Prior Distributions

Prior Distributions#

This page describes the prior probability distributions used in brutus. Priors encode astrophysical knowledge about the Galaxy and help break parameter degeneracies inherent in photometric fitting.

For detailed mathematical derivations, see Speagle et al. (2025) §2.4 and Appendix A.

Why Priors Matter#

Photometry alone cannot uniquely determine stellar properties. A nearby cool M dwarf and a distant reddened K giant can produce similar colors and magnitudes. Priors help resolve this ambiguity by incorporating knowledge about Galactic structure, stellar mass distributions, and dust extinction.

Prior impact depends on data quality:

  • Well-measured stars (bright, accurate parallax): Likelihood dominates; results are insensitive to priors

  • Poorly-measured stars (faint, no parallax): Priors strongly influence results

Always check prior sensitivity for your science case (see Testing Prior Sensitivity below).

Note

Parallax is not required but helps significantly. When parallax measurements are available, they break the distance-extinction degeneracy that is inherent in photometric fitting, leading to much tighter posteriors. Without parallax, brutus relies on photometry and Galactic priors alone, which still produces useful results but with broader uncertainties, especially for faint stars.

The Galactic Model#

brutus uses a 3-D Galactic model with factorized priors:

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

Prior Components#

Initial Mass Function#

The Kroupa (2001) IMF describes stellar masses at formation as a two-part power law:

\[\begin{split}\pi(M) \propto \begin{cases} M^{-1.3} & 0.08 < M/M_\odot < 0.5 \\ M^{-2.3} & 0.5 < M/M_\odot < 150 \end{cases}\end{split}\]

Low-mass stars strongly dominate; a randomly drawn star is ~10× more likely to be 0.5 M☉ than 1.0 M☉.

Implementation: brutus.priors.logp_imf()

Evolutionary State (EEP)#

The EEP prior is uniform, which accounts for varying evolutionary timescales. Stars spend most of their lives on the main sequence (EEP ~300-450), so the uniform EEP prior naturally weights toward main sequence stars when combined with the age prior.

3-D Stellar Density#

The spatial distribution combines three Galactic components, each with characteristic structure and stellar populations:

Component

Spatial Profile

Mean [Fe/H]

Mean Age

Thin disk

Exponential (h_R = 2.6 kpc, h_Z = 300 pc)

-0.2 dex

~5 Gyr

Thick disk

Exponential (h_R = 2.0 kpc, h_Z = 900 pc)

-0.7 dex

~8 Gyr

Halo

Flattened power-law (η = 4.2)

-1.6 dex

~12 Gyr

The halo follows a flattened (oblate) power-law profile with radius-dependent oblateness: highly flattened near the Galactic center (q = 0.2) and more spherical at large radii (q = 0.8), transitioning over a scale of ~6 kpc.

The combined prior weights each component by its stellar density at the 3-D position \((d, \ell, b)\). Near the Sun, the thin disk dominates; at high Galactic latitudes or large distances, the thick disk and halo contribute more.

Implementation: brutus.priors.logp_galactic_structure(), brutus.priors.logp_feh(), brutus.priors.logp_age_from_feh(), brutus.priors.logn_disk(), brutus.priors.logn_halo()

3-D Dust Extinction#

brutus can use 3-D dust maps to provide distance-dependent extinction priors. The default dust map file distributed with brutus is derived from Bayestar19 (Green et al. 2019). For a given sky position \((\ell, b)\) and distance \(d\):

\[\pi(A_V\,|\,d,\ell,b) \sim \mathcal{N}(\mu_{A_V}, \sigma_{A_V}^2)\]

where the mean \(\mu_{A_V}\) and uncertainty \(\sigma_{A_V}\) come from the dust map at that sightline and distance.

Note

The Bayestar 3D dust map is based on Pan-STARRS 1 photometry and is only available north of declination -30 degrees (the Pan-STARRS 1 footprint). For sightlines south of this limit, the dust map prior will not be applied.

Implementation: Enabled via dustfile parameter in fit()

R_V Variation#

The extinction curve shape R_V \(\equiv A_V / E(B-V)\) has a truncated Gaussian prior:

\[\pi(R_V) \sim \mathcal{N}(3.32, 0.18^2) \quad {\rm for} \quad 1.0 < R_V < 8.0\]

The default mean (3.32) and standard deviation (0.18) are based on Schlafly et al. (2016), who measured R_V variations toward tens of thousands of APOGEE stars across the Milky Way.

Implementation: Controlled via rv_gauss and rvlim parameters in fit()

Customizing Priors#

Disabling Priors#

For diagnostic purposes, priors can be disabled or made uninformative:

Galactic structure prior:

from brutus.analysis import BruteForce

fitter = BruteForce(grid)

# Fit without Galactic structure prior
fitter.fit(
    data, data_err, data_mask, labels, save_file='results.h5',
    lngalprior=lambda *args, **kwargs: 0.0,  # Uniform prior
)

Extinction priors:

# Fit with uninformative (wide) priors on A_V and R_V
fitter.fit(
    data, data_err, data_mask, labels, save_file='results.h5',
    avlim=(0.0, 10.0),            # Wide A_V range
    av_gauss=(0.0, 1e6),          # Effectively uniform A_V prior
    rvlim=(1.0, 8.0),             # Wide R_V range
    rv_gauss=(3.32, 1e6),         # Effectively uniform R_V prior
)

Warning

Disabling priors can lead to highly degenerate parameter estimates. Only disable when you understand the implications.

Custom Prior Functions#

Pass custom Galactic structure prior functions via lngalprior:

from brutus.priors import logp_galactic_structure

def custom_galactic_prior(dist, coord, labels=None):
    """Custom prior: uniform within 100 pc, default otherwise.

    ``dist`` is in kpc and ``coord`` is a single ``(l, b)`` tuple in
    degrees, matching how ``fit()`` calls ``lngalprior``.
    """
    if dist < 0.1:  # kpc
        # Return value at boundary to ensure continuity
        return logp_galactic_structure(0.1, coord, labels=labels)
    return logp_galactic_structure(dist, coord, labels=labels)

fitter.fit(
    data, data_err, data_mask, labels, save_file='results.h5',
    data_coords=coords,
    lngalprior=custom_galactic_prior,
)

When to Customize#

Consider customizing priors for:

  • Extragalactic objects: LMC/SMC stars need different Galactic priors

  • Special regions: Galactic bulge, Local Bubble, or spiral arms

  • Known populations: If you have independent age/metallicity constraints

For cluster modeling with fixed age/metallicity/distance, see Population-Based Modeling.

Testing Prior Sensitivity#

Compare results with and without priors to assess prior influence:

import h5py
import numpy as np

# Fit with and without Galactic prior
fitter.fit(data, data_err, data_mask, labels, save_file='with.h5',
           data_coords=coords)
fitter.fit(data, data_err, data_mask, labels, save_file='without.h5',
           lngalprior=lambda *args, **kwargs: 0.0)

# Compare
with h5py.File('with.h5', 'r') as f:
    d1 = np.median(f['samps_dist'][0])
with h5py.File('without.h5', 'r') as f:
    d2 = np.median(f['samps_dist'][0])

change = abs(d1 - d2) / d1
print(f"Fractional change: {change:.1%}")

Available Prior Functions#

Stellar Priors:

Galactic Priors:

Astrometric Priors:

Extinction Priors:

References#

Speagle et al. (2025), “Deriving Stellar Properties, Distances, and Reddenings using Photometry and Astrometry with BRUTUS”, arXiv:2503.02227 (§2.4, Appendix A)

Kroupa (2001), “On the variation of the initial mass function”, MNRAS, 322, 231

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

Jurić et al. (2008), “The Milky Way Tomography with SDSS. I. Stellar Number Density Distribution”, ApJ, 673, 864

Schlafly et al. (2016), “The Optical-Infrared Extinction Curve and its Variation in the Milky Way”, ApJ, 821, 78