The paper

NNPD implements the study of

Prior Dependence in Neural Ratio Estimation
Nabil Salama, Katherine Fraser, Po-Wen Chang, Benjamin Nachman
2026 Conference on Physics and AI (PAI26)

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Abstract

Neural ratio estimation is an effective technique for using neural networks to learn likelihood ratios from simulated data, which can be used for applications such as parameter inference and reweighting. While it can be theoretically shown that in principle the resulting likelihood-to-evidence ratio should be independent of the prior used to select the training data, that independence relies on achieving an idealized classifier, which is not possible in practice. It is therefore unclear to what extent prior independence survives under realistic training conditions, particularly for more complicated higher-dimensional problems. In this paper, we test the prior independence of parameter estimation for a simple Gaussian toy example. We plan to extend this study to realistic particle physics datasets in the future.

Cite

@inproceedings{salama2026prior,
  title     = {Prior Dependence in Neural Ratio Estimation},
  author    = {Salama, Nabil and Fraser, Katherine and Chang, Po-Wen and Nachman, Benjamin},
  booktitle = {2026 Conference on Physics and AI (PAI26)},
  year      = {2026},
  url       = {https://openreview.net/pdf?id=kdy81q0tOX}
}

From the paper to the code

Paper	Code	Output
Eq. (1), (3): joint versus factorized training classes	nnpd.nre.data.balanced_pairs, gaussian/sampling.py	training artifact
Eq. (2): classifier odds equal the likelihood-to-evidence ratio	nnpd.nre.training.fit_binary; the network returns the log ratio as a logit	model artifact
Section 2: Gaussian data, four priors, network and training	settings_training.py, gaussian/experiment.py	—
Eq. (4): ensemble of 15 observations, normalized over parameter space	nnpd.nre.inference.ensemble_log_ratio, normalized_mass	inference product
Eq. (5): likelihood-ratio maximizer and bias	ratio_mode in the inference product	bias metric
Eq. (6): HLD region	nnpd.nre.inference.summarize_density	inference product
Eq. (7): projected HLD bounds and widths	ratio_lower, ratio_upper	width metric
Eq. (8): mean coverage, width and bias	gaussian/metrics.py	coverage, width, bias metrics
Figure 1: one- and two-dimensional projections	inference plot	marginal-* and projection-* figures
Figure 2: estimates and HLD intervals per prior	inference plot	estimates-* figures
Figure 3: metrics across priors and dimensions	nnpd.results.comparison_plot	built from saved metrics, see below

The scientific notes map every quantity of the paper to its setting and explain where the implementation has to choose an interpretation: the exponential sign, the normal-prior scale, the truth design, the discrete integration measure, and coverage of the true parameter.

Run the study

The paper profile makes the paper’s choices explicit: exponential rate -0.1, no truth margin, the exact 25-point-per-axis truth grid where it is feasible (up to four inferred parameters, Sobol truths above), continuous candidates for the normalized ratio, and ratio and exact targets.

python run.py plan --profile paper
python run.py run --profile paper

Read the plan first: the full study trains 24 models, four priors in each of six dimensions. After the run, compare the priors across dimensions as in Figure 3:

from nnpd.results import comparison_plot, runs

records = runs("outputs/paper")
coverage = comparison_plot(records, "settings.problem.dimension", "metrics.coverage.projected_mean.0")
bias = comparison_plot(records, "settings.problem.dimension", "metrics.bias.mean_absolute")

projected_mean.0 is the coverage of the first level in settings_analysis.py, 68% by default. The runs use their own random streams and numerical choices, so they reproduce the study, not the paper’s exact figures.

The paper restricts itself to parameter inference. The analysis also computes the reweighting and normalization diagnostics that its conclusion anticipates for a longer version; see Products, metrics, and plots.