABI, and UA-SABI (from left to right) over 200 ground truth samples. In the ECDF difference plots, empirical ranks are shown in blue, 95% confidence bands assuming calibration are shown in grey.
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Figure 5: LogSin recovery plots (left) and ECDF difference plots (right) comparing ABI methods to corre- sponding MCMC methods over 200 ground truth samples. In the ECDF difference plots, empirical ranks are shown in blue, 95% confidence bands assuming calibration are shown in grey.
sampling of E-Post, with results shown in Appendix D. The analysis confirms that the difference can be attributed to non-convergence of many MCMC runs. Fxamining the calibration results reveals that while E-Post is slightly underconfident,, UA-SABI yields well-calibrated posteriors. This indicates the reliability and correctness of UA-SABI.
4.2.4 Runtime Comparison
Next, we justify the computational effort required to train UA-SABI. Specifically, we aim to determine the break-even point that is, the number of observation sets (i.c., mcasurement scries) after which training UA-SABI and performing (quasi-instant) inference becomes more efficient than repeatedly rerunning E-Post. Beyond that point, the one-time training cost of UA-SABI is amortized. All experiments performed
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standard laptop equipped with an Intel Core 17-1185G7 CPU. For ABI, we consider both the upfront training phase and the inference time for the given sets as part of the total runtime. For F-Post, we measure the inference runtime while parallelized across 8 cores. Unlike for UA-SABI, E-Post runtimes depend on the number of cores used for parallelization.
In Fig. 7a, we show the runtime of UA-SABI and F-Post for (5. 10, 15, 20} inference runs. We observe that CA-SABI's runtime is nearly constant for the number of inference runs, while E-Post scales linearly. Based this comparison, UA-SABI is already justified after around 9 inference runs.
4.3 Case Study 2: Carbon Dioxide (CO.) Storage Model
In the second case study, we test our method on a virtual benchmark that represents a real-world problem. Specifically, we consider a CO2 storage benchmark (Köppel el al., 2019). In this lest case, a non-linear hyperbolic partial differential equation models the two-phase flow of CO2 in brine. describes CO2 injection, plume migration, pressure build-up, and the influence of uncertain porous medium properties in deep saline aquifer. Given the CO2 saturation as measurements, injection rate of CO2 (IR), relative permeability degree in the fractional flux function (PM), and porosity (PR) of the formation. This test case allows us to analyze the performance of UA-SABI in inference for hydrosystem models. The underlying physical model exhibits strong discontinuities at certain combinations of parameter values and space/time coordinates, which inherently limit the approximation performance of PCE-based surrogate models. However, this scenario provides an opportunity to test UA-SABI under challenging conditions where the surrogate model is strongly misspecified. Five separate Bayesian PCE models with pparsit--inducing priors (Bürkner et al., 2023) are trained for five different instants at days {20, 40, 60, 80, 100} and at fixed location 30m from the injection well. A similar setup was studied in (Oladyshkin et al., 2020). We show parameter posterior results for porosity, the most informative parameter of the CO2 storage model.
aim is to infer three parameters of interest:
4.3.1 Setup
In general, the setup for the CO2 storage model follows the same structure as that described in Section 4.2.1. The priors of the input parameters are given as IR and PR Beta(2.4,9)) (Köppel et al., 2019). For PCE training, we considered 64 Sobol sequence evaluations scaled to the input parameter prior ranges and constructed the polynomial basis with arbitrary polynomial chaos (aPC) based on the priors (Oladyshkin & Nowak, 2012; Bürkner et al., 2023).
(xxx10xx((BBBt(((( 2)), PM 2x Beta(1.25, 1.25)-2,
In contrast to case study 1, we used an offiine ABI training set, where the training data is pre-generated from 10a parameter prior draws and fixed during the training process, allowing for direct comparison with a standard full-budee ball-budget ABI trained on pre-generated simulation data. Further computational details are given in Appendix C.
4.3.2 Validation of Parameter Posterior
We compare the inferred posterior samples obtained from a full-budget ABI (N₂ = 10*), a low-budget ABI (Nn = 64), SABI, and UA-SABI. Full-budget ABI was feasible due to the already available data (Köppel et al., 2017; Oladyshkin et al., 2020). Therefore, recovery plots were chosen to validate the quality of our results relative to standard ABI (Fig. 6).
Recovery plots in Fig. 6a again show that low-budget ABI struggles to recover the ground truth parameter, particularly near domain boundaries. SABI also produces poor estimates and, moreover, fails to capture uncertainty in the inferred parameter posteriors. This overconfidence results in miscalibrated posteriors, confirmed by the two corresponding ECDF difference plots in Fig. 6b.
In contrast, UA-SABI performs comparably to full-budget ABI while accounting for the additional uncer- tainty introduced by using surrogate to generate training data. According to the ECDF difference plots, it produces a well-calibrated posterior for porosity. Additionally, recovery plots between ABI and MCMC for barsd methods and ECDF difference plots for MCMC-based methods are provided in Appendix E.
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Despite the model's discontinuities and the resulting surrogate misspecification, our method successfully quantifies and propagates surrogate uncertainty to the parameter posterior.
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Figure 6: CO2 recovery plots (top) and ECDF difference plots (bottom) for full-budget ABI, low-budget ABI, SABI, and UA-SABI (from left to right) over 200 ground truth samples. In the ECDF diffcrence plots, empirical ranks are shown in blue, 95% confidence bands assuming calibration are shown in grey.
4.3.3 Runtime Comparison
Also, for the CO2 storago model, we compare the runtimes, following the same approach as in Section 4.2.4. The experiments were run computing cluster with two AMD EPYC 7551 CPUs (votaling 64 physical cores) to speed up E-Post through parallelization.
Figure 7b presents the measured runtimes of UA-SABI and E-Post for (5, 6, inference runs, whereby E- Post was parallelized on 16 cores. We observe break-even point between 6 and 7 inference runs, indicating that UA-SABI becomes justified after 7 runs. Despite using more cores for E-Post, the training costs are amortized earlier for more expensive model.
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Figure 7: Comparison of runtimes for UA-SABI (training and inference) and E-Post (inference) with break-
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4.4 Case Study 3: Microbially Induced Calcite Precipitation Model
In this case study, we transition from benchmarks to realistic conditions by using real measurement data. Case studies and 2 introduced a toy example and a virtual benchmark, respectively. Case study 3 now focuses posterior inference from observed measurements, where the primary goal is to evaluate the ap- plicability of UA-SABI under real-world conditions. Specifically, in addition to simulation-to-surron misspecification, we encounter simulation-to-real and thus surrogate-to-real misspecification. A runtime comparison for the MICP model on synthetic data is added in Appendix F.5.
The underlying model predicts microbially induced calcite precipitation (MICP), MICP is a reactive trans- port process, here taking place inside a porous medium. It includes at least biofilm, calcite, and the unreactive solid matrix solid phases, water, and, in some cases, additional fluid phase such as gas. Relevant in the aqueous phase are dissolved calcium, inorganic carbon, and urea. The complete set of relevant species is detailed in Hommel et al. (2015). The process is mediated by S. pasteurii, a bacterium that produces the enzyme urease. Urease catalyzes the hydrolysis of urea into ammonia and carbonic acid. In aqueous solution, ammonia consumes hydrogen ions (H)), which raises the pH. The increase in pH alters carbonate equilibria: carbonic acid dissociates further, releasing additional H+ and carbonate ions. This raises the concentration of dissolved carbonate species. When caleium ions are present, they combine with carbonate ions and trigger the precipitation of calcite within the pore space. A schematic illustration of the relevant process during MICP is given in Appendix F.1.
The experiment is performed on a 61cm sand-filled column with a 2.54cm diameter. Bacteria are first injected from the bottom, followed by an overnight no-flow period to allow biofilm formation. Biofilm growth is then stimulated with a 24h substrate injection. Afterwards, two pore volumes of a 0.33mol//L calcium-urea solution are injected at 10ml/min, repeated every 24h. Each injection is followed by a no-flow period for mineralization, and then substrate injection to reactivate the biofilm (Hommel et al., 2015). This cycle is repeated 30 times. A more detailed description can be found in Hommel et al. (2015).
Mcasurements of the volume fraction of calcite on eight spatial points located at 3.81, 11.43, 19.05, 26.67, 34.29, 41.91, 49.53, and 57.15em distance from the bottom are available. After the last cycle, we build our surrogate based on four parameters of interest: the coefficient for preferential attachment to biomass (c [s),),, the coefficient for attachment to arbitrary surfaces (Co,2, [s) the dry mass density of biofilm (ps. kg/mm)), and the enzyme content of biomass (kub, [kg/kg)
The MICP case provides opportunity to test UA-SABI
model that is highly computationally ex- pensive yet low-dinensional in parameter space, and for which real-world measurement data are available. We again employ PCE as the surrogate modeling technique, due to its demonstrated success for this model (Scheurer et al., 2021). In total, eight Bayesian PCEs are trained, one for each measurement location.
4.4.1 Setup
In general, the setup for the MICP model follows the same structure as that described in the former case studies. The priors of the imput parameters are given Ca,1 ~ U(10-10 10-7), Ca,2 ~ U(10-10, 10-9), Pf 151, and kus ~ 211005,55 10-1) (Scheurer et al., 2021). For PCE training, 25 model evaluations within the prior ranges of input parameters are available, which are not necessarily optimal for surrogate training. We constructed the polynomial basis with aPC based on the available model evaluations (Oladyshkin & Nowak, 2012; Bürkner al., 2023).
As in case study 2, we used an offline ABI training set of 10 parameter prior draws. In contrast case study 2, no pre-generated simulation data are available for these draws. Instead, output data can only be generated through the surrogate; thus, comparison with full-budget ABI is possible. Further computational details are given in Appendix C.
4.4.2 Validation of Surrogate Model and Parameter Inference Setup
Since inference will be performed using real measurement data, the first step is to evaluate how accurately the surrogate represents the underlying system. Figure 8a presents the measurement data, the model evaluations
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used for surrogate training, and the surrogate predictions obtained from 10* prior samples across the eight locations. The results indicate that the simulated model data-an consequently the surrogate do not cover the region occupied by the measurement data. This makes the measurement data out-of-distribution (OOD) relative to the simulation model's a priori predictions a phenomenon that is suspected to occur Trequently dynamic systems modeling, although rarely investigated in detail.
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Figure 8: Measurement data for calcite, 25 model evaluations, and surrogate model output obtained with original prior (left) and adjusted prior (right) for Rub of the MICP model across eight locations. Surrogate predictions are obtained from 10a prior samples; median and 90% confidence intervals are shown.
Neural networks are known to generalize poorly outside their training domain, resulting in poor posteriors obtained via ABI (Schmitt et al., 2024a), which is shown in Appendix F.2. Thus, seek to transfer this challenge to the surrogate instcad. To this end, we found that the surrogate priors used for paramcter inference can be adjusted to ensure that the measurement data fall within the surrogate's support. This approach leverages the generalization properties of the low-dimensional, regularized surrogate, avoiding dependence on the poor generalization abilities of neural networks, which are used within the ABI model during inference.
The results of sensitivity analysis (more detailed in Appendix F.3) the surrogate using Sobol indices (Sobol', 1990) shows that the surrogate's output at all locations is almost only sensitive to kub-
In Fig. 9, the surrogate output is shown exemplarily at location index 3 by varying kub. It can be observed that the uncertainty the surrogate increases when evaluating OOD inputs. Based on these results, we adapt the parameter inference prior range for kun, such that kub ~ U(10- 5- 10-3). This adjustment results in prior predictive that covers the measurement data, as illustrated Fig. 8b.
4.4.3 Comparison of Parameter Posterior
Since parameter inference is carried out on real measurement data, no ground truth is available. We therefore compare only the inferred posteriors obtained using low-budget ABI (Ng = 25). SABI, UA-SABI, Point, and E-Post. For easy visual comparison, we applied kernel density estimation (KDE) with Scott's bandwidth method to the posterior samples from each approach. Since kub is the only sensitive paramcter, we present its posterior estimates in Fig. 10 to avoid redundancy, while posterior estimates for the remaining parameters are provided in Appendix F.4.
Figure 10 demonstrates that low-budget ABI, which is shown a point estimate due to quasi-nonexistent variance in the posterior, exhibits a pronounced bias and substantially underestinates posterior uncertainty. This confirms again that training an ABI model with very limited data not meaningful. Comparing the surrogato-basee mothods further highlights that explicitly quantifying surrogate uncertainty and propagating it through the inference process is reflected in the wider posterior estimates. However, E-Post and UA-SABI posteriors exhibit slight difference, which attribute to non-convergence of some MCMC runs in E-Post.
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To assess the validity of the obtained posteriors and the appropriateness of the quantified and propagated uncertainty, Fig. 11 presents the posterior predictive distributions for calcite across the four surrogate-base methods.
Figure 1la and Fig. 1lb clearly illustrate that not quantifying and propagating uncertainty results in over- confident posterior predictives. In contrast, shown in Fig. 11c and Fig. 11d, explicitly accounting for uncertainty produces confidence intervals that successfully cover the measurement data. Both methods yield comparable intervals, with UA-SABI producing slightly broader interval its higher posterior density in the out-of-distribution region of the most sensitive parameter kub (Fig. 10) yields increased surrogate predictive uncertainty.
Overall, this demonstrates that quantifying and propagating the uncertainty surrogate model enables ABI for highly computationally expensive models. Even when the available model simulations are subop- timal for surrggate training and far from the measurement data, accounting for uncortainty and promoting generalization in the surrogate model leads to more trustworthy results.
5 Summary and Outlook
In this work, we introduced Uncertainty-Awar Surrogate-based Amortized Bayesian Inference (UA-SABI) framework designed to enable efficient and reliable ABI for computationally expensive models. UA-SABI combines surrogate modcling and ABI while explicitly quantifying and propagating surrogate uncertainties limitation of existing approaches: while surrogate
through the inference process. This addresses
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Figure 11: MICP posterior predictives for calcite for the four surrogate- basod methods. Surrogate predictions are obtained from 4,000 posterior samples, median and 90% confidence intervals are shown.
models reduce the cost of generating training data for ABI, ignoring their approximation error often leads to overconfident and misleading posteriors. By incorporating uncertainly awareness, UA-SABI enables better-calibrated and reliable inference, even under tight computational constraints.
We validated UA-SABI in both a simple toy example and two real-world problems of modcling CO2 storage and MICP, highlighting its ability to produce well-calibrated posterior estimates that match those obtained with MCMC-based methods. Our experiments demonstrate the importance of sufficient training data, as low-budget ABI produces erroneous posteriors. Also, they show the importance of uncertainty propagation when using a surrogate: Surrogate-based ABI results in overconfident posteriors, whereas UA-SABI correctly reflects model uncertainty in its predictions. Moreover, showed that the upfront computat ional effort of training UA-SABI is quickly offiee in scenarios involving repeated inference, with amortization becoming beneficial already after a few inference runs. Additionally, the MICP experiment poses a challenging real- world scenario by having 1) only a limited number model runs available, and 2) real measurement data that are not adequately captured by these model runs.
Overall, UA-SABI offers an efficient and reliable solution in settings that require repeated inference for computationally expensive models. In general, conclude that using surrogates to generate training data can be an effective strategy for ABT when simulations are scarce or computationally expensive. Morcover, explicitly accounting for surrogate uncertainty in the training data improves ABI results, particularly by reducing overconfidence.
In this work, we applied UA-SABI only to low-dimensional, computationally expensive problems to keep results interpretable and avoid challenges that would distract from our scope. While this focus allowed clearly demonstrate the method's advantages, the question of scalability to higher dimensions naturally
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arises. In general, standard ABI performs well in high-dimensional settings when sufficient simulated train- ing data is available (e.g. Zhou et al., 2025). However, data-scarcity necessitates the use of surrogate models to augment the limited training data. Extending UA-SABI to high-dimensional problems thus hinges on developing surrogates that satisfy two critical requirements: robustness to high-dimensionality and minimal training data requirements. If these conditions cannot be met, slandard ABI remains the more praclical choice. High-dimensional surrogate modeling, however, presents fundamental challenges. In the PCE frame- work, for example, high-dimensional parameter space results in a high-dimensional coefficient space and thus digh-dimensional integration problem, which suffers from the curse of dimensionality. Consequently, the surrogate type must be chosen to accommodate this challenge. Neural network surrogates could be one option, but they require substantial training data again (similar to standard ABI) so nothing is gained here. In PCE, existing approaches try lo address high-ddimensionality through input dimensionality reduction tech- niques (e.g. Li & Tartakovsky, 2020) or sparsity methods PCE (e.g. Bürkner et al., 2023; Luthen et al., 2021) to reduce either the parameter space or the integration space. Combining these methods would be a valuable direction for future research.
Additionally, the surrogate is technically misspecified, the simulator is not fully contained within the surrogate class. Thus, future work could integrate UA-SABI with recently proposed methods for detecting and mitigating model misspecification in ABI (Schmitt et al., 2024a; Dellaporta et al., 2022; Elsemilllrr et al., 2025), which are all naturally compatible with our framework.
Author Contributions
This work represents an equal contribution of Stefania Scheurer and Philipp Reiser. Following the CRediT
taxonomy:
Stefania Scheurer: Conceptualliation, Methodology, Software, Writing Original Draft, Writing Review & Editing, Philipp Reiser: Conceptualization, Methodology, Software, Writing - Original Draft, Writing
Review & Editing, Tim Brünnette: Writing Review & Editing, Wolfgang Nowak: Writing Review
& Editing, Anneli Guthke: Writing Review & Editing, Paul-Christian Bürkner: Conceptualization,
Writing Review & Editing.
Acknowledgments
We thank the Deutsche Forschungsgemenncchaftt (DFG, German Research Foundation) for supporting this work by funding EXC2075 390740016 under Germany's Excellence Strategy and the Collaborative Re- search Centre SFB 1313, Project Number 327154368. We acknowledge the support by the Stuttgart Center for Simulation Science (Sim'Tech). We further acknowledge the support of the DFG Collaborative Research Center 391 (Spatio-Temporal Statistics for the Transition of Energy and Transport) - 520388526. Addition- ally, we thank Dr. Iija Kröker for his help and to both Dr. Kröker and Dr. Johannes Hommel for providing data.
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A Algorithmic Overview
Algorithm 1 gives an algorithmic overview of UA-SABI training. Note that this is for online training. In case of offine training, inputs and parameters {(x(", w ( Na are fixed and not resampled in each epoch.
Algorithm Training of Surrogate and UA-SABI
Surrogate Training Phase
Require: Simulation model M (x,w), w), prior p(c,o), likelihood ply x, x,w,c,o)
Ensure: Posterior p(c, a Dr)
1: Choose imputs and parameters {(x(1) )
for i l....., Nr do
Evaluate simulation model y()) M (x(1), wl)))
end for
Construct training dataset Dr ((x(1)) w())) yl b
6: Perform Bayesian inference for surrogate parameters
plc, DT) x I(y 1x(),w()) ,x, p(,c)
UA-SABI Training Phase
Require:
Prior p(x, w), posterior p(c, a Dr)
Ensure: Trained summary and inference network parameters 0,
7: for each epoch do
8: for l=, NB do
Sample inputs and parameters from prior (x()),(()) wlt)) ~ p(x,w)
9:
Sample surrogate parameters from posterior (c()), o(i)) ~ plc,o Dr)
10:
Evaluate surrogate yz(1) = Mate (x(*), (+)
11:
Sample correctod surrogate output y! ~ p(ye ()(1)
12:
Pass (x()) through summary network s = Se ())(()))))) }}!
13:
Pass g(i) through inference network 40 (s())) which implies &o (w) ss))
14:
15: end for
16: Compute loss from Eq. (5) and update network parameters 0,6
17:
end for
B Proofs
We refer to ABI standard conditions as those under which ABT yields asymptotically correct posteriors, following Radev et al. (2020). Specifically, we assume (i) an infinitely large training dataset, (ii) a conditional neural density estimator 9 (w y) that is sufficiently expressive, (iii) convergence of the training procedure to the true conditional density, and (iv) inference data drawn from the same aata-generating process as the training data.
Likewise, MCMC standard conditions refer to ergodicity, irreducibility, and aperiodicity of the Markov chain (e.g., Robert & Casella, 2004), ensuring convergence of the chain to the target posterior distribution as the number of samples tends to infinity.
Proposition 1. Under ABI slandard conditions, and assuming an infovite number of samples used to prop- agale from the surrogale posterior p Dr), the posterior distribution largeled by UA-SABI converges to the E-Post posterior.
Proof. In UA-SABI, as per Fq. (11), sample from the joint distribution
plw.y.c.o Dr) ply ((w) Dr)
(17)
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to train the conditional neural density estimator 9p(W y). We only condition the data y and treat the surrogate parameters (c, as nuisance parameters ignored by lp(w y This means that we (implicitly) integrate over the distribution of (c,o). Hence, UA-SABI targets the following posterior:
PUA-SABI(4 y, DT) x If
Dfs ply w, plw) DT) dedo.
(18)
On the other hand, E-Post largets the following posterior, as per Eq. (13):
II plw y. v.c,o) dedo
PE PE-Post Post (w yy, Dr)
(19)
x If ply folyw,.c,o (. Dr) deda,
(20)
which shows PUA PUA-SABI(W
PE-Post (w LyDrr)
Corollary 1. For an infinile number of samples from the surrogale posterior p(c, a Dr) and under ABI and MCMC standard conditions, the empirical distributions of UA-SABI samples and MCMC samples from E-Post converge to each other.
C Additional Case Study Details
C.1 Surrogate Models
In case study 1, we train a Bayesian PCE with 2-dimensional Legendre polynomials and maximum total degree of 3, resulting in J/ = 10 polynomials. We set a normal prior for the surrogate coefficients, p(c) ((0,,) 5), and a half-normal prior for the approximation error paramcter, p(a) = (a.f-
In case study 2 and 3, we consider a Bayesian PCE with 3- and 4-dimensional aPC polynomials for the CO₂ model (see Section 4.3) and a maximun total degree of 3, resulting in J 19 and 35 polynomials rospectively. Following Bürkner et al. (2023), place sparsity- coefficients e with R ~ Beta(0.5,.2 2). For the approximation error set the prior as p(o) Haaff-V(0.1..
R2D2 prior the surrogate
C.2 Neural Posterior Estimation
For the first two case studies, we used the same NPE setup. The summary network So(x,y) is DeepSel Zaheer al. (2017) composed of two hidden layers, each containing 10 neurons. It outputs 10-dimensional summary vector. For the inference network (s), employ coupling flow as implemented in Radev et al. (2023b). For the third case study, we did not use summary network as the measurement locations in the experiment were fixed. For the inference network Ip(s), we used a spline coupling flow (Durkan et al., 2019) as implemented in Radev et al. (2023b). The training process employed cosine learning rate scheduler with an initial learning rate of 5 10-4 and minimum learning rale fraction of a = 10-6 The scheduler operated over total of 12,800 steps, corresponding to 128 batches per epoch over 100 epochs. All NPE models were trained using the Adam optimizer (Kingma & Ba, 2017) for 100 epochs.
D Additional Results for Case Study 1: LogSin Model
This section shows the MCMC convergence analysis of E-Post for the LogSin model to explain the differences between UA-SABI and E-Post. We follow the same setup as described in Section 4.2.1, but used more samples and chains to check convergence. We analyzed the convergence for each of the 200 ground truth parameters and for each surrogate posterior sample respectively. We run MCMC separately for each of the 1,000 surrogate posterior samples with chains and 1,000 warm-up and 1,000 sampling iterations, respectively, resulting in 1,000 x 1,000 draws per ground truth parameter used for analysis.
For each surrogate posterior sample we can check the convergence of the MCMC run using standard diagnos- ties R. As convergence threshold, follow the standard recommendation R 1.01 (Vehtari et al., 2021).
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Figure 12 shows the ratio of non-converged runs (R > 1.01) started by each surrogate posterior sample (SPS) vs. total number of runs for each of the 200 ground truth parameters. It is evident that for most ground truth parameter values more than 10% of the runs have not converged, with some having more than 40% of the runs with an R > 1.01. Notably, for most ground truth parameter values, more than 10% of the runs have not converged, with some parameters even exceeding more than 40% for that ratio. This indicates that the E-Post posterior not reliable, which results deviating posteriors for E-Post and UA-SABI, as shown in Fig. 5.
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Figure 12: Ratio of non-converged runs (N > 1.01) started by each surrogate posterior sample (SPS) vs. total number runs for each of the 200 ground truth paramcters for the LogSin model.
Additionally, Fig. 13 shows the trace plots of single F-Post runs (4 chains) for two exemplary draws of the surrogate posterior with bad R given single ground truth paramcter for the LogSin model. These traces indicate that the chains fail to mix properly between the two modes of the underlying posterior. When the modes have little overlap, MCMC struggles to recover their correct proportions.
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Figure 13: Trace plots of single E-Post MCMC runs for two exemplary draws of the surrogate postcrior given single ground bruth parameter for the LogSin model. The different colors indicate different chains.
E
Additional Results for Case Study 2: CO2 Storage Model
This section shows the recovery and ECDF difference plots for porosity of the CO2 storage model shown for SABT vs. Point and UA-SABI vs. E-Post.
In Fig. 14 we show that SABT and Point as well as UA-SABI and E-Post yield similar results in both recovery and calibration. The notably wide intervals observed for Point, especially compared to SABI in the recovery plots suggest convergence problems the MCMC, likely due to the low standard deviation in the likelihood. These convergence problems may explain the inconsistencies between SABI and Point. In comparison, their uncertainty-aware counterparts UA-SABI and E-Post produce highly similar estimates.
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Figure 14: CO2 recovery plots comparing to ground truth and MCMC full reference solution, ECDF difference plots (from left to right) for 200 ground truth samples: SABI Point (top) and UA-SABI vs. E-Post (bottom). For ECDF difference plots, empirical ranks are shown in blue, 95% confidence bands assuming calibration are shown in grey.
F Additional Results for Case Study 3: MICP Model
F.1 Illustration
water
suspended biomass
urcolysis solute
calcite
precipitation/ dissolution
biofilm
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decay
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rock grain
Figure 15: Schemaatc illustration of relevant processes during MICP. Figure adapted from Scheurer et al. (2021), licensed under CC BY 4.0.
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F.2 Parameter Posterior with Original Priors
Performing inference with the original prior for lub leads, regardless of the inference method, to point-like posteriors at the upper boundary of the prior. The reason is the same for all methods: the maximal value of bab yields the highest output values. Since the measurement data are always higher than the surrogate output under the original prior, the posterior collapses to a point-like estimate at the maximum admissible value of kub, as exemplified Fig.
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Figure 16: KDEs of 4,000 parameter posterior samples for kub of the MICP model obtained with low-budget ABI, Point, E-Post, SABI and UA-SABI using the original parameter prior.
F.3 Sensitivity Analysis
We conducted a sensitivity analysis on the surrogate to identify the parameters to which the surrogate output is most sensitive. Sobol indices (Sobol', 1990) decompose the output variance into contributions from each parameter (including their interactions), with the total Sobol index of a parameter quantifying its overall influence on the output variance. Figure presents the distributions of the total Sobol indices (Te Gratiet et al., 2017) for cach parameter each location, noting that each location has own surrogate. Since the Bayesian PCE yields samples of the coefficients, the corresponding Sobol indices are also obtained in the form of distributions. Figure 17 clearly shows that almost all the variance in the output at all locations is determined by lib-
F.4 Parameter Posterior for Remaining Parameters
Figure 18 shows the KDEs of 4,000 parameter samples for the 3 remaining parameters, Ca Ca,2, and pr obtained with low-budget ABI, Point, E-Post, SABI, and UA-SABI. Two distinct scenarios emerge: 1) the posterior collapses to point estimate (often at the boundary of the prior), or 2) the posterior remains close to
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Figure 17: Distribution of total Sobol indices all locations for the four parameters Ca.1, P, kub of the MICP model.
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the prior distribution. Point-like posteriors are produced by low-budget ABI or Point, while surrogate- based methods yield posteriors resembling the prior. Both outcomes can be explained by the lack of sensitivity of the output to these parameters, as shown by the total Sobol indices in Fig. 17. This missing sensitivity makes reliable inference very difficult.
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Figure 18: KDEs of 4,000 parameter posterior samples for the remaining parameters Ca.1, Ca,2, and Py of the MTCP model obtained with low-budget. ABI, Point, F-Post, SABI, and UA-SABI.
F.5 Runtime Comparison
To compare runtimes in case studies and 2, inference must be performed multiple datasets. Since only one dataset of real measurements is available, conducted the runtime analysis on synthelic datasets. The experiments were also run on computing cluster with two AMD EPYC 7551 CPUs (totaling 64 physical corcs) to speed up E-Post through parallclization.
Figure 19 shows the measured runtimes of UA-SABI and E-Post for 2,3, 4, inference runs, with R-Post parallelized across 16 cores. We observe break-even point between and runs, meaning that UA-SABI becomes advantagcous starting from 4 runs. Despite E-Post leveraging more cores, the training costs of UA-SABI are amortized earlier when the model is more expensive.
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Figure 19: Comparison of runtimes for MICP for (2,3,4,5) runs for UA-SABI (training and inference) and E-Post (inference) with break-even point.
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Reviews:
[{"review_id": "WvGCbJuwRF", "reviewer": "Reviewer_MQqn", "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": "", "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": "This paper introduces a simple but seemingly effective uncertainty-aware extension of surrogate-based amortised Bayesian inference (SABI), following the same principles introduced previously by Reiser et al. (2025) for expensive sampling-based inference (MCMC). Like the latter, the proposed method explicitly propagates the approximation uncertainty from the surrogate model, while its amortised nature trades off a one-time training cost for extremely efficient and well calibrated inference.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The method is evaluated on (1) a fully synthetic, univariate, single-parameter task, (2) a real-world-inspired virtual benchmark with 3 target parameters, and (3) a 4-parameter experiment with real measurements. The experimental setup follows the precedent from Reiser et al. (2025): these problems are of similar or higher complexity than those originally supporting E-Post, the same class of surrogates is used (PCE), and evaluation follows the same simulation-based calibration (SBC) protocols.\n\nWithin the scope of these experiments, the reported results corroborate the claims of calibrated uncertainty propagation and higher efficiency than the MCMC-based E-post for more than a few inference runs.\n\nOn the other hand, although the size of the case studies may be representative of many important real-world problems, higher-dimensional problems were not deliberated in sufficient detail. I understand and agree that running such experiments might be out of scope, though I believe a deeper discussion is warranted about the expected behaviour, challenges, and implications of scaling UA-SABI to higher dimensions.", "audience_yn": "Yes", "audience_explanation": "While methodologically incremental, the main contributions of this work are in unlocking cheap approximate inference from few expensive simulations with quantifiable, calibrated uncertainty. Therefore, I believe it could be of interest to methods researchers working on Bayesian inference or to applied researchers in complex domains that often rely on expensive simulators (e.g. geosciences, materials, computational biology).", "requested_changes": "1. Please elaborate on the expected behaviour, challenges, and implications of scaling UA-SABI to higher dimensions. This will help readers judge its wider applicability and consider future extensions if appropriate.\n2. As a partly unfamiliar reader, I found parts of the text rather dry and not self-contained:\n    1. [Introduction] The motivations for the work were not very concrete, and would benefit from some examples. What kinds of simulation problems are we considering? How expensive are they to run and why? What kinds of parameters are we trying to infer, from what kinds of observations?\n    2. [Sec. 4.1] As the SBC analysis is the main evaluation, it would be really helpful to explain its principles and interpretation in some more detail.\n3. Other parts also need clarification:\n    1. [Sec. 2.2] The relationship between $\\mathbf{y}$, $\\tilde{\\mathbf{y}}$, and $\\tilde{\\mathbf{y}}_e$ was confusing, especially given that the simulated $\\mathbf{y}^{(i)}$ is used in eq. (6) in place of the error-adjusted surrogate $\\tilde{\\mathbf{y}}_e$ from eq. (5). (Using $e$ subscript instead of $\\epsilon$ due to a rendering glitch.)\n    2. [Sec. 2.1, 4.2.1] It was not clear whether the summary network $S_\\theta$ ingests a single pair of inputs-outputs, as suggested in the equations and pseudocode, or a set thereof, implied by its DeepSet implementation.", "broader_impact_concerns": "N/A", "additional_comments": "- [Sec. 2.2] It could be helpful to give readers some pointers (specific changes, and/or references) for how to account for heteroscedastic (non-i.i.d.) approximation error, even if it's not demonstrated in this paper.\n- Consider using more diverse colours and/or markers in the comparison plots, as the shades of blue are hard to distinguish.\n- [Sec. 4] Optionally, some illustrations and/or plots for the problem settings in the case studies could make them much easier to follow.\n- [Sec. 4.4.2] While marginally interesting on its own, the exploration of prior misspecification of the surrogate is a distraction from the main focus of the experiment. Moving it to the appendix might help streamline the narrative.", "extra_scores": {}, "extra_text": {}}, {"review_id": "NY5Zl1vyod", "reviewer": "Reviewer_Cuvo", "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": "", "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 introduces uncertainty‑aware surrogate‑based amortised Bayesian inference to enable amortised posterior inference when simulators are too costly to generate sufficient data for training neural posterior estimators. UA‑SABI learns a surrogate from a small set of high‑fidelity simulations and trains the NPE on surrogate‑generated data while propagating both epistemic and approximation uncertainty through training and inference. The authors prove asymptotic equivalence to marginalising over the surrogate posterior (E-Post) and demonstrate, on a toy problem and two real-world case studies that UA-SABI attains accurate posteriors with fewer simulator calls and lower inference cost than MCMC-based alternatives.\n\nStrengths: Clear and well-written, core contribution convincingly demonstrated, interesting real-data examples\n\nWeaknesses: not highly novel - more a (useful) incremental combination and implementation of established ideas", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes each claim is supported: UA-SABI’s uncertainty-aware training and target equality with E-Post are formalised, double amortisation is demonstrated. Evidence includes calibrated recovery/SBC plots and posterior predictives across some compelling case studies with SABI and low-budget ABI consistently overconfident justifying the UA-SABI approach.", "audience_yn": "Yes", "audience_explanation": "Yes. The paper covers amortised Bayesian inference with computationally expensive simulators, a common constraint in SBI, and will be of interest to TMLR readers interested in ABI/SBI (e.g., I found this interesting) as well as practitioners in simulation-heavy domains.", "requested_changes": "Requested change:\n\nReduce claims: “we eliminate the training error that arises from insufficient data during ABI training” -> “reduce the training error”. \n\nOther notes (won’t impact acceptance, your discretion to respond to / change in the paper):\n\nWhile surrogate construction is not the focus here, could there be any further brief guidance on when the use of the surrogate tends to work versus when it is likely to struggle? My main personal reservation if I were to use this myself is just concern if key features of the simulator will be missed by using a surrogate with so few simulations - but I think the case studies address this pretty well (I am surprised and impressed at the decent inference with such few simulations).\n\nIs it possible to contextualise this error modelling approach and its relevance to model misspecification more broadly? Would the error model here also have benefits for simulator misspecification?\n\nWhen surrogate uncertainty is large, do you ever obtain outputs or parameter regions that violate basic physical bounds? Might this be an issue with the surrogate approach?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "LBM3IvkiBG", "reviewer": "Reviewer_BKxS", "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": "", "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": "**Summary**: The paper introduces an amortized Bayesian inference method that explicitly quantifies and propagates surrogate uncertainty into amortized posterior inference, claiming asymptotic equivalence relative to surrogate-marginalizing MCMC (E-post), and empirically shows calibrated uncertainty and amortization benefits on three low-dimensional examples.\nOverall, I find that the paper makes a timely and relevant methodological contribution to the simulation-based inference domain and supplements recent work (E-post) by explicitly qualifying and propagating surrogate uncertainty but now using ABI for the final inference. The paper addresses the same basic problem as E-post which relies on MCMC for the final inference step, but the proposal to use ABI appears to be a new direction that provides better scalability than E-post on the inference side. \n\n\n**Strengths**\n  - The scope is clear and sufficiently aligned with the actual content with claims largely supported by evidence (a few unexplained differences between E-post vs. UA-SBI posteriors). \n  - Addresses an important practical challenge of conducting Bayesian inference when simulators are expensive.\n   - Method-wise I find the combination of surrogate uncertainty with amortized inference sufficiently new/unexplored to be of interest to the community and it compliment the recent E-post method. \n  - The methodology is clear with training setup and inference workflow well-defined and the  pseudocode and actual source code aiding reproducibility and impact.\n  - Empirical evidence on three problems using relevant ECDF-difference/rank shows a clear advantage over methods not accounting for surrogate uncertainty. The results qualitatively match the E-post method albeit with slight differences, but trades in training cost for reduced inference cost.\n\n**Weaknesses / Limitations**\n  - ***Convergence of the UA-SABI posterior to E-post***. A key claim in the paper is that E-post and UA-SABI “targets” the same posterior, but most if not all of the empirical results (4b, 9, 10, E3) shows that there is a small but quantitative difference between the posteriors which is not adequately explained as far as I can tell. \n  - ***Possibly too limited scope?*** The authors deliberately and clearly outscope many of the detailed elements/studies, questions and experiments I’d normally expect to see in a method paper of this sort, including:\n    - Model assumptions are not really explored, leaving doubts about the method’s robustness to model choices, misspecification and general practicality.\n    - Only low-dimensional problems are explored (a deliberate choice) leaving the performance on high-dimensional/complex posteriors unknown. \n    - ....and similar low-level experiments that would provide insights into the important elements in the setup; however, I’d be interested in the other reviewers’ and the action editors view on this before coming to a conclusion on whether additional experiments need to be included for publication at TMLR.\n  - ***Other*** less critical limitations are addressed with suggestion below.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims are ***mostly*** supported by evidence but I'd like to see more explanation/discussion of the gap between the UA-SABI and E-post posterior (and ideally a few more optional experiments) and links to the theoretical claim. See Requested Changes for details.", "audience_yn": "Yes", "audience_explanation": "Yes, people in the SABI community that worries about reliable posteriors would be interested in this.", "requested_changes": "Note: items listed with **suggest/consider** are not things that would prevent me from potentially recommend acceptance;  **strongly suggest** or **please fix** indicate things that require a response and/or correction.\n\n\n\n  -  ***Overview figure***: Figure 1 goes some way in communicating the very high-level concept; however, I **strongly suggest** extending Figure 1 or adding another figure that clearly visualizes the details in a coherent way (perhaps a graphical model) making it easier to appreciate the many elements and various distributions. \n  -  ***Objective/loss ( Eq.(2) ) and Algorithm 1***: \n     -\tThe usual SABI loss is stated very briefly in eq. 2. I’d **suggest** providing a bit more context e.g. perhaps simply writing out the derivation in a few more details.\n     -\tEq. 2.: Please verify if the expectation should be with respect to p(y) – I believe it to be with respect to p(x,y)? \n     - Algorithm 1, line 15 (in the for-loop over N_B observations) mentions that the loss is computed from eq 2; however, this is already a sum over NB terms - **please correct** this.\n  - ***Proposition 1***: The wording (i.e. “targets”) seems rather informal. I’d **strongly suggest**; a) improving clarity to more formally state the proposition as an asymptotic convergence result if possible, and b) explicitly listing the corresponding assumptions (expressive estimators, infinite data etc.). \n  - ***Discrepancy between E-post and UA-SABI*** As mentioned, I believe the results shown in fig. 4, 9 and 10 (and results in E.3) deserve some explanation/discussion given the key claim that UA-SABI “targets\" the same posterior as E-post. I’d **strongly suggest** contrasting the E-post and UA-SABI posteriors and discussing how that gap relates to proposition 1 (see also suggestions for further experiments), i.e. what explains the gap (finite setting, limited expressiveness etc.)? Ideally, I’d **suggest** supporting the discussion by a small empirical study on the convergence of the UA-SABI posterior to the E-post posterior to further back up the claims; however, I’ll not insist on this.\n  - ***Supporting experiments*** As previously mentioned, there are several detailed technical experiments I would like to see included. I appreciate that the paper outscoped many of these to provide high-level evidence that the method provides a benefit. Regardless, I’d **suggest** including demonstration/results on at least synthetic high-dimensional problem(s), impact of model choices including misspecification, impact of surrogate model expressiveness and training set size, but I am in doubt whether these should be optional or required, so I will not insist on it at this point.\n\n\n**Minor comments***:\n  - Eq 12: Strictly speaking, I believe this should be $\\mathcal{N}( y \\mid … )$ instead of $\\mathcal{N}( M(x,w) | … )$ - if so, **please correct** it. \n  - The code (at least the readme.md) has traces of a NeurIPS submission, I’d **suggest** correcting it before publishing.", "broader_impact_concerns": "No concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)