bayes-yplus 1.3.1

A Bayesian Model of Radio Recombination Line Emission

bayes_yplus

A Bayesian Model of Radio Recombination Line Emission

bayes_yplus implements models to infer the helium abundance (y+) from radio recombination line (RRL) observations.

• Installation
  • Basic Installation
  • Development Installation
• Notes on Physics & Radiative Transfer
• Models
  • Model Notes
  • YPlusModel
  • ordered
• Syntax & Examples
• Issues and Contributing
• License and Copyright

Installation

Basic Installation

Install with pip in a conda virtual environment:

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

Development Installation

Alternatively, download and unpack the latest release, or fork the repository and contribute to the development of bayes_yplus!

Install in a conda virtual environment:

cd /path/to/bayes_yplus
conda env create -f environment.yml
conda activate bayes_yplus-dev
pip install -e .

Notes on Physics & Radiative Transfer

All models in bayes_yplus assume the emission is optically thin. The helium RRL is assumed to have a fixed centroid velocity -122.15 km/s from that of the hydrogen RRL.

Models

The models provided by bayes_yplus 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_yplus.

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 fwhm_L much larger than the channel size should be scrutinized carefully.
2. By default, the spectral RMS noise is not inferred, rather it is taken from the noise attribute of the passed SpecData datasets. If prior_rms is not None, then the spectral RMS noise of each dataset is inferred.

YPlusModel

The basic model is YPlusModel. The model assumes that the emission can be explained by hydrogen and helium RRL emission from discrete clouds. 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}
H_area	H RRL line area	mK km s-1	$\int T_{B, \rm H} dV \sim {\rm HalfNormal}(\sigma=p)$	1000.0
H_center	H RRL center velocity	km s-1	$V_{\rm LSR, H} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 25.0]
H_fwhm	H RRL FWHM line width	km s-1	$\Delta V_{\rm H} \sim {\rm Gamma}(\mu=p_0, \sigma=p_1)$	[25.0, 10.0]
He_H_fwhm_ratio	He/H FWHM line width ratio	``	$\Delta V_{\rm He}/\Delta V_{\rm H} \sim {\rm Gamma}(\mu=p_0, \sigma=p_1)$	[1.0, 0.1]
yplus	He$^+/$H$^+$ abundance by number	``	$y^+ \sim {\rm HalfNormal}(\sigma=p)$	0.05

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	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	1.0
rms	Spectral rms noise	mK	${\rm rms} \sim {\rm HalfNormal}(\sigma=p)$	0.01
baseline_coeffs	Normalized polynomial baseline coefficients	``	$\beta_i \sim {\rm Normal}(\mu=0.0, \sigma=p_i)$	[1.0]*(baseline_degree + 1)

ordered

An additional parameter to set_priors for these models is ordered. By default, this parameter is False, in which case the order of the clouds is arbitrary. Sampling from these models can be challenging due to the labeling degeneracy: if the order of clouds does not matter (i.e., the emission is optically thin), then each Markov chain could decide on a different, equally-valid order of clouds.

If we assume that the emission is optically thin, then we can set ordered=True, in which case the order of clouds is restricted to be increasing with velocity. When ordered=True, the velocity prior is defined differently:

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
H_center	H RRL center velocity	km s-1	$V_i \sim p_0 + \sum_0^{i-1} V_i + {\rm Gamma}(\alpha=2, \beta=1.0/p_1)$	[0.0, 25.0]

Syntax & Examples

See the various notebooks under examples for demonstrations of these models.

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)