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:

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

Note that the age prior conditions on the star’s metallicity as well as its position: each Galactic component’s age distribution is weighted by that component’s metallicity-updated membership probability, so (for example) a nearby metal-poor star is correctly weighted toward old, halo-like ages.

Prior Components#

Initial Mass Function#

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

\[\begin{split}\pi(M) \propto \begin{cases} M^{-1.3} & 0.08 < M/M_\odot < 0.5 \\ M^{-2.3} & 0.5 < M/M_\odot < 100 \end{cases}\end{split}\]

Low-mass stars strongly dominate; a randomly drawn star is ~5× 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. The Gaussian is truncated and renormalized over the fitted A_V range (avlim), so distances where the map profile sits near an A_V boundary are not artificially disfavored.

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 prior falls back to being uniform (uninformative). In addition, Bayestar queries by default honor the map’s per-pixel reliability metadata (converged, DM_reliable_min/DM_reliable_max): distance bins outside a sightline’s reliable range, and sightlines whose fits did not converge, also degrade to a uniform prior. Pass Bayestar(..., apply_reliability_mask=False) to disable this masking.

Implementation: Enabled via the dustfile parameter in fit(), which accepts a path to a Bayestar HDF5 file (str or pathlib.Path) or a pre-loaded brutus.dust.Bayestar object; the prior itself is evaluated by brutus.priors.logp_extinction()

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:

import numpy as np

from brutus.analysis import BruteForce

fitter = BruteForce(grid)

# Fit without Galactic structure prior. The function receives an array
# of distances (kpc) and must return an array of the same shape.
fitter.fit(
    data, data_err, data_mask, labels, save_file='results.h5',
    lngalprior=lambda dist, *args, **kwargs: np.zeros_like(dist),  # Uniform
)

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, 20.0),            # Full A_V range (the default)
    av_gauss=(0.0, 1e6),          # Effectively uniform A_V prior (the default)
    rvlim=(1.0, 8.0),             # Full R_V range (the default)
    rv_gauss=(3.32, 1e6),         # Effectively uniform R_V prior
)

Note that av_gauss is applied in addition to any dust-map prior from dustfile (they are not exclusive), so passing an informative av_gauss together with dustfile counts the A(V) constraint twice.

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:

import numpy as np

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 an array of distances in kpc and ``coord`` is a single
    ``(l, b)`` tuple in degrees, matching how ``fit()`` calls
    ``lngalprior``.
    """
    # Evaluate distances inside 100 pc at the boundary value so the
    # prior is flat there and continuous at 0.1 kpc
    return logp_galactic_structure(np.maximum(dist, 0.1), 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 dist, *args, **kwargs: np.zeros_like(dist))

# 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:

logp_imf() - Initial mass function (Kroupa two-part power law)
logp_ps1_luminosity_function() - Pan-STARRS r-band luminosity function

Galactic Priors:

logp_galactic_structure() - Combined 3-D stellar density (thin disk + thick disk + halo)
logn_disk() - Exponential disk number density
logn_halo() - Flattened power-law halo number density
logp_feh() - Metallicity distribution by Galactic component
logp_age_from_feh() - Age distribution conditioned on metallicity

Astrometric Priors:

logp_parallax() - Parallax prior with distance-to-parallax conversion
logp_parallax_scale() - Parallax constraint on flux density scale factors
convert_parallax_to_scale() - Convert parallax to flux density scale factor

Extinction Priors:

logp_extinction() - 3-D dust extinction from Bayestar maps

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