Bayesian Inference Framework

Bayesian Inference Framework#

This page provides an overview of the statistical framework underlying brutus. For detailed mathematical derivations, see Speagle et al. (2025).

Tip

For quick definitions of key terms, see the Glossary.

Overview#

brutus performs Bayesian inference to derive stellar properties from photometric and astrometric observations. Given multi-band photometry and (optionally) parallax measurements, brutus estimates:

  • Distance and extinction (\(A_V\), \(R_V\))

  • Stellar parameters: mass, age, metallicity, temperature, luminosity, surface gravity

  • Full posterior distributions with properly quantified uncertainties

The name “brutus” reflects the brute force computational approach: rather than using MCMC sampling, brutus systematically evaluates the posterior over a pre-computed grid of stellar models, then marginalizes to obtain parameter estimates. This guarantees complete parameter space coverage and avoids convergence issues. See Grid-Based Fitting for details on the computational approach.

The Forward Model#

brutus uses a forward modeling approach: predict what observations should look like given stellar parameters, then compare to actual data.

Stellar Parameters#

The model separates intrinsic parameters (properties of the star itself) from extrinsic parameters (environmental effects):

Intrinsic parameters \(\theta\):

Extrinsic parameters \(\phi\):

  • \(d\): Distance (parsecs)

  • \(A_V\): V-band extinction (magnitudes)

  • \(R_V\): Extinction curve shape parameter

Note

brutus parameterizes stellar evolution using EEP rather than age. EEP provides a mass-independent coordinate that increases monotonically along evolutionary tracks, avoiding degeneracies where multiple ages produce similar observables. Age is a derived quantity computed from (mass, EEP, metallicity). See Stellar Models and Photometry for details.

Predicted Magnitudes#

The predicted apparent magnitude in each photometric band is:

\[m_{\rm band} = M_{\rm band}(\theta) + \mu(d) + A_V \times \left[ R_{\rm band}(\theta) + R_V \times R'_{\rm band}(\theta) \right]\]

where:

  • \(M_{\rm band}(\theta)\) is the absolute magnitude from stellar models

  • \(\mu(d) = 5 \log_{10}(d/10\,{\rm pc})\) is the distance modulus

  • \(R_{\rm band}\) and \(R'_{\rm band}\) are reddening vectors encoding wavelength-dependent extinction

The reddening vectors depend on both the dust extinction law and the stellar spectrum, allowing accurate extinction modeling across different stellar types.

Model Assumptions#

The forward model assumes:

  • Single stars: No unresolved companions (binaries bias distances and masses)

  • Non-rotating: MIST models do not include rotational effects

  • Solar-scaled abundances: Detailed abundance patterns beyond [Fe/H] are not modeled

  • Fitzpatrick & Massa (2009) extinction law: R_V-dependent parameterization

These assumptions are appropriate for most field stars but may break down for rapid rotators, chemically peculiar stars, or close binaries.

Statistical Framework#

Observation Model#

Real observations include measurement uncertainties. brutus models:

Photometry as Gaussian in flux space:

\[\hat{F}_{\rm band} \sim \mathcal{N}\left[ F_{\rm band}, \sigma_{F,{\rm band}}^2 \right]\]

where \(F = 10^{-0.4\,m}\) converts magnitudes to linear flux density (in “maggies”, where 1 maggie = flux of a 0th magnitude source). Working in flux space ensures Gaussian errors—magnitude uncertainties are asymmetric and non-Gaussian.

Parallax as Gaussian:

\[\hat{\varpi} \sim \mathcal{N}\left[ 1000/d, \sigma_\varpi^2 \right]\]

where \(\varpi\) is in milliarcseconds and \(d\) in parsecs. Parallax provides a direct distance constraint that helps break the distance-extinction degeneracy.

Posterior Distribution#

Bayes’ theorem combines the likelihood with prior information:

\[P(\theta, \phi \,|\, \hat{\mathbf{F}}, \hat{\varpi}) \propto \underbrace{\mathcal{L}_{\rm phot}(\theta, \phi)}_{\text{photometric likelihood}} \times \underbrace{\mathcal{L}_{\rm plx}(\phi)}_{\text{parallax likelihood}} \times \underbrace{\pi(\theta, \phi)}_{\text{prior}}\]

The posterior represents our knowledge of stellar parameters given the observed data and astrophysical priors.

Prior Distributions#

brutus uses informative priors based on Galactic structure:

\[\pi(\theta, \phi) = \pi(M_{\rm init}) \times \pi({\rm EEP}) \times \pi([{\rm Fe/H}], t_{\rm age} \,|\, d, \ell, b) \times \pi(A_V \,|\, d, \ell, b) \times \pi(R_V)\]

The components are:

  1. Initial Mass Function: Kroupa IMF favoring low-mass stars

  2. EEP prior: Uniform in EEP (accounts for varying evolutionary timescales)

  3. Galactic structure: 3-D density model (thin disk + thick disk + halo) with position-dependent age and metallicity distributions

  4. 3-D dust: Extinction constraints from Bayestar dust maps

  5. R_V variation: Gaussian prior centered on \(R_V = 3.32\) with \(\sigma = 0.18\)

These priors encode that: low-mass stars are more common than high-mass stars; nearby stars are likely disk members with near-solar metallicity; distant stars may be older halo members; and extinction increases with distance along dusty sightlines.

See also

See Prior Distributions for detailed descriptions and customization options.

Outputs#

For each fitted star, brutus produces:

  • Posterior samples: Draws from P(distance, \(A_V\), \(R_V\) | data)

  • Model indices: Which grid models contribute to the posterior (for deriving mass, age, Teff, etc.)

  • Diagnostics: Minimum χ², log-evidence, number of bands used

See Understanding Results for interpretation guidance.

Limitations#

Model Systematics

Stellar models have known systematic uncertainties:

  • Temperature scale: MIST temperatures may be offset by 50–200 K depending on mass and evolutionary state

  • Radius inflation: Magnetic activity inflates radii of low-mass stars by 5–15%

  • Bolometric corrections: Missing opacities cause ~0.02–0.05 mag errors in some bands

See Empirical Calibrations for empirical calibration procedures.

Prior Sensitivity

For well-measured stars (bright, accurate parallax), priors have minimal impact. For poorly-measured stars (faint, no parallax), results can be prior-dominated. Always check prior sensitivity for your science case.

Binary Contamination

Unresolved binaries appear overluminous, biasing distances (too near) and masses (too high). Gaia RUWE > 1.4 often indicates binarity.

Computational Cost

Fitting scales with grid size and number of photometric bands. Typical performance: ~0.1–1 second per star with pre-computed grids. Large surveys benefit from parallelization.

References#

Speagle et al. (2025), “Deriving Stellar Properties, Distances, and Reddenings using Photometry and Astrometry with BRUTUS”, arXiv:2503.02227

Choi et al. (2016), “Mesa Isochrones and Stellar Tracks (MIST). I. Solar-scaled Models”, ApJ, 823, 102

Dotter (2016), “MESA Isochrones and Stellar Tracks (MIST) 0: Methods for the Construction of Stellar Isochrones”, ApJS, 222, 8

Green et al. (2019), “A 3D Dust Map Based on Gaia, Pan-STARRS 1, and 2MASS”, ApJ, 887, 93

Fitzpatrick & Massa (2009), “An Analysis of the Shapes of Interstellar Extinction Curves. VI. The Near-IR Extinction Law”, ApJ, 699, 1209