Learning theory informed priors for Bayesian inference: A case study with early dark energy
Bibliographic record
Abstract
Cosmological models are often motivated and formulated in the language of particle physics, using quantities such as the axion decay constant, but tested against data using ostensibly physical quantities, such as energy density ratios, assuming uniform priors on the latter. This approach neglects priors on the model from fundamental theory, including from particle physics and string theory, such as the preference for sub-Planckian axion decay constants. We introduce a novel approach to learning theory informed priors for Bayesian inference using normalizing flow (NF), a flexible generative machine learning technique that generates priors on model parameters when analytic expressions are unavailable or difficult to compute. As a test case, we focus on early dark energy (EDE), a model designed to address the Hubble tension. Rather than using uniform priors on the phenomenological EDE parameters ${f}_{\mathrm{EDE}}$ and ${z}_{c}$, we train a NF on EDE cosmologies informed by theory expectations for axion masses and decay constants. Our method recovers known constraints in this representation while being $\ensuremath{\sim}300$, 000 times more efficient in terms of total CPU compute time. Applying our NF to Planck and BOSS data, we obtain the first theory informed constraints on EDE, finding ${f}_{\mathrm{EDE}}\ensuremath{\lesssim}0.02$ at 95% confidence with an ${H}_{0}$ consistent with Planck, but in $\ensuremath{\sim}6\ensuremath{\sigma}$ tension with SH0ES. This yields the strongest constraints on EDE to date, additionally challenging its role in resolving the Hubble tension.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.019 | 0.086 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".