Global Bayesian Mode-Fitting Framework

• Global Bayesian mode-fitting framework is a hierarchically structured approach that estimates and compares parametrized models while fully propagating uncertainties to robustly detect subtle signals.
• It employs nested sampling to compute model evidence, balancing fit quality and model complexity with Bayesian Occam’s razor.
• The framework leverages physically motivated parametrization and forward-modeling to separate weak global signals from dominant, structured foregrounds.

A global Bayesian mode-fitting framework is a hierarchically structured inferential methodology designed to estimate and compare parametrized models, including mode parameters, on observational or experimental data by fully propagating uncertainties within a Bayesian paradigm. This framework identifies optimal model complexity, robustly quantifies uncertainties via marginalization, and enables rigorous model selection through Bayesian evidence calculations, often relying on advanced algorithms such as nested sampling. The approach is fundamental in contexts where physically motivated models compete with confounding signals or when the inference target is subtle and the global mode—whether a parameter peak, structure, or field—is of scientific interest.

1. Hierarchical Bayesian Model Specification

A global Bayesian mode-fitting framework begins with a joint probabilistic description for the observed data $D = \{y_k\}$ and the parameter set $\theta$, which includes mode-defining parameters as well as nuisance, background, and noise terms. The generic structure is: $$P(\theta | D, N) \propto L(D|\theta) \, \pi(\theta)$$ where $L(D|\theta)$ is the likelihood, $\pi(\theta)$ the prior, and $N$ denotes chosen model complexity (e.g., number of regions, modes, or basis functions).

A concrete example is found in global 21cm experiments (Anstey et al., 2020). The parameter vector $\theta$ comprises $N$ region spectral indices $\{\beta_i\}$ describing the foreground, plus signal amplitude and width parameters $(A_\mathrm{sig}, \nu_c, \sigma)$ characterizing the global 21 cm signal, and a noise rms $\sigma_n$. The modeled data spectrum is

$$T_\mathrm{model}(\nu; \theta) = T_\mathrm{fg}(\nu; \{\beta_i\}) + T_{21}(\nu; A_\mathrm{sig}, \nu_c, \sigma)$$

with $T_\mathrm{fg}$ generated via forward modeling and beam convolution; $T_{21}$ often parameterized as a Gaussian trough.

The posterior encapsulates all uncertainties. Model evidence

$Z_N = \int L(D|\theta)\,\pi(\theta)\,d\theta$

enables robust model comparison, including the detection of statistically significant modes or features via Bayes factors $\Delta\ln Z$.

2. Physics-Motivated Parametric Structure

Critical to the framework is construction of a physics-based, forward-parametrized model for both the foreground/background and signal. In the 21cm context, the foreground is built by partitioning the sky into $N$ regions, each with a uniform spectral index $\beta_i$, using masks $M_i(\Omega)$: $$T_s(\nu, \Omega) = [T_0(\Omega) - T_\mathrm{CMB}] \left(\frac{\nu}{\nu_0}\right)^{-\beta_i} + T_\mathrm{CMB}, \quad \Omega \in R_i$$ After convolution with the frequency-dependent antenna beam $D(\Omega, \nu)$ and time averaging: $$K_i(\nu) = \frac{1}{4\pi}\int D(\Omega, \nu) M_i(\Omega) [T_0(\Omega)-T_\mathrm{CMB}] \, d\Omega$$ and the modeled spectrum becomes: $$T_\mathrm{fg}(\nu) = \sum_{i=1}^N K_i(\nu) \left(\frac{\nu}{\nu_0}\right)^{-\beta_i} + T_\mathrm{CMB}$$

The global signal is frequently parameterized as a template (e.g., a Gaussian absorption profile): $$T_{21}(\nu) = -A_\mathrm{sig} \exp\left[ -\frac{(\nu - \nu_c)^2}{2\sigma^2} \right]$$

All observables are subjected to instrumental, chromatic, and noise effects, which enter the modeling forward path and are key to the fidelity of the fit.

3. Inference Mechanism: Nested Sampling and Model Evidence

Global Bayesian mode-fitting typically relies on sampling-based Bayesian inference (e.g., PolyChord, MultiNest) to efficiently handle high-dimensional, multi-modal posteriors.

• Nested sampling introduces an ensemble of "live" points that traverse the likelihood space, yielding both posterior samples and the marginalized model evidence $Z_N$.
• Each choice of complexity parameter $N$ (e.g., number of sky regions) is evaluated independently, and evidence maximization balances fit quality against "Occam penalty."
• Model selection via Bayes factors proceeds as

$$B = \frac{Z_{N,\,\mathrm{signal+fg}}}{Z_{N,\,\mathrm{fg}}}$$

with $\Delta\log Z \gg$ few representing decisive detection or discrimination.

Implementation steps involve:

1. Precomputing all forward-model building blocks (masking, beam-averaged kernels $K_i(\nu)$).
2. Specifying parameter priors (often uniform or log-uniform for spectral indices, signal template parameters, and noise).
3. Running the sampler, accumulating posterior samples and evidence.
4. Comparing evidences to adjudicate between models of different complexity and to formally detect modes or target signals.

4. Priors, Hierarchical Structure, and Complexity Control

The prior structure ensures physical interpretability, regularization, and identifiability:

• Foreground indices: $$\beta_i \sim \mathrm{Uniform}(\beta_\mathrm{min}, \beta_\mathrm{max})$$ ensures coverage of physically plausible values as derived from the base map (e.g., GSM: $[2.46, 3.15]$).
• Signal template parameters: Amplitude $A_\mathrm{sig} \sim [0, 0.25]$ K, center $\nu_c \sim [50, 200]$ MHz, width $\sigma \sim [10, 20]$ MHz; boundaries ensure avoidance of unphysical solutions.
• Noise: $$\sigma_n \sim \mathrm{LogUniform}(10^{-4}, 10^{-1})$$ K, allowing for variations in measurement noise.
• No hyperpriors are required beyond shared bounds (exchangeable $\beta_i$), though extension to further hierarchical levels is possible.

Complexity control is inherent: the optimal $N$ is chosen via evidence maximization over $N = 5 \ldots 13$, directly controlling overfitting via Bayesian Occam's razor.

5. Detection, Model Comparison, and Quantitative Results

Inference outputs multiple comparable models, which are adjudicated by evidence:

• The foreground$+$signal model is directly tested against a foreground-only baseline.
• A strong Bayes factor in favor of the former is taken as a confident detection of a subtle mode or signal in presence of overwhelming foregrounds.
• Anstey et al. (Anstey et al., 2020) demonstrate that, for a relatively smooth conical log spiral antenna, the global 21 cm signal is reliably recovered in synthetic data, while for a more chromatic conical sinuous antenna, the Bayesian evidence readily distinguishes between a true signal and systematic artifact—thus enabling robust falsification.

6. Implementation Guidance and Application Scope

The global Bayesian mode-fitting framework is widely extensible and its implementation is systematic:

1. Preprocessing: Compute region masks and beam convolution kernels before inference; this yields computational efficiency since map and beam operations are the bottleneck in each model iteration.
2. Dimension selection: Scan model complexity (e.g., region number) with evidence maximization, enforcing model parsimony.
3. Signal detection: Always compare the full (foreground+signal) and nested (foreground-only) hypotheses as a control.
4. Broader applicability: The methodology generalizes to any domain where a global or spatially-invariant signal is embedded in complex, structured, or spatially-inhomogeneous backgrounds—e.g., CMB spectral distortions, global recombination lines, or searches for faint modes in astronomical time/frequency series.

7. Advantages, Limitations, and Extensions

• Advantages:
  • Rigorous uncertainty quantification and marginalization.
  • Physical interpretability through explicit, parametric foreground modeling, including instrumental effects.
  • Automatic complexity control via Bayesian evidence.
  • Transparent detection criterion based on Bayes factors.
• Limitations:
  • Computational demand scales strongly with model dimension $N$ and number of foreground regions; sampling cost grows rapidly.
  • Success depends on the quality of the base map and accuracy of the beam model; model misspecification can induce bias, especially if not handled by more flexible error modeling (Pagano et al., 2022).
  • For highly chromatic or poorly characterized instruments, identifiability may break down; the Bayesian framework will reveal the associated lack of evidence for detection.
• Extensions:
  • Inclusion of spatially-dependent amplitude errors in the base map or additional error-correlated structure (Pagano et al., 2022).
  • Simultaneous multi-temporal or multi-instrument fitting improves both signal and background constraints, exploiting structure in richer data sets (Anstey et al., 2022).
  • Incorporation of more complex signals and physically motivated priors on the target mode.

The global Bayesian mode-fitting framework has become foundational in modern scientific inference for situations where faint signals must be discerned against foregrounds of orders-of-magnitude greater power, controlled by physical, instrumental, and statistical uncertainties, and where mode (structure, parameter, or feature) detection must be evidence-based and robust to model specification.