pith. sign in

arxiv: 2606.23838 · v1 · pith:MYPCF37Unew · submitted 2026-06-22 · 💻 cs.LG · astro-ph.IM· physics.comp-ph· physics.data-an· stat.ML

The Degeneracy Distillery

T. Lucas Makinen , Deaglan J. Bartlett , Niall Jeffrey , Benjamin D. Wandelt This is my paper

Pith reviewed 2026-06-26 08:59 UTC · model grok-4.3

classification 💻 cs.LG astro-ph.IMphysics.comp-phphysics.data-anstat.ML
keywords degeneracyFisher information matrixsymbolic transformationsneural posterior estimationinformation geometrysimulation-based inferenceparameter estimation
0
0 comments X

The pith

The degeneracy distillery finds symbolic transformations that flatten the Fisher information matrix globally from parameter-simulation pairs alone.

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

The paper presents a method that detects and resolves degenerate parameter combinations automatically and symbolically using only parameter-simulation pairs. It does so by estimating the Fisher information matrix and discovering coordinate transformations that make different parameters produce independent effects on the data. These transformations flatten the matrix in expectation across the whole parameter space rather than at one point. The result is a lower simulation budget for neural posterior estimation, up to 10 times fewer runs at matched calibration, together with direct physical insight into which parameter combinations matter.

Core claim

By exploring the information geometry of the likelihood, degeneracies are characterized as an intrinsic property of the physical model requiring no realised data observation; symbolic coordinate transformations derived from parameter-simulation pairs then identify independent parameter effects and flatten the Fisher information in expectation globally.

What carries the argument

Estimation and symbolic flattening of the Fisher information matrix from parameter-simulation pairs to remove degeneracies.

If this is right

  • Degeneracies become identifiable without any realised data observations.
  • Neural posterior estimation requires up to 10 times fewer simulations while maintaining calibration.
  • The resulting coordinates provide physical insight into which parameter combinations act independently on the data.
  • Flattening holds globally in expectation rather than only at a single point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Precomputed transformations could speed up repeated inference tasks on similar model families.
  • The same machinery might expose structural deficiencies in a model by revealing degeneracies that cannot be removed.
  • The approach may transfer to other simulation-heavy inverse problems where distinguishability of parameters limits performance.

Load-bearing premise

That symbolic coordinate transformations exist which can flatten the Fisher information matrix in expectation globally for the models of interest and can be reliably estimated from parameter-simulation pairs alone.

What would settle it

A test case in which no symbolic transformation flattens the expected Fisher information or in which the method requires at least as many simulations as standard posterior estimation at matched calibration.

Figures

Figures reproduced from arXiv: 2606.23838 by Benjamin D. Wandelt, Deaglan J. Bartlett, Niall Jeffrey, T. Lucas Makinen.

Figure 1
Figure 1. Figure 1: Degeneracy distillery pipeline in three steps. (1) Parameter-data pairs [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Rosenbrock validation case. Left: Learned Fisher curvature (top) is flattened by learning 𝜂 (bottom). Right: Knowing the symbolic, flat geometry A) makes learning posteriors easier even with increasing simulation budget B) yields faithful posteriors in target 𝜃 coordinates and C) is better calibrated than learning in degenerate 𝜃 space. complex, or have many or finely tuned parameters, and is an approximat… view at source ↗
Figure 3
Figure 3. Figure 3: Left: Neural and symbolic regression components outperform (i) hand-picked (ad-hoc) coordinates and (ii) flatten geometry far from the fiducial point Eq. (74), without the need for an analytic Fisher matrix to derive geodesic normal coordinates. Right: Inference validation log 𝑝(·|𝑦) shows little degredation for NPE estimating distilled 𝜂 = Î 𝑖 𝜃𝑖 ∈ R for the chain-product heater simulator with intrinsic d… view at source ↗
Figure 4
Figure 4. Figure 4: Epidemiology example. Distilled coordinates flatten learned curvature ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Main GW result (IMRPhenomD). Left: Example Fisher matrices in (𝑚1, 𝑚2) and in the learned 𝜂 coordinates. Middle: posterior contours from neural posterior estimators trained in 𝜃 = (𝑚1, 𝑚2), in the 𝜂 basis, and in the conventional (M𝑐, 𝑞) basis. The 𝜂-trained posterior is round and centred on the truth, while both 𝜃-space and (M𝑐, 𝑞) posteriors stretch into bananas along the M𝑐 direction (grey contours). Ri… view at source ↗
Figure 6
Figure 6. Figure 6: Top panels: Fishnets provides an unbiased estimate of a) unique Cholesky factors and b) normalised shape of the true Rosenbrock Fisher Matrix. Bottom panel: Learned flat coordinates 𝜂 and their ensembled-estimated uncertainties 𝛿𝜂 approximately follow a Gaussian sampling distribution. 0 20 40 Time 0.00 0.05 0.10 0.15 0.20 I(t)/N Example infection curves R₀=1.34 R₀=2.42 R₀=1.66 R₀=2.12 R₀=2.02 0.25 0.50 0.7… view at source ↗
Figure 7
Figure 7. Figure 7: Example SIR epidemic (𝑅0 > 1.0) time-series data and parameter values, coloured by 𝑅0. inference coordinate; we generate 2 500 training and 2 500 test simulations. For the epidemic-regime analysis we retain samples with 𝑅0 ≥ 1+𝛿 with 𝛿 = 0.15, and discard the band 0.85 ≤ 𝑅0 < 1.15; the corresponding non-epidemic subset is 𝑅0 < 0.85. We display example simulations and parameters in [PITH_FULL_IMAGE:figures… view at source ↗
Figure 8
Figure 8. Figure 8: Supplementary details for the SIR experiment. [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Example NPE posteriors using learned versus conventional weak lensing coordinates. [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Benchmark GW result, TaylorF2 (inspiral-only, 2PN, non-spinning). Same panel layout as [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
read the original abstract

When two or more parameters or labels produce similar data, they are degenerate, or hard to distinguish. Degeneracies render both label prediction and inverse problems difficult, since both machine learning algorithms and probabilistic samplers rely on the distinguishability of data and its gradients with respect to parameters. However, identifying degeneracies in physical models or real-world datasets can be elucidating about the choice of model or the underlying process that produces the data. We present the degeneracy distillery, a method that (1) detects and (2) resolves degenerate parameter combinations (a) automatically and (b) symbolically, from parameter-data (or parameter-simulation) pairs alone, through estimation and flattening of the Fisher information matrix. By exploring the information geometry of the likelihood, we characterize degeneracies as an intrinsic property of the physical model, requiring no realised data observation. We demonstrate our approach on a range of synthetic and real-world problems, discovering symbolic coordinate transformations that identify the combinations of parameters of a model which yield independent effects on the data. The resulting coordinates flatten the Fisher information in expectation globally, in contrast to posterior-based methods that flatten only at a single point, and substantially reduce the simulation budget required for downstream neural posterior estimation. In test cases we require up to $10\times$ fewer simulations for posterior estimation at matched validation calibration whilst simultaneously gaining physical insight on the system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents the Degeneracy Distillery, a method that detects and resolves degenerate parameter combinations automatically and symbolically from parameter-simulation pairs alone by estimating the Fisher information matrix and finding coordinate transformations that flatten it in expectation. The central claims are that the resulting coordinates achieve global flattening (unlike posterior-based methods that act only at a single point) and reduce the simulation budget for downstream neural posterior estimation by up to 10× at matched validation calibration, while also yielding physical insight.

Significance. If the claims on global flattening and simulation reduction hold with rigorous validation, the work would provide a useful tool for reparameterizing degenerate models in simulation-based inference, combining information geometry with symbolic methods to improve efficiency and interpretability without requiring realized observations.

major comments (2)
  1. [Abstract] Abstract: the claim that the coordinates 'flatten the Fisher information in expectation globally' is load-bearing for the novelty relative to posterior-based methods. The stress-test correctly identifies that an estimate formed by averaging over parameter-simulation pairs recovers only an averaged metric; the manuscript must supply the explicit derivation or proof (presumably in the methods) showing that the discovered symbolic transformation renders the local metric flat at every point rather than merely its expectation, particularly when the underlying information manifold has non-vanishing curvature.
  2. [Results] Results (or equivalent section reporting the 10× claim): the quantitative statement 'up to 10× fewer simulations for posterior estimation at matched validation calibration' requires supporting tables or figures with the exact calibration metric (e.g., coverage or C2ST), the baseline method, and error bars across the synthetic and real-world test cases; without these, the efficiency gain cannot be assessed as load-bearing evidence.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'synthetic and real-world problems' is vague; a parenthetical list of the specific models or datasets would improve immediate readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address the major comments point-by-point below. Where revisions are needed to clarify or strengthen the manuscript, we indicate our plans to update the next version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the coordinates 'flatten the Fisher information in expectation globally' is load-bearing for the novelty relative to posterior-based methods. The stress-test correctly identifies that an estimate formed by averaging over parameter-simulation pairs recovers only an averaged metric; the manuscript must supply the explicit derivation or proof (presumably in the methods) showing that the discovered symbolic transformation renders the local metric flat at every point rather than merely its expectation, particularly when the underlying information manifold has non-vanishing curvature.

    Authors: We thank the referee for this insightful comment, which helps clarify the scope of our contribution. The degeneracy distillery derives a symbolic transformation by estimating and diagonalizing the expected Fisher information matrix from parameter-simulation pairs. This yields a global coordinate system in which the expected information matrix is flattened (i.e., diagonal and scaled to identity in expectation). The 'globally' qualifier refers to the fact that the transformation is not point-dependent, unlike methods that reparameterize based on a local posterior approximation. We do not claim that the local Fisher metric is exactly flat at every individual point when the manifold has curvature; rather, the expectation over the parameter distribution is flattened. The derivation of the transformation from the averaged FIM is provided in the Methods. To prevent misinterpretation, we will revise the abstract to emphasize 'flattens the expected Fisher information globally' and add a clarifying paragraph in the Methods discussing the distinction from pointwise flattening and the role of manifold curvature. revision: yes

  2. Referee: [Results] Results (or equivalent section reporting the 10× claim): the quantitative statement 'up to 10× fewer simulations for posterior estimation at matched validation calibration' requires supporting tables or figures with the exact calibration metric (e.g., coverage or C2ST), the baseline method, and error bars across the synthetic and real-world test cases; without these, the efficiency gain cannot be assessed as load-bearing evidence.

    Authors: We agree that the 10× efficiency claim requires more detailed quantitative support to be fully convincing. While the manuscript presents results demonstrating reduced simulation budgets with matched calibration in figures, we will add a dedicated table in the Results section. This table will report the exact calibration metrics used (C2ST and coverage), the baseline (standard NPE without degeneracy distillery), the achieved simulation reduction factors, and error bars from repeated experiments for all synthetic and real-world cases. This will make the evidence for the efficiency gains explicit and reproducible. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper describes a procedure that estimates the expected Fisher information matrix from parameter-simulation pairs and then applies symbolic search to discover coordinate transformations that flatten this matrix. The flattening result is the explicit output of the search step rather than a quantity presupposed by the inputs; the claim of global flattening in expectation follows directly from the optimization objective and is not equivalent to the input data by construction. No self-citation load-bearing step, uniqueness theorem imported from the same authors, or fitted parameter renamed as a prediction appears in the abstract or described method. The approach builds on the standard definition of the Fisher matrix as a measure of local distinguishability and therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the domain assumption that the Fisher information matrix estimated from simulations captures intrinsic degeneracies and that symbolic transforms can globally diagonalize it; no free parameters or invented entities are introduced in the abstract description.

axioms (1)
  • domain assumption The Fisher information matrix estimated from parameter-simulation pairs characterizes degeneracies as an intrinsic property of the physical model.
    This is the core premise enabling the method without realized data.

pith-pipeline@v0.9.1-grok · 5790 in / 1261 out tokens · 31682 ms · 2026-06-26T08:59:30.590239+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

51 extracted references · 29 canonical work pages

  1. [1]

    Aasi et al

    J. Aasi et al. Advanced LIGO.Class. Quantum Grav., 32:074001, 2015. doi: 10.1088/ 0264-9381/32/7/074001

    2015

  2. [2]

    Fast likelihood-free cosmology with neural density estimators and active learning.Monthly Notices of the Royal Astronomical Society, Jul 2019

    Justin Alsing, Tom Charnock, Stephen Feeney, and Benjamin Wandelt. Fast likelihood-free cosmology with neural density estimators and active learning.Monthly Notices of the Royal Astronomical Society, Jul 2019. ISSN 1365-2966. doi: 10.1093/mnras/stz1960. URLhttp: //dx.doi.org/10.1093/mnras/stz1960

    work page doi:10.1093/mnras/stz1960 2019

  3. [3]

    Priors for symbolic regression

    Deaglan Bartlett, Harry Desmond, and Pedro Ferreira. Priors for symbolic regression. In ProceedingsoftheCompanionConferenceonGeneticandEvolutionaryComputation,GECCO ’23 Companion, page 2402–2411. ACM, July 2023. doi: 10.1145/3583133.3596327. URL http://dx.doi.org/10.1145/3583133.3596327

    work page doi:10.1145/3583133.3596327 2023

  4. [4]

    Bartlett, Harry Desmond, and Pedro G

    Deaglan J. Bartlett, Harry Desmond, and Pedro G. Ferreira. Exhaustive symbolic regression. IEEE Transactions on Evolutionary Computation, 28(4):950–964, August 2024. ISSN 1941-

    2024

  5. [5]

    URLhttp://dx.doi.org/10.1109/TEVC.2023

    doi: 10.1109/tevc.2023.3280250. URLhttp://dx.doi.org/10.1109/TEVC.2023. 3280250

    work page doi:10.1109/tevc.2023.3280250 2023

  6. [6]

    Weak lensing statistics as a probe of𝜔and power spectrum.Astronomy & Astrophysics, 322:1–18, 1997

    Francis Bernardeau, Ludovic van Waerbeke, and Yannick Mellier. Weak lensing statistics as a probe of𝜔and power spectrum.Astronomy & Astrophysics, 322:1–18, 1997

    1997

  7. [7]

    C. M. Biwer, Collin D. Capano, Soumi De, Miriam Cabero, Duncan A. Brown, Alexander H. Nitz, and V. Raymond. PyCBC Inference: A Python-based parameter estimation toolkit for compact binary coalescence signals.Publ. Astron. Soc. Pac., 131:024503, 2019. doi: 10.1088/1538-3873/aaef0b

    work page doi:10.1088/1538-3873/aaef0b 2019

  8. [8]

    2014, Living Reviews in Relativity, 17, 2, doi: 10.12942/lrr-2014-2

    Luc Blanchet. Gravitational radiation from post-Newtonian sources and inspiralling compact binaries.Living Rev. Relativ., 17:2, 2014. doi: 10.12942/lrr-2014-2

    work page doi:10.12942/lrr-2014-2 2014

  9. [9]

    Iyer, Evan Ochsner, Yi Pan, and B

    Alessandra Buonanno, Bala R. Iyer, Evan Ochsner, Yi Pan, and B. S. Sathyaprakash. Com- parison of post-Newtonian templates for compact binary inspiral signals in gravitational-wave detectors.Phys. Rev. D, 80:084043, 2009. doi: 10.1103/PhysRevD.80.084043

    work page doi:10.1103/physrevd.80.084043 2009

  10. [10]

    Operon c++: An efficient genetic programming framework for symbolic regression

    Bogdan Burlacu, Gabriel Kronberger, and Michael Kommenda. Operon c++: An efficient genetic programming framework for symbolic regression. InProceedings of the 2020 Genetic and Evolutionary Computation Conference Companion, GECCO ’20, page 1562–1570, New York, NY, USA, 2020. Association for Computing Machinery. ISBN 9781450371278. doi: 10.1145/3377929.3398...

    work page doi:10.1145/3377929.3398099 2020

  11. [11]

    What does a cosmologicalexperimentreallymeasure? Covariantposteriordecompositionwithnormalizing flows.Physical Review D, 105(6):063529, March 2022

    Tara Dacunha, Marco Raveri, Minsu Park, Cyrille Doux, and Bhuvnesh Jain. What does a cosmologicalexperimentreallymeasure? Covariantposteriordecompositionwithnormalizing flows.Physical Review D, 105(6):063529, March 2022. doi: 10.1103/PhysRevD.105.063529

    work page doi:10.1103/physrevd.105.063529 2022

  12. [12]

    DensityestimationusingrealNVP

    LaurentDinh,JaschaSohl-Dickstein,andSamyBengio. DensityestimationusingrealNVP. In International Conference on Learning Representations, 2017. URLhttps://openreview. net/forum?id=HkpbnH9lx

    2017

  13. [13]

    Wichrowski, George A

    Nikolaos Evangelou, Noah J. Wichrowski, George A. Kevrekidis, Felix Dietrich, Mahdi Kooshkbaghi, Sarah McFann, and Ioannis G. Kevrekidis. On the Parameter Combinations That Matter and on Those That do Not.arXiv e-prints, art. arXiv:2110.06717, October 2021. doi: 10.48550/arXiv.2110.06717

    work page doi:10.48550/arxiv.2110.06717 2021

  14. [14]

    GeometricVariationalInference.Entropy, 23(7):853, July 2021

    PhilippFrank,ReimarLeike,andTorstenA.Enßlin. GeometricVariationalInference.Entropy, 23(7):853, July 2021. doi: 10.3390/e23070853

    work page doi:10.3390/e23070853 2021

  15. [15]

    Ali Hassani, Steven Walton, Jiachen Li, Shen Li, and Humphrey Shi

    Tilmann Gneiting and Adrian E Raftery. Strictly Proper Scoring Rules, Prediction, and Es- timation.Journal of the American Statistical Association, 102(477):359–378, March 2007. ISSN 0162-1459. doi: 10.1198/016214506000001437. URLhttps://doi.org/10.1198/ 016214506000001437. _eprint: https://doi.org/10.1198/016214506000001437. 11

    work page doi:10.1198/016214506000001437 2007

  16. [16]

    Gutenkunst, Joshua J

    Ryan N. Gutenkunst, Joshua J. Waterfall, Fergal P. Casey, Kevin S. Brown, Christopher R. Myers, and James P. Sethna. Universally Sloppy Parameter Sensitivities in Systems Biology Models.PLoS Computational Biology, 3(10):e189, January 2007. doi: 10.1371/journal.pcbi. 0030189

    work page doi:10.1371/journal.pcbi 2007

  17. [17]

    Hethcote

    Herbert W. Hethcote. The mathematics of infectious diseases.SIAM Review, 42(4):599–653,

  18. [18]

    doi: 10.1137/S0036144500371907

    work page doi:10.1137/s0036144500371907

  19. [19]

    Lovell, T

    Matthew Ho, Deaglan Bartlett, Nicolas Chartier, Carolina Cuesta-Lazaro, Simon Ding, Axel Lapel, Pablo Lemos, Christopher C. Lovell, T. Lucas Makinen, Chirag Modi, Viraj Pandya, Shivam Pandey, Lucia Perez-Granado, Hong-Yu Shao, Georgios Valogiannis, and Benjamin Wandelt. Ltu-ili: Anall-in-oneframeworkforimplicitinferenceinastrophysicsandcosmology. arXiv:24...

    arXiv 2024

  20. [20]

    Householder

    Alston S. Householder. Unitary Triangularization of a Nonsymmetric Matrix.J. ACM, 5 (4):339–342, October 1958. ISSN 0004-5411. doi: 10.1145/320941.320947. URLhttps: //dl.acm.org/doi/10.1145/320941.320947

    work page doi:10.1145/320941.320947 1958

  21. [21]

    Frequency-domain gravitational waves from nonprecessing black-hole binaries

    Sascha Husa, Sebastian Khan, Mark Hannam, Michael Pürrer, Frank Ohme, Xisco Jiménez Forteza, and Alejandro Bohé. Frequency-domain gravitational waves from nonprecessing black-hole binaries. I. New numerical waveforms and anatomy of the signal.Phys. Rev. D, 93: 044006, 2016. doi: 10.1103/PhysRevD.93.044006

    work page doi:10.1103/physrevd.93.044006 2016

  22. [22]

    Incropera, David P

    Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, and Adrienne S. Lavine.Funda- mentalsofHeatandMassTransfer. Wiley,Hoboken,NJ,7edition,2011. Lumped-capacitance method, Ch. 5; canonical derivation of the first-order thermal step response used in our simu- lator

    2011

  23. [23]

    Cosmologicalmodelpredictionsforweaklensing: Linearand nonlinear regimes.The Astrophysical Journal, 484(2):560–573, 1997

    BhuvneshJainandUrošSeljak. Cosmologicalmodelpredictionsforweaklensing: Linearand nonlinear regimes.The Astrophysical Journal, 484(2):560–573, 1997. doi: 10.1086/304372

    work page doi:10.1086/304372 1997

  24. [24]

    W.Kabsch. Asolutionforthebestrotationtorelatetwosetsofvectors.ActaCrystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography, 32(5):922– 923, September 1976. ISSN 0567-7394. doi: 10.1107/S0567739476001873. URLhttps: //journals.iucr.org/a/issues/1976/05/00/a12999/. Publisher: International Union of Crystallography

    work page doi:10.1107/s0567739476001873 1976

  25. [25]

    McKendrick

    William Ogilvy Kermack and Anderson G. McKendrick. A contribution to the mathematical theoryofepidemics.ProceedingsoftheRoyalSocietyofLondon.SeriesA,115(772):700–721,

  26. [26]

    doi: 10.1098/rspa.1927.0118

    work page doi:10.1098/rspa.1927.0118 1927

  27. [27]

    Frequency-domain gravitational waves from nonprecessing black-hole binaries

    Sebastian Khan, Sascha Husa, Mark Hannam, Frank Ohme, Michael Pürrer, Xisco Jiménez Forteza, and Alejandro Bohé. Frequency-domain gravitational waves from nonprecessing black-hole binaries. II. A phenomenological model for the advanced detector era.Phys. Rev. D, 93:044007, 2016. doi: 10.1103/PhysRevD.93.044007

    work page doi:10.1103/physrevd.93.044007 2016

  28. [28]

    Cosmology with cosmic shear observations: a review.Reports on Progress in Physics, 78(8):086901, 2015

    Martin Kilbinger. Cosmology with cosmic shear observations: a review.Reports on Progress in Physics, 78(8):086901, 2015. doi: 10.1088/0034-4885/78/8/086901

    work page doi:10.1088/0034-4885/78/8/086901 2015

  29. [29]

    Winkler, and Michael Affenzeller.Symbolic Regression

    Gabriel Kronberger, Bogdan Burlacu, Michael Kommenda, Stephan M. Winkler, and Michael Affenzeller.Symbolic Regression. Chapman & Hall / CRC Press, 2024

    2024

  30. [30]

    Springer Berlin Heidelberg, New York, NY,

    John Lee.Introduction to Riemannian manifolds. Springer Berlin Heidelberg, New York, NY,

  31. [31]

    ISBN 978-3-319-91754-2

  32. [32]

    Sampling- based accuracy testing of posterior estimators for general inference, 2023

    Pablo Lemos, Adam Coogan, Yashar Hezaveh, and Laurence Perreault-Levasseur. Sampling- based accuracy testing of posterior estimators for general inference, 2023. URLhttps: //arxiv.org/abs/2302.03026

    arXiv 2023

  33. [33]

    pmwd: ADifferentiableCosmolog- ical Particle-Mesh N-body Library, 2022

    Yin Li, Libin Lu, Chirag Modi, Drew Jamieson, Yucheng Zhang, Yu Feng, Wenda Zhou, NgaiPokKwan,FrançoisLanusse,andLeslieGreengard. pmwd: ADifferentiableCosmolog- ical Particle-Mesh N-body Library, 2022. _eprint: 2211.09958. 12

    arXiv 2022

  34. [34]

    Lucas Makinen, Justin Alsing, and Benjamin D

    T. Lucas Makinen, Justin Alsing, and Benjamin D. Wandelt. Fishnets: Scalable neural com- pression with optimal aggregation.arxiv, 2023, Aug 2023

    2023

  35. [35]

    Lucas Makinen, Alan Heavens, Natalia Porqueres, Tom Charnock, Axel Lapel, and Benjamin D

    T. Lucas Makinen, Alan Heavens, Natalia Porqueres, Tom Charnock, Axel Lapel, and Benjamin D. Wandelt. Hybrid summary statistics: neural weak lensing inference be- yond the power spectrum.Journal of Cosmology and Astroparticle Physics, 2025(01): 095, January 2025. ISSN 1475-7516. doi: 10.1088/1475-7516/2025/01/095. URL http://dx.doi.org/10.1088/1475-7516/2...

    work page doi:10.1088/1475-7516/2025/01/095 2025

  36. [36]

    P., Paprocki, M., et al

    Aaron Meurer, Christopher P. Smith, Mateusz Paprocki, Ondřej Čertík, Sergey B. Kirpichev, Matthew Rocklin, AMiT Kumar, Sergiu Ivanov, Jason K. Moore, Sartaj Singh, Thilina Rathnayake, Sean Vig, Brian E. Granger, Richard P. Muller, Francesco Bonazzi, Harsh Gupta, Shivam Vats, Fredrik Johansson, Fabian Pedregosa, Matthew J. Curry, Andy R. Terrel, Štěpán Rou...

    work page doi:10.7717/peerj-cs.103 2017

  37. [37]

    An Elementary Introduction to Information Geometry.Entropy, 22(10):1100, October 2020

    Frank Nielsen. An Elementary Introduction to Information Geometry.Entropy, 22(10):1100, October 2020. ISSN 1099-4300. doi: 10.3390/e22101100. URLhttps://www.mdpi.com/ 1099-4300/22/10/1100

    work page doi:10.3390/e22101100 2020

  38. [38]

    Maskedautoregressiveflowfordensity estimation

    GeorgePapamakarios,TheoPavlakou,andIainMurray. Maskedautoregressiveflowfordensity estimation. InAdvances in Neural Information Processing Systems (NeurIPS), 2017

    2017

  39. [39]

    Normalizing flows for probabilistic modeling and inference.Journal of Machine Learning Research, 22(57):1–64, 2021

    George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshminarayanan. Normalizing flows for probabilistic modeling and inference.Journal of Machine Learning Research, 22(57):1–64, 2021

    2021

  40. [40]

    Kevrekidis, and Ronald R

    Erez Peterfreund, Ofir Lindenbaum, Felix Dietrich, Tom Bertalan, Matan Gavish, Ioannis G. Kevrekidis, and Ronald R. Coifman. Local conformal autoencoder for standardized data coor- dinates.Proceedings of the National Academy of Science, 117(49):30918–30927, December

  41. [41]

    doi: 10.1073/pnas.2014627117

    work page doi:10.1073/pnas.2014627117

  42. [42]

    Information and the accuracy attainable in the estimation of statistical parameters.Bulletin of the Calcutta Mathematical Society, 37:81–91, 1945

    Calyampudi Radhakrishna Rao. Information and the accuracy attainable in the estimation of statistical parameters.Bulletin of the Calcutta Mathematical Society, 37:81–91, 1945

    1945

  43. [43]

    H. H. Rosenbrock. An Automatic Method for Finding the Greatest or Least Value of a Function.The Computer Journal, 3(3):175–184, January 1960. ISSN 0010-4620. doi: 10.1093/comjnl/3.3.175. URLhttps://doi.org/10.1093/comjnl/3.3.175. _eprint: https://academic.oup.com/comjnl/article-pdf/3/3/175/988633/030175.pdf

    work page doi:10.1093/comjnl/3.3.175 1960

  44. [44]

    Charles J. Stone. Optimal rates of convergence for nonparametric estimators.The Annals of Statistics, 8(6):1348–1360, 1980. doi: 10.1214/aos/1176345206

    work page doi:10.1214/aos/1176345206 1980

  45. [45]

    Transtrum and Peng Qiu

    Mark K. Transtrum and Peng Qiu. Model reduction by manifold boundaries.Phys. Rev. Lett., 113:098701, Aug 2014. doi: 10.1103/PhysRevLett.113.098701. URLhttps://link.aps. org/doi/10.1103/PhysRevLett.113.098701

    work page doi:10.1103/physrevlett.113.098701 2014

  46. [46]

    Transtrum, Benjamin B

    Mark K. Transtrum, Benjamin B. Machta, and James P. Sethna. Geometry of nonlinear least squareswithapplicationstosloppymodelsandoptimization.PhysicalReviewE,83(3):036701, March 2011. doi: 10.1103/PhysRevE.83.036701

    work page doi:10.1103/physreve.83.036701 2011

  47. [47]

    Transtrum, Benjamin B

    Mark K. Transtrum, Benjamin B. Machta, Kevin S. Brown, Bryan C. Daniels, Christopher R. Myers,andJamesP.Sethna. Perspective: Sloppinessandemergenttheoriesinphysics,biology, and beyond.J. Chem. Phys., 143(1):010901, July 2015. doi: 10.1063/1.4923066. The appendices are organised as follows. Section A proves that the Fishnet objective recovers the posterior...

    work page doi:10.1063/1.4923066 2015

  48. [48]

    But given thatsgn(sinh𝛽/𝛽)=1, fromtheearlierformula,weknowthatthismustbetrue

    We can now substitute this into our earlier expression forsinh𝛽 sinh𝛽= 𝛽 𝜉 𝜇−𝜇 ★ 𝜎 = sgn(𝜉) 𝑣★ √︄ 1+ 𝜂2 𝜉2 𝜇−𝜇 ★ 𝜎 .(73) Since𝛽 >0, we require the conditionsgn(𝜉)=sgn(𝜇−𝜇 ★). But given thatsgn(sinh𝛽/𝛽)=1, fromtheearlierformula,weknowthatthismustbetrue. Wethereforefindthatwecanobtain𝛽from sinh𝛽= √︂ 1 2 (𝜇−𝜇 ★)2 + (𝜎−𝜎 ★)2 1 2 (𝜇−𝜇 ★)2 + (𝜎+𝜎 ★)2 2𝜎★𝜎 ,(74)...

  49. [49]

    If𝜇=𝜇 ★ and𝜎=𝜎 ★, then𝜉=𝜂=0

  50. [50]

    Otherwise,𝛽can be obtained from evaluating Eq. (74). The signs of𝜉and𝜂are given by Eq. (75). To obtain the magnitudes of𝜉and𝜂, one can use the results: (a) If𝜇=𝜇 ★, then|𝜂|=𝑣 ★𝛽and𝜉=0. (b) Otherwise, use Eq. (72) D.6 Metric and Jacobian for Geodesic Normal Coordinates Now we have functions for𝑢and𝑣as a function of𝜉and𝜂, let us find the metric in these new...

  51. [51]

    To verify that this has the desired behaviour, let us expand the metric around the point𝜉=𝜂=0. In this case, we obtain ˜𝑔𝑎𝑏 ≈1+ 1 6 𝜂2 −𝜂𝜉 −𝜂𝜉 𝜉 2 + O |𝑥| 3 ,(81) 23 so indeed, in these coordinates, the metric is flat at zeroth and linear order, with the correction occurring at quadratic order. Using the series expansion of Eq. (68), this can be written a...