bayes-hfs 1.1.1

Bayesian Hyperfine Spectroscopy Model

bayes_hfs

A Bayesian Molecular Hyperfine Spectroscopy Model

bayes_hfs implements a probabilistic model to infer the physics of the interstellar medium from molecular hyperfine spectroscopy observations.

This replaces a similar model tuned for CN and $^{13}$CN hyperfine spectroscopy observations called bayes_cn_hfs. bayes_hfs aims to be a more general purpose model.

• Installation
  • Basic Installation
• Notes on Physics & Radiative Transfer
• Models
  • Model Notes
  • HFSModel
  • HFSRatioModel
• Syntax & Examples
• Issues and Contributing
• License and Copyright

Installation

Basic Installation

Install with pip in a conda virtual environment:

conda create --name bayes_hfs -c conda-forge pymc pip
conda activate bayes_hfs
pip install bayes_hfs

Notes on Physics & Radiative Transfer

All models in bayes_hfs apply the same physics and equations of radiative transfer.

The transition optical depth and source function are taken from Magnum & Shirley (2015) section 2 and 3.

The radiative transfer is calculated explicitly assuming an off-source background temperature bg_temp (see below) similar to Magnum & Shirley (2015) equation 23. By default, the clouds are ordered from nearest to farthest, so optical depth effects (i.e., self-absorption) may be present. We do not assume the Rayleigh-Jeans limit; the source radiation temperature is predicted explicitly and can account for observation effects, i.e., the models can predict brightness temperature ($T_B$) or corrected antenna temperature ($T_A^*$).

Non-constant excitation temperature (CTEX) effects are modeled by considering the column densities of all states and self-consistently solving for the excitation temperature of each transition.

For the HFSRatioModel, we can either assume or not assume CTEX for either species. If both species do not assume CTEX, then the CTEX variance hyperparameter is shared between species.

Notably, since these are forward models, we do not make assumptions regarding the optical depth or the Rayleigh-Jeans limit. These effects, and the subsequent degeneracies and biases, are predicted by the model and thus captured in the inference.

Models

The models provided by bayes_hfs are implemented in the bayes_spec framework. bayes_spec assumes that the source of spectral line emission can be decomposed into a series of "clouds", each of which is defined by a set of model parameters. Here we define the models available in bayes_hfs.

Model Notes

1. The velocity of a cloud can be challenging to identify when spectral lines are narrow and widely separated. We overcome this limitation by modeling the line profiles as a "pseudo-Voight" profile, which is a linear combination of a Gaussian and Lorentzian profile. The parameter fwhm_L is a latent hyper-parameter (shared among all clouds) that characterizes the width of the Lorentzian part of the line profile. When fwhm_L is zero, the line is perfectly Gaussian. This parameter produces line profile wings that may not be physical but nonetheless enable the optimization algorithms (i.e, MCMC) to converge more reliably and efficiently. Model solutions with non-zero fwhm_L should be scrutinized carefully. This feature can be turned off by supplying None (default) to prior_fwhm_L, in which case the line profiles are assumed Gaussian.
2. Hyperfine anomalies are treated as deviations from the LTE populations of each state. The value passed to prior_log10_Tex_CTEX sets the CTEX statistical weights, CTEX_weights. Deviations from these weights are modeled as a Dirichlet distribution with a concentration parameter len(states) * CTEX_weights / 10**log10_CTEX_variance, where log10_CTEX_variance is a cloud parameter that describes the scatter in state weights around the LTE values. A small log10_CTEX_variance implies a large concentration around CTEX_weights.
3. To prevent masers, which have a different equation of radiative transfer than is assumed by the model, we clip the statistical weights to be in the range [clip_weights, 1.0 - clip_weights] (i.e., the weights can't be 0 or 1), and we clip the optical depth below clip_tau.

HFSModel

The basic model is HFSModel, a general purpose model for modelling molecular hyperfine spectroscopic observations. The model assumes that the emission can be explained by the radiative transfer of emission through a series of isothermal, homogeneous clouds as well as a polynomial spectral baseline. The following diagram demonstrates the relationship between the free parameters (empty ellipses), deterministic quantities (rectangles), model predictions (filled ellipses), and observations (filled, round rectangles). Many of the parameters are internally normalized (and thus have names like _norm). The subsequent tables describe the model parameters in more detail.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_Ntot	Total column density across all upper and lower states	cm-2	$\log_{10}N_{\rm tot} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[13.5, 0.25]
fwhm2	Square FWHM line width	km s-1	$\Delta V^2 \sim ChiSquared(\nu=1)$	1.0
velocity	Velocity (same reference frame as data)	km s-1	$V \sim p[0] + p[1] {\rm Beta}(\alpha=2, \beta=2)$	[-10.0, 10.0]
log10_Tex_CTEX	CTEX excitation temperature	K	$\log_{10}T_{{\rm ex}, ul} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.75, 0.1]
log10_CTEX_variance	CTEX variance	``	$\log_{10} \sigma_{\rm CTEX}^2 \sim p[0] + {\rm HalfNormal}(\sigma=p[1])$	[-4.0, 1.0]

The log10_CTEX_variance parameter is only relevant when assume_CTEX is False. Otherwise, all transitions are assumed to have the same excitation temperature.

Hyper Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
fwhm_L	Lorentzian FWHM line width	km s-1	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	None
baseline_coeffs	Normalized polynomial baseline coefficients	``	$\beta_i \sim {\rm Normal}(\mu=0.0, \sigma=p_i)$	[1.0]*baseline_degree

HFSRatioModel

bayes_hfs also implements HFSRatioModel, a model to infer the column density ratio between two species under the assumption that they originate in the same slabs. Different assumptions about the excitation conditions can be made.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_Ntot1	Total column density of the first species	cm-2	$\log_{10}N_{\rm tot, 1} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[13.5, 0.25]
ratio	Column density ratio of second to first species	``	$\log_{10}N_{\rm tot, 2}/\log_{10}N_{\rm tot, 1} \sim {\rm HalfNormal}(\sigma=p)$	0.1
fwhm2	Square FWHM line width	km s-1	$\Delta V^2 \sim ChiSquared(\nu=1)$	1.0
velocity	Velocity (same reference frame as data)	km s-1	$V \sim p[0] + p[1] {\rm Beta}(\alpha=2, \beta=2)$	[-10.0, 10.0]
log10_Tex_CTEX	CTEX excitation temperature	K	$\log_{10}T_{{\rm ex}, ul} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.75, 0.1]
log10_CTEX_variance	CTEX variance	``	$\log_{10} \sigma_{\rm CTEX}^2 \sim p[0] + {\rm HalfNormal}(\sigma=p[1])$	[-4.0, 1.0]

The log10_CTEX_variance parameter is only relevant when either assume_CTEX1 is False or assume_CTEX2 is False. If they are both False, then both species are assumed to be drawn from the same distribution of state densities set by the log10_CTEX_variance parameter.

Hyper Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
fwhm_L	Lorentzian FWHM line width	km s-1	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	None
baseline_coeffs	Normalized polynomial baseline coefficients	``	$\beta_i \sim {\rm Normal}(\mu=0.0, \sigma=p_i)$	[1.0]*baseline_degree

Syntax & Examples

See the various tutorial notebooks under docs/source/notebooks. Tutorials and the full API are available here: https://bayes-hfs.readthedocs.io.

Issues and Contributing

Anyone is welcome to submit issues or contribute to the development of this software via Github.

License and Copyright

Copyright(C) 2024-2025 by Trey V. Wenger

This code is licensed under MIT license (see LICENSE for details)