Physics-Based Likelihoods

• Physics-based likelihoods are inference models derived from first-principle laws that integrate simulation and analytical methods for precise data analysis.
• They leverage Monte Carlo simulators, analytical forms, and machine learning surrogates to construct high-dimensional, non-Gaussian models for experimental observables.
• Robust validation and standardized publication of these likelihoods enable reproducible hypothesis testing and reinterpretation across diverse physics experiments.

Physics-based likelihoods are mathematical models for statistical inference that are constructed directly from, or heavily constrained by, underlying first-principles physical laws, domain-specific phenomenology, or detailed mechanistic simulators. They play a foundational role in the quantitative analysis of experimental data in high energy physics, astroparticle physics, cosmology, and other physical sciences where the forward modeling of experimental observables traces back to established theories, complex detector responses, or Monte Carlo–based simulators. The proper construction, validation, and communication of physics-based likelihoods is essential for rigorous hypothesis testing, parameter estimation, and reproducible reinterpretation across experiments and theoretical models.

1. Formal Definition and Mathematical Structure

A physics-based likelihood is a function $L(\theta)$, typically defined as $L(\theta) = p(x_{\mathrm{obs}}|\theta)$, where $x_{\mathrm{obs}}$ denotes the observed data vector and $\theta$ denotes the set of physical (and often nuisance) parameters that govern the generative process for $x$ (Cousins, 2020). The statistical model $p(x|\theta)$ is derived from physical first principles, e.g., quantum field theory cross-sections, classical event rates, or stochastic differential equations, and is augmented by models for detector effects, instrumental uncertainties, and possible systematic errors (Balázs et al., 2017, Workgroup et al., 2020).

In prototypical high energy physics measurements, the likelihood may take the form of an extended product over observed data and auxiliary constraints: $$L(\theta) = \mathrm{Pois}(n_{\mathrm{obs}} | \nu(\theta)) \prod_{i=1}^{n_{\mathrm{obs}}} f(x_i|\theta) \prod_{j} \pi_j(\psi_j)$$ where $n_{\mathrm{obs}}$ is the observed count, $\nu(\theta)$ is the model prediction for total rate, $f(x|\theta)$ is the probability density for observables, $\psi_j$ are nuisance parameters, and $\pi_j$ are constraint distributions reflecting systematics (Cousins, 2020, Balázs et al., 2017).

In physics experiments, these likelihoods are frequently high-dimensional, non-Gaussian, and may incorporate hierarchical models combining multiple data sources, e.g., background, signal, and auxiliary measurements (Bernlochner et al., 2017, Gärtner et al., 2024).

2. Construction Methodologies: Simulation, Analytical, and Machine Learning Surrogates

Construction of physics-based likelihoods can employ analytical calculations, direct simulation, or data-driven surrogates:

• Analytical and Semi-Analytical Forms: In some domains (e.g., Gaussian CMB anisotropies, Poisson counting), analytic forms are tractable and standard Gaussian or Poisson likelihoods are used. Model predictions can be propagated through linear (or occasionally nonlinear) detector response functions with the likelihood built from multivariate covariances (Workgroup et al., 2020, Bernlochner et al., 2017).
• Monte Carlo Simulators: Where the physics or instrument is too complex for closed forms, forward simulators (e.g., Pythia8 for LHC events, NEST for noble-element detectors, CLASS/AlterBBN for cosmological observables) generate realizations $x \sim p(x|\theta)$. These may inform binned or unbinned surrogate likelihoods through kernel density estimation, template fits, or other methods (James et al., 2022, Balázs et al., 2017, Workgroup et al., 2020).
• Normalizing Flows and DNN Surrogates: Physics-based likelihoods are increasingly parameterized by normalizing flows (invertible deep neural networks) trained on MC or experimental samples (Araz et al., 13 Feb 2025, Reyes-Gonzalez et al., 2023, Gavrikov et al., 31 Jul 2025). Given samples $\{x_i\}$ from an intractable distribution, flows learn a bijection $f_\theta: x \to z$, mapping $x$ into a tractable latent space (e.g., Gaussian), with the target density reconstructed via the change-of-variables formula:

$p_\theta(x) = p_0(f_\theta(x)) \left|\det \J_f(x)\right|$

where $p_0(z)$ is the base (usually standard normal) density and $\J_f(x)$ is the Jacobian. Training employs maximum likelihood over the observed data:

$\mathcal{L}(\theta) = \sum_i \bigl[\ln p_0(f_\theta(x_i)) + \ln |\det \J_f(x_i)|\bigr]$

(Araz et al., 13 Feb 2025). These surrogates are validated with rigorous statistical tests such as the multivariate Kolmogorov–Smirnov test.

• Classification-based Ratio Estimation: Neural classifiers, trained to distinguish between samples generated under competing physical hypotheses $(\theta_0, \theta_1)$, can be used to directly estimate likelihood ratios via the "likelihood-ratio trick" (Rizvi et al., 2023). The optimal classifier output can be mapped to the ratio $\Lambda(x) = p(x|\theta_0)/p(x|\theta_1)$.

3. Validation and Statistical Testing of Learned Likelihoods

Physics-based likelihoods—especially those supplied via generative models or machine-learning-based surrogates—require robust validation beyond superficial agreement in summary statistics:

• Multivariate Kolmogorov–Smirnov Test: By mapping data via the learned flow to latent $z$ space, where the reference distribution is known (e.g., standard normal), a one-sample KS test of the squared-radius distribution ($t_j = ||z_j||^2$) against the $\chi^2(D)$ null yields a p-value; a low p-value indicates significant deviation and model rejection (Araz et al., 13 Feb 2025).
• Binned $\chi^2$ Tests: Used as auxiliary validation; equiprobable bins under the reference measure are checked for consistency between predicted and empirical counts.
• Coverage Checks and Feldman–Cousins Intervals: For complex or intractable likelihoods, coverage of constructed confidence intervals/sets is checked via Monte Carlo generation under the physical model (Algeri et al., 2019, Kadhim et al., 2023).
• Likelihood Principle and its Violations: Particle physics often employs procedures that, strictly speaking, violate the likelihood principle (e.g., procedures based on tail-areas or "number of sigma"), motivated by standard frequentist practice (Cousins, 2020).

4. Applications: High Energy Physics, Astroparticle Physics, Cosmology

Physics-based likelihoods are foundational in diverse domains, with representative implementations including:

• Collider Data Analyses: The SModelS/pyhf interface interprets ATLAS full-likelihood JSON workspaces, combining Poisson terms for observed counts in signal/control regions and hierarchical constraints for nuisance parameters to build composite likelihoods (Alguero et al., 2020). These are deployed in reinterpretation tools and global fits, achieving agreement with official limits within ≲10 GeV mass reach (Alguero et al., 2020).
• Flavour Physics: FlavBit models numerous correlated observables via a joint Gaussian likelihood with a full experimental + theoretical covariance structure, enabling global analysis of $B, D, K, \pi$ decays with precise uncertainty propagation (Bernlochner et al., 2017).
• Cosmology: Modular frameworks like CosmoBit orchestrate numerous backends (e.g., CLASS, AlterBBN, MontePython, plc) to compute the cosmological observables and their theoretical predictions, assembling full joint likelihoods for CMB, BAO, LSS, and BBN constraints (Workgroup et al., 2020).
• Astroparticle Physics: Explicit per-event likelihoods in direct detection (e.g., FlameNEST) or approximate differentiable surrogates (via probabilistic programming) for dark matter searches enable unbinned inference, joint channel analyses, and automatic propagation of uncertainties (Qin et al., 2024, James et al., 2022).
• Simulation-based Practice: In regimes with intractable likelihoods, simulation-based inference (SBI) leverages amortized neural surrogates, joint normalizing flows, or DNN-based confidence set learning to maintain coverage and statistical rigor (Gavrikov et al., 31 Jul 2025, Kadhim et al., 2023).

5. Storage, Sharing, and Reinterpretation

Ensuring open communication and reinterpretation of physics-based likelihoods is increasingly vital:

• Standardized Formats and Surrogates: Analytical likelihoods, or those parameterized by normalizing flows (e.g., “.nabu” files), encapsulate both model weights and input transformations to enable downstream users to compute $p_\theta(x)$ and derive new samples (Araz et al., 13 Feb 2025, Reyes-Gonzalez et al., 2023).
• Model-Agnostic Template Publication: Publishing the MC joint map $n^0_{cbz}$, kinematic weights, covariance matrices, and bin specifications is essential for robust reinterpretation, notably in the context of SMEFT and flavor physics analyses (Gärtner et al., 2024).
• Digital Likelihood Workspaces: Pyhf/HistFactory–style JSON representations now serve as de facto standards for distributing binned Poisson+constraint models, with full systematics and correlations encoded (Alguero et al., 2020, Gärtner et al., 2024).
• Tools and Automation: Open-source codebases (e.g., nabu, pyhf, FlameNEST, UltraNest for nested sampling) and probabilistic programming backends facilitate flexible reanalysis, gradient-based optimization, and data exchange (Araz et al., 13 Feb 2025, James et al., 2022, Gavrikov et al., 31 Jul 2025).

6. Limitations, Outlook, and Best Practices

While physics-based likelihoods offer a principled foundation for inference, several challenges persist:

• Expressivity and Scalability: In high dimensions (e.g., D ≫ 10), standard flow architectures may require innovations in coupling-layer design, hybrid models (flows+diffusions), or neural surrogates for efficient training and tail coverage (Araz et al., 13 Feb 2025).
• Validation in Tails and Correlations: Routine use of joint-distribution and tail-sensitive tests, rather than marginal or visual agreement alone, is essential for scientific reliability (Araz et al., 13 Feb 2025).
• Propagation of Uncertainties: Appropriate marginalization or profiling of nuisance parameters, using either analytic integration, MC, or profiling under boundary conditions, remains critical (Balázs et al., 2017, Algeri et al., 2019).
• Communication and Cataloguing: Publishing minimal sufficient information (joint MC, binning, modifier definitions, covariance matrices) is necessary for true model-agnostic reinterpretation and forward compatibility (Gärtner et al., 2024).
• Philosophical and Practical Subtleties: Standard practice in particle physics frequently violates the likelihood principle (e.g., via trial factors, reading off p-values from tail probabilities, etc.), requiring practitioners to exercise judgment regarding coverage, statistical validity, and reporting (Cousins, 2020).

Future work will focus on improved neural architectures for high-dimensional likelihood surrogates, incorporation of diffusion and flow-based models, more automated and standard infrastructure for likelihood publication and cataloguing (e.g., in EOS/Spey), and refined methods for quantifying coverage and robustness, particularly in low-signal or high-systematics regimes (Araz et al., 13 Feb 2025, Gavrikov et al., 31 Jul 2025, Kadhim et al., 2023).

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References:

• "Communicating Likelihoods with Normalising Flows" (Araz et al., 13 Feb 2025)
• "A SModelS interface for pyhf likelihoods" (Alguero et al., 2020)
• "What is the likelihood function, and how is it used in particle physics?" (Cousins, 2020)
• "Approximate Differentiable Likelihoods for Astroparticle Physics Experiments" (Qin et al., 2024)
• "ColliderBit: a GAMBIT module for the calculation of high-energy collider observables and likelihoods" (Balázs et al., 2017)
• "Constructing model-agnostic likelihoods, a method for the reinterpretation of particle physics results" (Gärtner et al., 2024)
• "The NFLikelihood: an unsupervised DNNLikelihood from Normalizing Flows" (Reyes-Gonzalez et al., 2023)
• "Simulation-based inference for Precision Neutrino Physics through Neural Monte Carlo tuning" (Gavrikov et al., 31 Jul 2025)
• "Dynamic Likelihood Filter" (Restrepo, 2017)
• "CosmoBit: A GAMBIT module for computing cosmological observables and likelihoods" (Workgroup et al., 2020)
• "FlavBit: A GAMBIT module for computing flavour observables and likelihoods" (Bernlochner et al., 2017)
• "Searching for new physics with profile likelihoods: Wilks and beyond" (Algeri et al., 2019)
• "Learning Likelihood Ratios with Neural Network Classifiers" (Rizvi et al., 2023)
• "Bayesian model comparison with un-normalised likelihoods" (Everitt et al., 2015)
• "FlameNEST: Explicit Profile Likelihoods with the Noble Element Simulation Technique" (James et al., 2022)
• "Amortized Simulation-Based Frequentist Inference for Tractable and Intractable Likelihoods" (Kadhim et al., 2023)
• "Rising Above Chaotic Likelihoods" (Du et al., 2014)