stat.MLOct 16, 2026

ENFR: Evidence Networks with Flow-Matched References for Amortized Simulation-Based Model Inference

Authors: Kai Lehman, Jameson Dong, Carolina Cuesta-Lazaro, Niall Jeffrey, Benjamin D. Wandelt, Francisco Villaescusa-Navarro, Sven Krippendorf, Jochen Weller

Organizations: Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 München, Germany · Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA · Excellence Cluster ORIGINS, Boltzmannstr. 2, 85748 Garching, Germany · Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD, 21218, USA · 

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Abstract

The evidence, Z=p(x∣M)\mathcal{Z} = p(x | \mathcal{M}), of a data-generating model M\mathcal{M} is a powerful tool in Bayesian frameworks: it enables model inference, model misspecification tests, and goodness-of-fit assessment as an alternative to standard discrepancy statistics, e.g. χ2χ^2. While versatile, the evidence is notoriously difficult to compute, even when the likelihood is available. In both likelihood-based and simulation-based inference, estimating Z\mathcal{Z} provides an immediate solution to high-dimensional goodness-of-fit or model misspecification tests: using Z(x)\mathcal{Z}(x) as the test statistic induces a minimum-volume level-αα test. Such tests are essential to assess model fidelity, which is necessary to trust simulation-based parameter inference. Existing learned approaches largely target Bayes factors, which only measure evidence ratios. Instead, we provide accurate, fast, and amortized estimates of the absolute evidence. We present ENFR: Evidence Networks with Flow-matched References. Building on the Evidence Network framework of Jeffrey & Wandelt (2024), the proposed method first learns Bayes factors with respect to a flow-matched reference, from which the absolute evidence is recovered. We show improvement in estimation of Z\mathcal{Z} over the flow-based baseline in all applications considered and the best performance of all density estimation methods considered in our benchmarks. We furthermore introduce and provide the proof for an architecture that jointly learns the individual evidences of a set of models {Mi}\{\mathcal{M}_i\}. We demonstrate the performance and scaling behavior of these approaches on data from a variety of scientific applications.

Figures & tables

Figure 1: A visualization of our proposed SingleENFR and MultiENFR architectures. Given simulated data from one or multiple models, we train flow matching once over all models. In a second step, we use the simulated data and samples from the flow to train an evidence network classifier, which learns all Bayes factors. Those can be combined with the evaluated flow density to obtain the evidence under a given model.
Figure 2: Scaling of density-estimation accuracy with (a) data dimensionality and (b) number of training simulations on the Gaussian mixture model of Appendix A.1 , for which the true density is known analytically. We report the RMSE between estimated and true log-density over 200200 held-out test points. In (a) the simulation budget is fixed at 5×1065\times 10^{6} ; in (b) the dimensionality is fixed at 64. The line shows the mean across three trained models and the errorbars show the estimated standard deviation. SingleENFR (orange) improves on the flow it is built on (blue) at every point and is the most accurate of all estimators considered, while CTSM-VP (yellow) is consistently the least accurate.
SimulationflowSingleENFR (ours)MultiENFR (ours)CTSM-SB + FlowRef
Λ\Lambda CDM1.340.530.522.74
ww CDM1.400.760.672.46
w0waw_{0}w_{a} CDM1.160.830.813.79
ν\nu CDM1.410.470.461.77
Ωk\Omega_{k} CDM1.520.540.782.29
Table 1: RMSE of our learned log-evidence estimates against the likelihood-based log-evidence estimate over 100 held out BAO test simulations (lower is better). Our Models SingleENFR and MultiENFR consistently outperform both the flow and CTSM-SB with a flow reference.
ModelLikelihood-based analysisMultiENFRΔ\DeltaMultiENFR log⁡K\log K
ΛCDM\Lambda\mathrm{CDM}-13.99 ±\pm 0.16-14.07 ±\pm 0.34-0.080.00
wCDMw\mathrm{CDM}-16.41 ±\pm 0.18-16.00 ±\pm 0.350.40-1.94
w0waCDMw_{0}w_{a}\mathrm{CDM}-15.63 ±\pm 0.18-15.46 ±\pm 0.400.17-1.40
νCDM\nu\mathrm{CDM}-14.06 ±\pm 0.16-14.01 ±\pm 0.300.040.06
ΩkCDM\Omega_{k}\mathrm{CDM}-15.88 ±\pm 0.17-15.72 ±\pm 0.230.16-1.65
Table 2: Evidence estimates using MultiENFR on the DESI DR2 observed data. For MultiENFR we also quote the estimate of the standard deviation over the ensemble of 5 members. We can reproduce the likelihood-based evidence estimates of the DESI DR2 data with our simulation-based framework within the ensemble scatter.
Figure 3: (a) Evidence estimation for noisy observations of an ODE driven time series using flow matching and our ENFR models. MultiENFR outperforms the other models. (b) Diffusion drift application of standard flow matching and our ENFR networks. MultiENFR shows the best performance and good evidence estimation over a large dynamical range.
Figure 4: Blind coverage testing for the simulation-based applications for (a) the Hodgkin-Huxley application, and (b) the Galaxy assembly bias application. As we do not have access to a likelihood-based evidence estimate for our simulation based models we test whether the resulting Bayes factors are well calibrated. We find that our MultiENFR architecture is well calibrated in both cases. The error bars show 2σ2\sigma binomial errors.
Figure 5: Evidence estimates of the MultiENFR model on prior predictive simulations and the DESI DR2 data under the Λ\Lambda CDM model. While 46.1% of simulations have lower evidence than the data, no simulation has lower evidence than the toy w(z)w(z) model, and we can clearly pick up on the model misspecification.
Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

ParameterΛ\Lambda CDMww CDMw0waw_{0}w_{a} CDMν\nu CDMΩk\Omega_{k} CDM
Base parameters (shared)
H0H_{0}[20, 100][20,\,100]
Ωbh2\Omega_{b}h^{2}[0.005, 0.1][0.005,\,0.1]
Ωch2\Omega_{c}h^{2}[0.001, 0.99][0.001,\,0.99]
τ\tau[0.01, 0.8][0.01,\,0.8]
log⁡(1010As)\log(10^{10}A_{s})[1.61, 3.91][1.61,\,3.91]
Appendix
Table 3: Boundaries of the uniform priors for the parameters of the five cosmological models used in our BAO example. The six base parameters of the Λ\Lambda CDM model are shared by every model. Each extension introduces either one or two more parameters.
ParameterPriorHarmonicDampedDuffing
x0x_{0}U(−2, 2)\mathcal{U}(-2,\,2)✓✓✓
v0v_{0}U(−2, 2)\mathcal{U}(-2,\,2)✓✓✓
ω\omegaU(0.4, 2.5)\mathcal{U}(0.4,\,2.5)✓✓✓
γ\gammaU(0, 0.4)\mathcal{U}(0,\,0.4) ✓✓
β\betaU(−0.8, 0.8)\mathcal{U}(-0.8,\,0.8)  ✓
Appendix
Table 4: Prior distributions for the ODE oscillator models. All priors are independent uniforms; parameters shared between models have identical priors.
MethodharmonicdampedDuffingAll
flow6.133.693.914.71
SingleENFR2.542.442.602.53
MultiENFR2.360.942.171.93
flow + CTSM-SB (ens.)135.4726.237.9679.80
Appendix
Table 5: RMSE of the learned evidence against nested sampling (dynesty) on the ODE oscillator benchmark, over 10 held-out simulations per model.
ParameterDescriptionPriorM0\mathcal{M}_{0}M1\mathcal{M}_{1}
vvdrift rateU(0, 5)\mathcal{U}(0,\,5)✓✓
λ\lambdaleak rateU(−20, −5)\mathcal{U}(-20,\,-5) ✓
BBinitial boundary heightU(0.3, 2)\mathcal{U}(0.3,\,2)✓✓
κ\kappaboundary collapse rateU(0.5, 1.5)\mathcal{U}(0.5,\,1.5)✓✓
tndt_{\mathrm{nd}}non-decision time [s]U(0.1, 0.3)\mathcal{U}(0.1,\,0.3)✓✓
Appendix
Table 6: Prior distributions for the drift-diffusion models.
ModelRMSE under M0\mathcal{M}_{0}RMSE under M1\mathcal{M}_{1}
flow4.05.5
SingleENFR2.43.5
MultiENFR1.81.7
flow + CTSM-SB20.3115.4
Appendix
Table 7: Estimated evidence against nested sampling (dynesty) on the DDM benchmark, over 50 held-out datasets from each model.
ParameterPriorUnit
gLg_{L}U(10−6, 3×10−4)\mathcal{U}(10^{-6},\,3\times 10^{-4})S cm−2\mathrm{S\,cm^{-2}}
ELE_{L}U(−80, −60)\mathcal{U}(-80,\,-60)mV\mathrm{mV}
gKg_{\mathrm{K}}U(1.5×10−3, 1.5×10−2)\mathcal{U}(1.5\times 10^{-3},\,1.5\times 10^{-2})S cm−2\mathrm{S\,cm^{-2}}
VtV_{t}U(−70, −50)\mathcal{U}(-70,\,-50)mV\mathrm{mV}
gNag_{\mathrm{Na}}U(8×10−3, 8×10−2)\mathcal{U}(8\times 10^{-3},\,8\times 10^{-2})S cm−2\mathrm{S\,cm^{-2}}
σ\sigmaU(10−4, 0.15)\mathcal{U}(10^{-4},\,0.15)μA cm−2 ms1/2\mu\mathrm{A\,cm^{-2}\,ms^{1/2}}
Appendix
Table 8: Prior distributions for the Hodgkin–Huxley models.
QuantityValueUnit
ENaE_{\mathrm{Na}}5050mV\mathrm{mV}
EKE_{\mathrm{K}}−90-90mV\mathrm{mV}
ECaE_{\mathrm{Ca}}120120mV\mathrm{mV}
CmC_{m}11μF cm−2\mu\mathrm{F\,cm^{-2}}
Compartment length / radius1010 / 11μm\mu\mathrm{m}
Step current density (225–1225 ms)22μA cm−2\mu\mathrm{A\,cm^{-2}}
Appendix
Table 9: Fixed parameters for the Hodgkin–Huxley simulations.
ParameterM0\mathcal{M}_{0}M1\mathcal{M}_{1}
Ωm\Omega_{\rm m}U(0.1,0.5)\mathcal{U}(0.1,0.5)U(0.1,0.5)\mathcal{U}(0.1,0.5)
Ωb\Omega_{\rm b}U(0.03,0.07)\mathcal{U}(0.03,0.07)U(0.03,0.07)\mathcal{U}(0.03,0.07)
hhU(0.5,0.9)\mathcal{U}(0.5,0.9)U(0.5,0.9)\mathcal{U}(0.5,0.9)
nsn_{\rm s}U(0.8,1.2)\mathcal{U}(0.8,1.2)U(0.8,1.2)\mathcal{U}(0.8,1.2)
σ8\sigma_{8}U(0.6,1.0)\mathcal{U}(0.6,1.0)U(0.6,1.0)\mathcal{U}(0.6,1.0)
log⁡10Mmin⁡\log_{10}M_{\min}NGC empirical priorNGC empirical prior
Appendix
Table 10: Priors for the Quijote galaxy-bias comparison. U(a,b)\mathcal{U}(a,b) denotes a uniform distribution and TN(0,0.22;[−1,1])\mathcal{TN}(0,0.2^{2};[-1,1]) a zero-mean Gaussian of standard deviation 0.20.2 , truncated to [−1,1][-1,1] . Mass logarithms are base ten with masses in h−1M⊙h^{-1}M_{\odot} . Numerical constants denote fixed values. The HOD mass M1M_{1} is distinct from the model label M1\mathcal{M}_{1} .
ExampleFlowCTSM-VPSingleENFRCTSM-SB + FlowRef
GMM 64D0.762.140.451.06
GMM 128D2.307.701.322.46
GMM 256D6.6127.873.046.57
GMM 512D15.7965.3614.4415.87
GMM 1024D34.99163.2133.4435.01
Appendix
Table 11: RMSE values of the estimated density log⁡q(x)\log q(x) with respect to the true density log⁡p(x)\log p(x) over 200 held out test draws (lower is better) averaged over the three seeds. The SingleENFR combination consistently outperforms the other models. The flow is shared at each seed.
Figure 6: The performance on test points in 128 dimensions for the flow and SingleENFR.
ModelLikelihood Based AnalysisMultiENFRΔ\Delta
wCDMw\mathrm{CDM}-2.42 ±\pm 0.24-1.94+0.48
w0waCDMw_{0}w_{a}\mathrm{CDM}-1.64 ±\pm 0.24-1.40+0.25
νCDM\nu\mathrm{CDM}-0.07 ±\pm 0.220.06+0.13
ΩkCDM\Omega_{k}\mathrm{CDM}-1.89 ±\pm 0.23-1.65+0.24
Appendix
Table 12: Bayes factors w.r.t. Λ\Lambda CDM on the DESI DR2 data. The error of our simulation-based estimate is largely comparable to the error of the likelihood-based estimate in three of the four models.
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