Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Posterior SBC: Simulation-Based Calibration Checking Conditional on Data

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Posterior SBC re-bases simulation-based calibration at the observed-data posterior, letting modellers check whether their inference algorithm is trustworthy for the dataset they actually have.

desk verdict A clean, useful extension of SBC that conditions on the observed data, but its guarantee is self-consistency under updating, not absolute posterior correctness; the paper slightly oversells it in Section 3. read the letter →

arxiv 2502.03279 v2 pith:MW7DAD65 submitted 2025-02-05 stat.ME stat.COstat.ML

Teemu Säilynoja , Marvin Schmitt , Paul-Christian Bürkner , Aki Vehtari This is my paper · ORCID
classification stat.MEstat.COstat.ML MSC 62F1562-0865C05
keywords posteriorSBCsimulation-basedcalibrationBayesianworkflowdata-conditionalamortizedinferencePITuniformityhierarchicalmodelsdrift-diffusionmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Standard simulation-based calibration (SBC) checks an inference algorithm by drawing parameters from the prior, simulating data, and testing whether posterior ranks are uniform on average. The paper argues this answers the wrong question after data arrive: it can miss miscalibration that is confined to the region of parameter space that actually matters for the observed dataset, or flag problems that vanish once the data are conditioned on. The authors introduce posterior SBC, which re-bases the same self-consistency check at the observed-data posterior: draw a parameter from the posterior, simulate new data from it, draw again from the augmented posterior, and verify that the two draws are exchangeable. Three case studies show it detecting parameterization-dependent miscalibration in a hierarchical model, avoiding irrelevant prior-driven warnings in a Lotka-Volterra model, and exposing bad neural-network posterior approximation in amortized inference that prior SBC missed.

What carries the argument

The load-bearing object is the posterior-rebased SBC equality (Equation 3), which applies the chain rule of Bayesian updating to the joint distribution of the observed data, a posterior predictive draw, and two parameter draws. It turns data-conditional calibration into a rank-uniformity test: compare draws $\theta'$ from the original posterior with draws $\theta''$ from the augmented posterior that conditions on both the observed data and a posterior-predictive simulation, and check the empirical PIT values for discrete uniformity using the graphical ECDF-difference test. The same test quantities as in prior SBC, including the joint log-likelihood, carry over.

What would settle it

Run posterior SBC with an inference algorithm that ignores the likelihood and always returns the prior as its posterior, on a dataset with fairly informative data. If the parameter PIT values come out uniform, that demonstrates the failure mode the paper's heuristic assumption dismisses; a complete version of the check should use a test quantity such as the joint log-likelihood, which should expose the bias.

Watch

Extended reading notes

Core claim

The paper's central claim is that calibration of Bayesian inference should be assessed conditional on the data actually observed, and that the standard SBC self-consistency argument can be shifted to make this possible. Treating the observed-data posterior $\pi(\theta \mid y_{\mathrm{obs}})$ as the new prior, the identity in Equation (3) shows that for $\theta' \sim \pi(\theta \mid y_{\mathrm{obs}})$, $y \sim \pi(y \mid \theta')$, and $\theta'' \sim \pi(\theta \mid y_{\mathrm{obs}}, y)$, the two parameter draws $\theta'$ and $\theta''$ have the same conditional distribution. Hence the empirical PIT values of $\theta'$ with respect to draws from the augmented posterior are uniform if and only if the inference algorithm is calibrated in the region of parameter space that matters for the observed data. The paper demonstrates the diagnostic in three cases where prior SBC either misses real miscalibration or reports irrelevant miscalibration.

Load-bearing premise

The load-bearing premise is that a biased inference algorithm will not stay biased in the same way when it is conditioned on additional data; if an algorithm's error is invariant under further conditioning, posterior SBC can show uniform PIT values despite the posterior being wrong.

Editorial extensions

If this is right

  • Modellers can validate the inference algorithm for their actual dataset, detecting miscalibration that prior SBC misses when problematic parameter regions are small or cancellation occurs.
  • Posterior SBC gives per-parameter information about the direction and magnitude of bias, allowing informed model reparameterization or algorithm changes.
  • In the Lotka-Volterra case it is computationally cheaper than prior SBC, since the prior's problematic regions are avoided, so more iterations can be run within the same budget.
  • For amortized Bayesian inference, posterior SBC provides a near-free data-conditional diagnostic where no MCMC convergence diagnostics exist.
  • Passing posterior SBC is a necessary but not sufficient condition for trustworthy inference, so it complements rather than replaces convergence diagnostics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The self-consistency assumption could be stress-tested by deliberately injecting a bias that is invariant to adding data (for instance, an algorithm that ignores the likelihood); parameter-based posterior SBC would show uniform PIT values, revealing the method's blind spot and motivating likelihood-based test quantities.
  • The method could be extended to a K-fold or data-splitting version, conditioning the initial posterior on only part of the data so the augmented posterior is closer in size to the original; the paper mentions this as a possible adjustment but does not develop it.
  • Data-conditional miscalibration detected by posterior SBC may sometimes reflect model misspecification rather than algorithm failure; as a diagnostic it could therefore also guide model refinement in the Bayesian workflow, not just algorithm choice.
  • For amortized inference, posterior SBC could become a standard acceptance gate before a trained neural posterior is used on a new dataset, since the per-dataset cost is seconds.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper introduces posterior SBC, a variant of simulation-based calibration checking that conditions on an observed dataset. The method draws parameter values from the posterior given the observed data, generates new data from the likelihood, draws from the posterior conditioned on both the observed and generated data, and tests uniformity of the resulting PIT values. The central derivation (Eq. 3) is a correct application of the Bayesian chain rule, and the paper demonstrates the method on three case studies: a hierarchical model, a Lotka-Volterra model, and an amortized Bayesian inference model. The authors position posterior SBC as a data-conditional diagnostic that complements prior SBC and is especially useful for amortized inference.

Significance. Posterior SBC is a useful and natural extension of SBC that focuses calibration checking on the region of parameter space relevant to an observed dataset. The derivation is clean, involves no free parameters, and the case studies are well executed with public code. The method is particularly valuable for amortized Bayesian inference, where standard MCMC diagnostics are unavailable. However, the paper overstates what is tested: posterior SBC verifies self-consistency of the inference algorithm under further Bayesian updating, not the absolute accuracy of the posterior at the observed data. A coherent Bayesian updater with an arbitrary initial prior will pass posterior SBC even when its posterior is far from the specified posterior. This limitation is acknowledged in the Discussion but is not reflected in the abstract or Section 3.

major comments (4)
  1. [Section 3, first full paragraph] The claim that 'if the sampling fails for either the original posterior or for the augmented posteriors, the SBC equality in Equation 3 does not hold, leading to non-uniform PIT values' is too strong. Uniformity of the posterior SBC PIT values holds for any inference algorithm that is exactly self-consistent under Bayesian updating with the true likelihood, even if the algorithm's posterior at the observed data is arbitrarily different from the specified posterior. For example, in the normal model of Section 2, an algorithm that is exact Bayes under a different prior (say N(10,0.1) instead of N(0,1)) produces uniform PIT values for both parameter and log-likelihood test quantities. The paper should state the precise self-consistency condition that the PIT test verifies, e.g., A(theta|y_obs) = E_{theta'~A(.|y_obs), y~p(y|theta')} A(theta|y_obs,y), and qualify the abstract and this paragraph accordingly.
  2. [Section 3, paragraph on the consistency of biased inference] The sentence 'it is very unlikely that we would encounter such biased inference that would be consistent when conditioned on more data' is an unsupported heuristic rather than a derived guarantee. The manuscript should replace this with a precise characterization of the class of failures that posterior SBC can detect: it detects failures of self-consistency under additional conditioning, but it cannot detect a coherent Bayesian updater that starts from an arbitrary initial posterior. This point affects the interpretation of all three case studies and should be addressed in Section 3 rather than deferred to the Discussion.
  3. [Abstract and Section 1] The abstract states that posterior SBC can 'validate the inference conditionally on observed data', and Section 1 similarly claims that it can 'validate the model implementation and inference algorithm when conditioned on some fixed set of data'. Given the limitation described above, such claims overstate what is tested: the method checks the update step from y_obs to (y_obs, y), not the absolute accuracy of the posterior at y_obs. The authors should either qualify these statements or define a precise sense in which the posterior at y_obs is validated. The Discussion's 'necessary but not sufficient' caveat should be introduced in the abstract or the introduction.
  4. [Section 3, Eq. (3)] The factorization in Eq. (3) treats y_obs and y as if both are random variables in a joint distribution, but y_obs is fixed. The chain rule should be written for the conditional distribution e.g., pi(y, theta', theta'' | y_obs) = pi(theta' | y_obs) pi(y | theta') pi(theta'' | y, y_obs), and the assumption that y is conditionally independent of y_obs given theta' should be stated explicitly.
minor comments (5)
  1. [Section 2, Eq. (2)] The notation u_i = p(theta''_i < theta'_i | y_i) is nonstandard; it should be written as P(theta''_i < theta'_i | y_i) or as the CDF F_{theta''|y_i}(theta'_i).
  2. [Figure 1 and Section 1] The caption of Figure 1 would benefit from a more explicit description: the gray area denotes the prior support and the colored regions denote posterior regimes with biased inference; in panel (b), 'the inference is calibrated outside the colored region' should be clarified as 'calibrated for parameter values outside the colored region'.
  3. [Section 4.1, Setup] The description of the two observed datasets could be clearer: the sentence 'The first case has a strong prior and a weak likelihood' refers to the dataset with tau=0.06 and sigma=1.96, but the connection between the prior on tau and the likelihood's strength may confuse readers; consider adding a sentence explaining the funnel geometry in each parameterization.
  4. [Abstract and Section 1] There are typographical errors in the abstract and introduction, such as 'conditionalonobserveddata' and 'the both the code for implementing the model'; these should be corrected.
  5. [Appendix A] The supplementary text refers to 'Section 3.2 of the article' when discussing the Lotka-Volterra multimodality outliers, but the relevant section in the main text is Section 4.2; this cross-reference should be fixed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: posterior SBC is a direct chain-rule application of the existing SBC self-consistency identity.

full rationale

The paper's central derivation is Eq. (3), which is an exact factorization of the joint distribution by the chain rule of probability: pi(y_obs, y, theta', theta'') = pi(theta'|y_obs) pi(y|theta', y_obs) pi(theta''|y, y_obs). No fitted parameters, hand-tuned constants, or data-dependent calibration enter this identity. The uniformity of the PIT values follows from the standard SBC self-consistency property, now applied with the observed-data posterior in place of the prior. The method is therefore derived from first principles rather than from the case-study results. The paper does cite prior work by overlapping authors, notably the graphical uniformity test of Säilynoja et al. (2022) and the log-likelihood test quantity of Modrák et al. (2023), but these are used as independent statistical tools for evaluating PIT values; the validity of posterior SBC itself does not depend on those citations. The paper's statement that it is 'very unlikely' that a biased inference algorithm would remain consistent under further conditioning is a heuristic limitation of the method, not a circular step. The skeptical concern that a coherently wrong Bayesian updater can pass posterior SBC is a legitimate caveat about what the method certifies, but it does not make the derivation equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central method introduces no fitted constants; the case studies use standard model priors and neural network hyperparameters that do not enter the derivation of Eq. (3). These are configuration choices, not free parameters of the method. No new physical or model entities are postulated; posterior SBC reuses existing posterior and data-generating distributions.

assumptions (4)
  • standard math Self-consistency of Bayesian posterior: draws from prior, data generated from likelihood, and draws from posterior conditioned on that data are exchangeable (SBC equality, Eq. 1).
    Basis of SBC; posterior SBC extends this with the chain rule (Eq. 3).
  • domain assumption The inference algorithm being tested is the one used to draw both from the original posterior and from the augmented posteriors.
    Section 3 states posterior SBC draws theta' from pi(theta|y_obs) and theta'' from pi(theta|y_obs,y) using the algorithm under evaluation.
  • ad hoc to paper Biased inference is not consistent under additional conditioning data: if the algorithm fails to sample faithfully, the non-uniformity of PIT values will show it.
    Section 3 asserts 'it is very unlikely that we would encounter such biased inference that would be consistent when conditioned on more data'; this is a heuristic, not proven, and is load-bearing for the method as a diagnostic.
  • domain assumption Model code and inference algorithm implementation are correct; only finite-time inference bias is considered.
    Section 1 states the paper assumes correct code and focuses on biased inference with finite computation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Posterior SBC: Simulation-Based Calibration Checking Conditional on Data." pith.science (2026). https://pith.science/paper/MW7DAD65

@misc{pith2026250203279,
  author       = {Pith},
  title        = {Pith review of: Posterior SBC: Simulation-Based Calibration Checking Conditional on Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MW7DAD65}},
  note         = {Machine review of arXiv:2502.03279}
}
read the original abstract

Simulation-based calibration checking (SBC) refers to the validation of an inference algorithm and model implementation through repeated inference on data simulated from a generative model. In the original and commonly used approach, the generative model uses parameters drawn from the prior, and thus the approach is testing whether the inference works for simulated data generated with parameter values plausible under that prior. This approach is natural and desirable when we want to test whether the inference works for a wide range of datasets we might observe. However, after observing data, we are interested in answering whether the inference works conditional on that particular data. In this paper, we propose posterior SBC and demonstrate how it can be used to validate the inference conditionally on observed data. We illustrate the utility of posterior SBC in three case studies: (1) A simple multilevel model; (2) a model that is governed by differential equations; and (3) a joint integrative neuroscience model which is approximated via amortized Bayesian inference with neural networks.

Figures

Figures reproduced from arXiv: 2502.03279 by the authors.

Figure 1
Figure 1. Two conceptual illustrations of a model parameter space, with gray area denoting the prior. In (a), the colored regions are potential posteriors with bias in opposite directions, while the inference is well calibrated for parameter values outside these regions. Prior SBC will not show calibration issues due to cancellation of biases, while posterior SBC would indicate issues for posteriors intersecting the colored r… view at source ↗
Figure 2
Figure 2. An illustration of how PIT values are computed in prior SBC. The model is a normal distribution N (𝜃, 𝜎) with a known standard deviation 𝜎. The dashed line shows the prior distribution 𝑝(𝜃), and solid lines show three posteriors 𝑝(𝜃|𝑦𝑖) conditioning on the predictive draws 𝑦𝑖 ∼ 𝜋(𝑦𝑖 | 𝜃 ′ 𝑖 ). The PIT value of each prior draw 𝜃 ′ 𝑖 , shown as blue dots, equals the respective shaded blue area (0.03, 0.43, and 0.97 re… view at source ↗
Figure 3
Figure 3. Prior SBC checking for the hierarchical model with (a) the centered and (b) the non-centered parameterization using four different test quantities. Subplots show PIT-ECDF difference plots using four different test quantities and 95% simultaneous confidence intervals under the assumption of uniformity. As all blue PIT-ECDF difference lines are inside the 95% simultaneous confidence intervals, the inference seems to b… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Posterior SBC for the hierarchical model with an observed data with large 𝜏 and small 𝜎, corresponding to weak population prior and strong likelihood, implying a close to normal posterior with (a) the centered parameterization, and a funnel shaped posterior with (b) th…
Figure 5
Figure 5. Figure 5: Posterior SBC for the hierarchical model with an observed data with small 𝜏 and large 𝜎, corresponding to strong population prior and weak likelihood, implying a funnel shaped posterior with (a) the centered parametrization, and a close to normal posterior with (b) the…
Figure 6
Figure 6. Figure 6: Prior predictive draws of the observed pelt counts over time. Top left shows the true historical observation data. Many prior draws do not exhibit the observed periodicity, or are in other ways unrealistic. All of these parameters are constrained to be positive. We do …
Figure 7
Figure 7. Figure 7: Bimodal posterior of the two population growth parameters, inlaid with predictive means and predictive intervals of two posterior draws where (𝜃1, 𝜃3) fall in two different modes. The concentrated mode has most of the posterior mass, and corresponds to lower predictive…
Figure 8
Figure 8. Figure 8: PIT-ECDF plots of the posterior joint log-likelihood test quantity in the Lotka-Volterra model for both prior and posterior SBC. Prior SBC uses only 250 iterations, resulting in wider confidence bands. Posterior SBC could include 500 iterations due to faster computatio…
Figure 9
Figure 9. Figure 9: Conceptual overview of amortized Bayesian inference with normalizing flows. Stage 1: Training. Based on samples from the joint model 𝑝(𝜃, 𝑦), a neural network tandem simultaneously learns to extract sufficient summary statistics ℎ𝜓 (𝑦) and establish a conditional mappi…
Figure 10
Figure 10. Figure 10: Results of prior and posterior simulation-based calibration checking with amortized inference. The Bayesian models 𝑀1 and 𝑀2 are two joint integrative neuroscience models from Ghaderi-Kangavari et al. (2023), and the datasets for posterior SBC checking are two real da…
Figure 11
Figure 11. Figure 11: Per parameter results of prior SBC for the Lotka-Volterra model. γ δ H0 L0 σh σl α β PIT ECDF difference (a) Individual parameter calibration assessments. σh σl α β γ δ H0 L0 Ground truth Estimate (b) Parameter recovery [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Per parameter results of posterior SBC for the Lotka-Volterra model [PITH_FULL_IMAGE:figures/full_fig_p022_12.png]
Figure 13
Figure 13. Figure 13: Parameter recovery of the amortized approximator for model 𝑀1. Model 2 The second probabilistic model 𝑀2 represents “model 6” by Ghaderi-Kangavari et al. (2023), which implements a drift-diffusion model with collapsing boundary (Drugowitsch et al., 2012; Hawkins et al…
Figure 14
Figure 14. Figure 14: Parameter recovery of the amortized approximator for model 𝑀2. Paralleling the previous evaluation for model 𝑀1, [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Diffusion Models in Simulation-Based Inference: A Tutorial Review

    stat.ML 2025-12 conditional novelty 5.0 of 10

    Design choices — noise schedule, parameterization, sampler, and model family — measurably change posterior accuracy in diffusion-based SBI; variance-preserving EDM diffusion with adaptive solvers leads on low-dimensio...

Reference graph

Works this paper leans on

53 extracted references · 47 canonical work pages · cited by 1 Pith paper

  1. [1]

    Betancourt, M. (2017). A conceptual introduction to hamiltonian monte carlo. arXiv:1701.02434

    arXiv 2017

  2. [2]

    Box, G. E. P. (1980). Sampling and Bayes inference in scientific modelling and robustness. Journal of the Royal Statistical Society. Series A , 143:383--430

    work page 1980

  3. [3]

    Bürkner, P.-C., Gabry, J., Kay, M., and Vehtari, A. (2024). posterior: Tools for working with posterior distributions. R package version 1.6.0

    work page 2024

  4. [4]

    Carpenter, B. (2018). Predator-prey population dynamics: the Lotka - Volterra model in Stan . Technical report, Columbia University

    work page 2018

  5. [5]

    Chan, J., Perrone, V., Spence, J., Jenkins, P., Mathieson, S., and Song, Y. (2018). A likelihood-free inference framework for population genetic data using exchangeable neural networks. Neural Information Processing Systems , 31

    work page 2018

  6. [6]

    U., Courville, A., and Zhu, Z

    Chen, Y., Zhang, D., Gutmann, M. U., Courville, A., and Zhu, Z. (2021). Neural approximate sufficient statistics for implicit models. In International Conference on Learning Representations

    work page 2021

  7. [7]

    R., Gelman, A., and Rubin, D

    Cook, S. R., Gelman, A., and Rubin, D. B. (2006). Validation of software for Bayesian models using posterior quantiles. Journal of Computational and Graphical Statistics , 15(3):675--692

    work page 2006

  8. [8]

    R., Macke, J

    Dax, M., Wildberger, J., Buchholz, S., Green, S. R., Macke, J. H., and Schölkopf, B. (2023). Flow matching for scalable simulation-based inference. In Neural Information Processing Systems

    work page 2023

Show all 53 references
  1. [9]

    K., Shadlen, M

    Drugowitsch, J., Moreno-Bote, R., Churchland, A. K., Shadlen, M. N., and Pouget, A. (2012). The cost of accumulating evidence in perceptual decision making. The Journal of Neuroscience , 32(11):3612–3628

    2012

  2. [10]

    and Mahr, T

    Gabry, J. and Mahr, T. (2024). Plotting for Bayesian Models . https://mc-stan.org/bayesplot/

    2024

  3. [11]

    Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., and Gelman, A. (2019). Visualization in Bayesian workflow. Journal of the Royal Statistical Society: Series A (Statistics in Society) , 182(2):389--402. tex.ids= gabryVisualizationBayesianWorkflow2019a arXiv: 1709.01449

    2019 arXiv

  4. [12]

    Geffner, T., Papamakarios, G., and Mnih, A. (2023). Compositional score modeling for simulation-based inference. In Krause, A., Brunskill, E., Cho, K., Engelhardt, B., Sabato, S., and Scarlett, J., editors, Proceedings of the 40th International Conference on Machine Learning ,...

    2023

  5. [13]

    Gelman, A. (2017). Correction to Cook , Gelman , and Rubin (2006). Journal of Computational and Graphical Statistics , 26(4):940--940

    2017

  6. [14]

    B., Stern, H

    Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis . CRC Press, third edition edition

    2013

  7. [15]

    Gelman, A., Meng, X.-L., and Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies. Statistica sinica , pages 733--760. Publisher: JSTOR

    1996

  8. [16]

    Gelman, A., Simpson, D., and Betancourt, M. (2017). The prior can often only be understood in the context of the likelihood. Entropy , 19(10):555

    2017

  9. [17]

    C., Carpenter, B., Yao, Y., Kennedy, L., Gabry, J., Bürkner, P.-C., and Modrák, M

    Gelman, A., Vehtari, A., Simpson, D., Margossian, C. C., Carpenter, B., Yao, Y., Kennedy, L., Gabry, J., Bürkner, P.-C., and Modrák, M. (2020). Bayesian workflow. arXiv:2011.01808

    2020 arXiv

  10. [18]

    K., Porcaro, C., Mayhew, S

    Georgie, Y. K., Porcaro, C., Mayhew, S. D., Bagshaw, A. P., and Ostwald, D. (2018). A perceptual decision making EEG/fMRI data set. bioRxiv 253047

    2018

  11. [19]

    A., and Nunez, M

    Ghaderi-Kangavari, A., Rad, J. A., and Nunez, M. D. (2023). A general integrative neurocognitive modeling framework to jointly describe EEG and decision-making on single trials. Computational Brain and Behavior , 6:317--376

    2023

  12. [20]

    E., Forstmann, B

    Hawkins, G. E., Forstmann, B. U., Wagenmakers, E.-J., Ratcliff, R., and Brown, S. D. (2015). Revisiting the evidence for collapsing boundaries and urgency signals in perceptual decision-making. The Journal of Neuroscience , 35(6):2476–2484

    2015

  13. [21]

    Hoffman, M. D. and Gelman, A. (2014). The No - U -turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo . Journal of Machine Learning Research , 15(1):1593--1623

    2014

  14. [22]

    H., Acerbi, L., and Kaski, S

    Huang, D., Bharti, A., Souza, A. H., Acerbi, L., and Kaski, S. (2023). Learning robust statistics for simulation-based inference under model misspecification. In Neural Information Processing Systems

    2023

  15. [23]

    H., Modr \' a k, M., Säilynoja, T., Fazio, L., and Stenz, T

    Kim, S., Moon, A. H., Modr \' a k, M., Säilynoja, T., Fazio, L., and Stenz, T. (2024). Simulation-based calibration: Sbc. https://hyunjimoon.github.io/SBC/

    2024

  16. [24]

    Mahaffy, J. M. (2010). Math 636 - Mathematical Modeling Fall Semester , 2010 Lotka - Volterra Models . http://jmahaffy.sdsu.edu/courses/f09/math636/lectures/lotka/qualde2.html

    2010

  17. [25]

    H., Kim, S., Bürkner, P., Huurre, N., Faltejsková, K., Gelman, A., and Vehtari, A

    Modrák, M., Moon, A. H., Kim, S., Bürkner, P., Huurre, N., Faltejsková, K., Gelman, A., and Vehtari, A. (2023). Simulation-based calibration checking for Bayesian computation: The choice of test quantities shapes sensitivity. Bayesian Analysis

    2023

  18. [26]

    Neal, R. M. (2003). Slice sampling. Annals of Statistics , 31(3):705--767

    2003

  19. [27]

    O'Brien, T., Moores, M., Warton, D., and Falster, D. (2025). Here be dragons: Bimodal posteriors arise from numerical integration error in longitudinal models. arXiv preprint arXiv:2502.11510

    2025

  20. [28]

    Odum, E. P. and Barrett, G. W. (2005). Fundamentals of ecology . Thomson Brooks/Cole, Belmont, CA, 5th edition

    2005

  21. [29]

    J., Mohamed, S., and Lakshminarayanan, B

    Papamakarios, G., Nalisnick, E., Rezende, D. J., Mohamed, S., and Lakshminarayanan, B. (2021). Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research , 22(1)

    2021

  22. [30]

    O., and Sköld, M

    Papaspiliopoulos, O., Roberts, G. O., and Sköld, M. (2007). A general framework for the parametrization of hierarchical models. Statistical Science , 22(1)

    2007

  23. [31]

    T., Mertens, U

    Radev, S. T., Mertens, U. K., Voss, A., Ardizzone, L., and K \"o the, U. (2020a). BayesFlow : Learning complex stochastic models with invertible neural networks. IEEE Transactions on Neural Networks and Learning Systems

    2020

  24. [32]

    T., Mertens, U

    Radev, S. T., Mertens, U. K., Voss, A., and K \"o the, U. (2020b). Towards end-to-end likelihood-free inference with convolutional neural networks. British Journal of Mathematical and Statistical Psychology , 73(1):23--43

    2020

  25. [33]

    u ller, L., Pratz, V., Sch \

    Radev, S. T., Schmitt, M., Schumacher, L., Elsem \"u ller, L., Pratz, V., Sch \"a lte, Y., K \"o the, U., and B \"u rkner, P.-C. (2023). BayesFlow : Amortized Bayesian workflows with neural networks. Journal of Open Source Software , 8(89):5702

    2023

  26. [34]

    and McKoon, G

    Ratcliff, R. and McKoon, G. (2008). The diffusion decision model: Theory and data for two-choice decision tasks. Neural Computation , 20(4):873--922

    2008

  27. [35]

    L., Brown, S

    Ratcliff, R., Smith, P. L., Brown, S. D., and McKoon, G. (2016). Diffusion decision model: Current issues and history. Trends in Cognitive Sciences , 20(4):260–281

    2016

  28. [36]

    Rubin, D. B. (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. Annals of Statistics , 12:1151--1172

    1984

  29. [37]

    J., Betancourt, M., and Vasishth, S

    Schad, D. J., Betancourt, M., and Vasishth, S. (2020). Toward a principled Bayesian workflow in cognitive science. arXiv:1904.12765

    2020 arXiv

  30. [38]

    J., Nicenboim, B., B \"u rkner, P.-C., Betancourt, M., and Vasishth, S

    Schad, D. J., Nicenboim, B., B \"u rkner, P.-C., Betancourt, M., and Vasishth, S. (2023). Workflow techniques for the robust use of bayes factors. Psychological methods , 28(6):1404

    2023

  31. [39]

    Schmitt, M., Bürkner, P.-C., Köthe, U., and Radev, S. T. (2023a). Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks . In 45th German Conference on Pattern Recognition (GCPR)

    2023

  32. [40]

    Schmitt, M., Li, C., Vehtari, A., Acerbi, L., B \"u rkner, P.-C., and Radev, S. T. (2024). Amortized Bayesian workflow (extended abstract). In NeurIPS Workshop on Bayesian Decision-Making and Uncertainty

    2024

  33. [41]

    Schmitt, M., Pratz, V., Köthe, U., Bürkner, P.-C., and Radev, S. T. (2023b). Consistency models for scalable and fast simulation-based inference. arXiv:2312.05440

    2023 arXiv

  34. [42]

    Sharrock, L., Simons, J., Liu, S., and Beaumont, M. (2024). Sequential neural score estimation: Likelihood-free inference with conditional score based diffusion models. In Salakhutdinov, R., Kolter, Z., Heller, K., Weller, A., Oliver, N., Scarlett, J., and Berkenkamp, F., edit...

    2024

  35. [43]

    Stan Users Guide and Reference Manual

    Stan Development Team (2023). Stan Users Guide and Reference Manual . https://mc-stan.org

    2023

  36. [44]

    S trumbelj, E., Bouchard-C\^ o té, A., Corander, J., Gelman, A., Rue, H., Murray, L., Pesonen, H., Plummer, M., and Vehtari, A. (2024). Past, present, and future of software for Bayesian inference. Statistical Science , 39(1):46--61

    2024

  37. [45]

    Säilynoja, T., Bürkner, P.-C., and Vehtari, A. (2022). Graphical test for discrete uniformity and its applications in goodness-of-fit evaluation and multiple sample comparison. Statistics and Computing , 32(2):32

    2022

  38. [46]

    Talts, S., Betancourt, M., Simpson, D., Vehtari, A., and Gelman, A. (2020). Validating Bayesian inference algorithms with simulation-based calibration. arXiv:1804.06788

    2020 arXiv

  39. [47]

    Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., and B \"u rkner, P.-C. (2021a). Rank-normalization, folding, and localization: An improved R for assessing convergence of MCMC . Bayesian Analysis , 16:667--718

    2021

  40. [48]

    Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., and Bürkner, P.-C. (2021b). Rank-normalization, folding, and localization: An improved R for assessing convergence of MCMC (with discussion). Bayesian Analysis , 16(2):667--718

    2021

  41. [49]

    Voss, A., Rothermund, K., and Voss, J. (2004). Interpreting the parameters of the diffusion model: An empirical validation . Memory & Cognition , 32(7):1206--1220

    2004

  42. [50]

    Yao, Y., Vehtari, A., Simpson, D., and Gelman, A. (2018). Yes, but did it work?: Evaluating variational inference. In Proceedings of the 35th International Conference on Machine Learning , pages 5581--5590

    2018

  43. [51]

    Zaheer, M., Kottur, S., Ravanbhakhsh, S., P\' o czos, B., Salakhutdinov, R., and Smola, A. J. (2017). Deep sets. In Neural Information Processing Systems , page 3394–3404

    2017

  44. [52]

    Zammit-Mangion, A., Sainsbury-Dale, M., and Huser, R. (2024). Neural methods for amortised inference. arXiv:2404.12484

    2024 arXiv

  45. [53]

    Zhang, L., Carpenter, B., Gelman, A., and Vehtari, A. (2022). Pathfinder: Parallel quasi- Newton variational inference. Journal of Machine Learning Research , 23(306):1--49

    2022

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.