{% raw %} Title: Create a Markdown Blog Post Integrating Research Details and a Featured Paper ==================================================================================== This task involves generating a Markdown file (ready for a GitHub-served Jekyll site) that integrates our research details with a featured research paper. The output must follow the exact format and conventions described below. ==================================================================================== Output Format (Markdown): ------------------------------------------------------------------------------------ --- layout: post title: "Accounting for Noise and Singularities in Bayesian Calibration Methods for Global 21-cm Cosmology Experiments" date: 2024-12-18 categories: papers --- ![AI generated image](/assets/images/posts/2024-12-18-2412.14023.png) Will HandleyIan RoqueSam LeeneyHarry BevinsDominic AnsteyEloy de Lera Acedo Content generated by [gemini-2.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2024-12-18-2412.14023.txt). Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2024-12-18-2412.14023.txt). ------------------------------------------------------------------------------------ ==================================================================================== Please adhere strictly to the following instructions: ==================================================================================== Section 1: Content Creation Instructions ==================================================================================== 1. **Generate the Page Body:** - Write a well-composed, engaging narrative that is suitable for a scholarly audience interested in advanced AI and astrophysics. - Ensure the narrative is original and reflective of the tone and style and content in the "Homepage Content" block (provided below), but do not reuse its content. - Use bullet points, subheadings, or other formatting to enhance readability. 2. **Highlight Key Research Details:** - Emphasize the contributions and impact of the paper, focusing on its methodology, significance, and context within current research. - Specifically highlight the lead author ({'name': 'Christian J. Kirkham'}). When referencing any author, use Markdown links from the Author Information block (choose academic or GitHub links over social media). 3. **Integrate Data from Multiple Sources:** - Seamlessly weave information from the following: - **Paper Metadata (YAML):** Essential details including the title and authors. - **Paper Source (TeX):** Technical content from the paper. - **Bibliographic Information (bbl):** Extract bibliographic references. - **Author Information (YAML):** Profile details for constructing Markdown links. - Merge insights from the Paper Metadata, TeX source, Bibliographic Information, and Author Information blocks into a coherent narrative—do not treat these as separate or isolated pieces. - Insert the generated narrative between the HTML comments: and 4. **Generate Bibliographic References:** - Review the Bibliographic Information block carefully. - For each reference that includes a DOI or arXiv identifier: - For DOIs, generate a link formatted as: [10.1234/xyz](https://doi.org/10.1234/xyz) - For arXiv entries, generate a link formatted as: [2103.12345](https://arxiv.org/abs/2103.12345) - **Important:** Do not use any LaTeX citation commands (e.g., `\cite{...}`). Every reference must be rendered directly as a Markdown link. For example, instead of `\cite{mycitation}`, output `[mycitation](https://doi.org/mycitation)` - **Incorrect:** `\cite{10.1234/xyz}` - **Correct:** `[10.1234/xyz](https://doi.org/10.1234/xyz)` - Ensure that at least three (3) of the most relevant references are naturally integrated into the narrative. - Ensure that the link to the Featured paper [2412.14023](https://arxiv.org/abs/2412.14023) is included in the first sentence. 5. **Final Formatting Requirements:** - The output must be plain Markdown; do not wrap it in Markdown code fences. - Preserve the YAML front matter exactly as provided. ==================================================================================== Section 2: Provided Data for Integration ==================================================================================== 1. **Homepage Content (Tone and Style Reference):** ```markdown --- layout: home --- ![AI generated image](/assets/images/index.png) The Handley Research Group stands at the forefront of cosmological exploration, pioneering novel approaches that fuse fundamental physics with the transformative power of artificial intelligence. We are a dynamic team of researchers, including PhD students, postdoctoral fellows, and project students, based at the University of Cambridge. Our mission is to unravel the mysteries of the Universe, from its earliest moments to its present-day structure and ultimate fate. We tackle fundamental questions in cosmology and astrophysics, with a particular focus on leveraging advanced Bayesian statistical methods and AI to push the frontiers of scientific discovery. Our research spans a wide array of topics, including the [primordial Universe](https://arxiv.org/abs/1907.08524), [inflation](https://arxiv.org/abs/1807.06211), the nature of [dark energy](https://arxiv.org/abs/2503.08658) and [dark matter](https://arxiv.org/abs/2405.17548), [21-cm cosmology](https://arxiv.org/abs/2210.07409), the [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1807.06209), and [gravitational wave astrophysics](https://arxiv.org/abs/2411.17663). ### Our Research Approach: Innovation at the Intersection of Physics and AI At The Handley Research Group, we develop and apply cutting-edge computational techniques to analyze complex astronomical datasets. Our work is characterized by a deep commitment to principled [Bayesian inference](https://arxiv.org/abs/2205.15570) and the innovative application of [artificial intelligence (AI) and machine learning (ML)](https://arxiv.org/abs/2504.10230). **Key Research Themes:** * **Cosmology:** We investigate the early Universe, including [quantum initial conditions for inflation](https://arxiv.org/abs/2002.07042) and the generation of [primordial power spectra](https://arxiv.org/abs/2112.07547). We explore the enigmatic nature of [dark energy, using methods like non-parametric reconstructions](https://arxiv.org/abs/2503.08658), and search for new insights into [dark matter](https://arxiv.org/abs/2405.17548). A significant portion of our efforts is dedicated to [21-cm cosmology](https://arxiv.org/abs/2104.04336), aiming to detect faint signals from the Cosmic Dawn and the Epoch of Reionization. * **Gravitational Wave Astrophysics:** We develop methods for [analyzing gravitational wave signals](https://arxiv.org/abs/2411.17663), extracting information about extreme astrophysical events and fundamental physics. * **Bayesian Methods & AI for Physical Sciences:** A core component of our research is the development of novel statistical and AI-driven methodologies. This includes advancing [nested sampling techniques](https://arxiv.org/abs/1506.00171) (e.g., [PolyChord](https://arxiv.org/abs/1506.00171), [dynamic nested sampling](https://arxiv.org/abs/1704.03459), and [accelerated nested sampling with $\beta$-flows](https://arxiv.org/abs/2411.17663)), creating powerful [simulation-based inference (SBI) frameworks](https://arxiv.org/abs/2504.10230), and employing [machine learning for tasks such as radiometer calibration](https://arxiv.org/abs/2504.16791), [cosmological emulation](https://arxiv.org/abs/2503.13263), and [mitigating radio frequency interference](https://arxiv.org/abs/2211.15448). We also explore the potential of [foundation models for scientific discovery](https://arxiv.org/abs/2401.00096). **Technical Contributions:** Our group has a strong track record of developing widely-used scientific software. Notable examples include: * [**PolyChord**](https://arxiv.org/abs/1506.00171): A next-generation nested sampling algorithm for Bayesian computation. * [**anesthetic**](https://arxiv.org/abs/1905.04768): A Python package for processing and visualizing nested sampling runs. * [**GLOBALEMU**](https://arxiv.org/abs/2104.04336): An emulator for the sky-averaged 21-cm signal. * [**maxsmooth**](https://arxiv.org/abs/2007.14970): A tool for rapid maximally smooth function fitting. * [**margarine**](https://arxiv.org/abs/2205.12841): For marginal Bayesian statistics using normalizing flows and KDEs. * [**fgivenx**](https://arxiv.org/abs/1908.01711): A package for functional posterior plotting. * [**nestcheck**](https://arxiv.org/abs/1804.06406): Diagnostic tests for nested sampling calculations. ### Impact and Discoveries Our research has led to significant advancements in cosmological data analysis and yielded new insights into the Universe. Key achievements include: * Pioneering the development and application of advanced Bayesian inference tools, such as [PolyChord](https://arxiv.org/abs/1506.00171), which has become a cornerstone for cosmological parameter estimation and model comparison globally. * Making significant contributions to the analysis of major cosmological datasets, including the [Planck mission](https://arxiv.org/abs/1807.06209), providing some of the tightest constraints on cosmological parameters and models of [inflation](https://arxiv.org/abs/1807.06211). * Developing novel AI-driven approaches for astrophysical challenges, such as using [machine learning for radiometer calibration in 21-cm experiments](https://arxiv.org/abs/2504.16791) and [simulation-based inference for extracting cosmological information from galaxy clusters](https://arxiv.org/abs/2504.10230). * Probing the nature of dark energy through innovative [non-parametric reconstructions of its equation of state](https://arxiv.org/abs/2503.08658) from combined datasets. * Advancing our understanding of the early Universe through detailed studies of [21-cm signals from the Cosmic Dawn and Epoch of Reionization](https://arxiv.org/abs/2301.03298), including the development of sophisticated foreground modelling techniques and emulators like [GLOBALEMU](https://arxiv.org/abs/2104.04336). * Developing new statistical methods for quantifying tensions between cosmological datasets ([Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio](https://arxiv.org/abs/1902.04029)) and for robust Bayesian model selection ([Bayesian model selection without evidences: application to the dark energy equation-of-state](https://arxiv.org/abs/1506.09024)). * Exploring fundamental physics questions such as potential [parity violation in the Large-Scale Structure using machine learning](https://arxiv.org/abs/2410.16030). ### Charting the Future: AI-Powered Cosmological Discovery The Handley Research Group is poised to lead a new era of cosmological analysis, driven by the explosive growth in data from next-generation observatories and transformative advances in artificial intelligence. Our future ambitions are centred on harnessing these capabilities to address the most pressing questions in fundamental physics. **Strategic Research Pillars:** * **Next-Generation Simulation-Based Inference (SBI):** We are developing advanced SBI frameworks to move beyond traditional likelihood-based analyses. This involves creating sophisticated codes for simulating [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1908.00906) and [Baryon Acoustic Oscillation (BAO)](https://arxiv.org/abs/1607.00270) datasets from surveys like DESI and 4MOST, incorporating realistic astrophysical effects and systematic uncertainties. Our AI initiatives in this area focus on developing and implementing cutting-edge SBI algorithms, particularly [neural ratio estimation (NRE) methods](https://arxiv.org/abs/2407.15478), to enable robust and scalable inference from these complex simulations. * **Probing Fundamental Physics:** Our enhanced analytical toolkit will be deployed to test the standard cosmological model ($\Lambda$CDM) with unprecedented precision and to explore [extensions to Einstein's General Relativity](https://arxiv.org/abs/2006.03581). We aim to constrain a wide range of theoretical models, from modified gravity to the nature of [dark matter](https://arxiv.org/abs/2106.02056) and [dark energy](https://arxiv.org/abs/1701.08165). This includes leveraging data from upcoming [gravitational wave observatories](https://arxiv.org/abs/1803.10210) like LISA, alongside CMB and large-scale structure surveys from facilities such as Euclid and JWST. * **Synergies with Particle Physics:** We will continue to strengthen the connection between cosmology and particle physics by expanding the [GAMBIT framework](https://arxiv.org/abs/2009.03286) to interface with our new SBI tools. This will facilitate joint analyses of cosmological and particle physics data, providing a holistic approach to understanding the Universe's fundamental constituents. * **AI-Driven Theoretical Exploration:** We are pioneering the use of AI, including [large language models and symbolic computation](https://arxiv.org/abs/2401.00096), to automate and accelerate the process of theoretical model building and testing. This innovative approach will allow us to explore a broader landscape of physical theories and derive new constraints from diverse astrophysical datasets, such as those from GAIA. Our overarching goal is to remain at the forefront of scientific discovery by integrating the latest AI advancements into every stage of our research, from theoretical modeling to data analysis and interpretation. We are excited by the prospect of using these powerful new tools to unlock the secrets of the cosmos. Content generated by [gemini-2.5-pro-preview-05-06](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/index.txt). Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/index.txt). ``` 2. **Paper Metadata:** ```yaml !!python/object/new:feedparser.util.FeedParserDict dictitems: id: http://arxiv.org/abs/2412.14023v2 guidislink: true link: http://arxiv.org/abs/2412.14023v2 updated: '2025-01-22T13:17:03Z' updated_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2025 - 1 - 22 - 13 - 17 - 3 - 2 - 22 - 0 - tm_zone: null tm_gmtoff: null published: '2024-12-18T16:40:37Z' published_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2024 - 12 - 18 - 16 - 40 - 37 - 2 - 353 - 0 - tm_zone: null tm_gmtoff: null title: "Accounting for Noise and Singularities in Bayesian Calibration Methods\n\ \ for Global 21-cm Cosmology Experiments" title_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: "Accounting for Noise and Singularities in Bayesian Calibration Methods\n\ \ for Global 21-cm Cosmology Experiments" summary: 'Due to the large dynamic ranges involved with separating the cosmological 21-cm signal from the Cosmic Dawn from galactic foregrounds, a well-calibrated instrument is essential to avoid biases from instrumental systematics. In this paper we present three methods for calibrating a global 21-cm cosmology experiment using the noise wave parameter formalisation to characterise a low noise amplifier including a careful consideration of how calibrator temperature noise and singularities will bias the result. The first method presented in this paper builds upon the existing conjugate priors method by weighting the calibrators by a physically motivated factor, thereby avoiding singularities and normalising the noise. The second method fits polynomials to the noise wave parameters by marginalising over the polynomial coefficients and sampling the polynomial orders as parameters. The third method introduces a physically motivated noise model to the marginalised polynomial method. Running these methods on a suite of simulated datasets based on the REACH receiver design and a lab dataset, we found that our methods produced a calibration solution which is equally as or more accurate than the existing conjugate priors method when compared with an analytic estimate of the calibrator''s noise. We find in the case of the measured lab dataset the conjugate priors method is biased heavily by the large noise on the shorted load calibrator, resulting in incorrect noise wave parameter fits. This is mitigated by the methods introduced in this paper which calibrate the validation source spectra to within 5% of the noise floor.' summary_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: 'Due to the large dynamic ranges involved with separating the cosmological 21-cm signal from the Cosmic Dawn from galactic foregrounds, a well-calibrated instrument is essential to avoid biases from instrumental systematics. In this paper we present three methods for calibrating a global 21-cm cosmology experiment using the noise wave parameter formalisation to characterise a low noise amplifier including a careful consideration of how calibrator temperature noise and singularities will bias the result. The first method presented in this paper builds upon the existing conjugate priors method by weighting the calibrators by a physically motivated factor, thereby avoiding singularities and normalising the noise. The second method fits polynomials to the noise wave parameters by marginalising over the polynomial coefficients and sampling the polynomial orders as parameters. The third method introduces a physically motivated noise model to the marginalised polynomial method. Running these methods on a suite of simulated datasets based on the REACH receiver design and a lab dataset, we found that our methods produced a calibration solution which is equally as or more accurate than the existing conjugate priors method when compared with an analytic estimate of the calibrator''s noise. We find in the case of the measured lab dataset the conjugate priors method is biased heavily by the large noise on the shorted load calibrator, resulting in incorrect noise wave parameter fits. This is mitigated by the methods introduced in this paper which calibrate the validation source spectra to within 5% of the noise floor.' authors: - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Christian J. Kirkham - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: William J. Handley - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Jiacong Zhu - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Kaan Artuc - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Ian L. V. Roque - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Samuel A. K. Leeney - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Harry T. J. Bevins - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Dominic J. Anstey - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Eloy de Lera Acedo author_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Eloy de Lera Acedo author: Eloy de Lera Acedo arxiv_comment: 9 pages, 5 figures links: - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: http://arxiv.org/abs/2412.14023v2 rel: alternate type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: title: pdf href: http://arxiv.org/pdf/2412.14023v2 rel: related type: application/pdf arxiv_primary_category: term: astro-ph.IM scheme: http://arxiv.org/schemas/atom tags: - !!python/object/new:feedparser.util.FeedParserDict dictitems: term: astro-ph.IM scheme: http://arxiv.org/schemas/atom label: null - !!python/object/new:feedparser.util.FeedParserDict dictitems: term: astro-ph.CO scheme: http://arxiv.org/schemas/atom label: null ``` 3. **Paper Source (TeX):** ```tex % mnras_template.tex % % LaTeX template for creating an MNRAS paper % % v3.0 released 14 May 2015 % (version numbers match those of mnras.cls) % % Copyright (C) Royal Astronomical Society 2015 % Authors: % Keith T. Smith (Royal Astronomical Society) % Change log % % v3.2 July 2023 % Updated guidance on use of amssymb package % v3.0 May 2015 % Renamed to match the new package name % Version number matches mnras.cls % A few minor tweaks to wording % v1.0 September 2013 % Beta testing only - never publicly released % First version: a simple (ish) template for creating an MNRAS paper %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Basic setup. Most papers should leave these options alone. \documentclass[fleqn,usenatbib]{mnras} % MNRAS is set in Times font. If you don't have this installed (most LaTeX % installations will be fine) or prefer the old Computer Modern fonts, comment % out the following line \usepackage{newtxtext,newtxmath} % Depending on your LaTeX fonts installation, you might get better results with one of these: %\usepackage{mathptmx} %\usepackage{txfonts} % Use vector fonts, so it zooms properly in on-screen viewing software % Don't change these lines unless you know what you are doing \usepackage[T1]{fontenc} % Allow "Thomas van Noord" and "Simon de Laguarde" and alike to be sorted by "N" and "L" etc. in the bibliography. % Write the name in the bibliography as "\VAN{Noord}{Van}{van} Noord, Thomas" \DeclareRobustCommand{\VAN}[3]{#2} \let\VANthebibliography\thebibliography \def\thebibliography{\DeclareRobustCommand{\VAN}[3]{##3}\VANthebibliography} %%%%% AUTHORS - PLACE YOUR OWN PACKAGES HERE %%%%% % Only include extra packages if you really need them. Avoid using amssymb if newtxmath is enabled, as these packages can cause conflicts. newtxmatch covers the same math symbols while producing a consistent Times New Roman font. Common packages are: \usepackage{graphicx} % Including figure files \usepackage{amsmath} % Advanced maths commands \usepackage{subcaption} \usepackage{dsfont} \usepackage{booktabs} \usepackage{orcidlink} \usepackage[colorinlistoftodos, color=lime]{todonotes} \DeclareMathAlphabet{\mathcal}{OMS}{cmsy}{m}{n} % Fix mathcal font %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% AUTHORS - PLACE YOUR OWN COMMANDS HERE %%%%% % Please keep new commands to a minimum, and use \newcommand not \def to avoid % overwriting existing commands. Example: %\newcommand{\pcm}{\,cm$^{-2}$} % per cm-squared \newcommand\overmat[2]{% \makebox[0pt][l]{$\smash{\color{white}\overbrace{\phantom{% \begin{matrix}#2\end{matrix}}}^{#1}}$}#2} \newcommand\bovermat[2]{% \makebox[0pt][l]{$\smash{\overbrace{\phantom{% \begin{matrix}#2\end{matrix}}}^{#1}}$}#2} \newcommand\partialphantom{\vphantom{\frac{\partial e_{P,M}}{\partial w_{1,1}}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% TITLE PAGE %%%%%%%%%%%%%%%%%%% % Title of the paper, and the short title which is used in the headers. % Keep the title short and informative. \title[Accounting for Noise and Singularities]{Accounting for Noise and Singularities in Bayesian Calibration Methods for Global 21-cm Cosmology Experiments} % The list of authors, and the short list which is used in the headers. % If you need two or more lines of authors, add an extra line using \newauthor \author[C. J. Kirkham et al.]{Christian J. Kirkham \textsuperscript{\orcidlink{0000-0001-5385-6329}},$^{1,2}$\thanks{E-mail: \href{mailto:cjk55@cam.ac.uk}{cjk55@cam.ac.uk}} William J. Handley \textsuperscript{\orcidlink{0000-0002-5866-0445}},$^{1,2}$ Jiacong Zhu \textsuperscript{\orcidlink{0009-0004-8965-0671}},$^{3,4}$ Kaan Artuc \textsuperscript{\orcidlink{0000-0001-8510-5159}},$^{1,2}$ Ian L. V. Roque \textsuperscript{\orcidlink{0000-0003-4874-9371}},$^{5}$\newauthor Samuel A. K. Leeney \textsuperscript{\orcidlink{0000-0003-4366-1119}},$^{1,2}$ Harry T. J. Bevins \textsuperscript{\orcidlink{0000-0002-4367-3550}},$^{1,2}$ Dominic J. Anstey \textsuperscript{\orcidlink{0000-0003-1742-7417}},$^{1,2}$ and Eloy de Lera Acedo \textsuperscript{\orcidlink{0000-0001-8530-6989}}$^{1,2}$ \\ % List of institutions $^{1}$Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK\\ $^{2}$Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK\\ $^{3}$National Astronomical Observatory, Chinese Academy of Sciences\\ $^{4}$School of Astronomy and Space Science, University of Chinese Academy of Sciences\\ $^{5}$SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA\\\\ } % These dates will be filled out by the publisher \date{Accepted XXX. Received YYY; in original form ZZZ} % Enter the current year, for the copyright statements etc. \pubyear{2025} % Don't change these lines \begin{document} \label{firstpage} \pagerange{\pageref{firstpage}--\pageref{lastpage}} \maketitle % Abstract of the paper \begin{abstract} Due to the large dynamic ranges involved with separating the cosmological 21-cm signal from the Cosmic Dawn from galactic foregrounds, a well-calibrated instrument is essential to avoid biases from instrumental systematics. In this paper we present three methods for calibrating a global 21-cm cosmology experiment using the noise wave parameter formalisation to characterise a low noise amplifier including a careful consideration of how calibrator temperature noise and singularities will bias the result. The first method presented in this paper builds upon the existing conjugate priors method by weighting the calibrators by a physically motivated factor, thereby avoiding singularities and normalising the noise. The second method fits polynomials to the noise wave parameters by marginalising over the polynomial coefficients and sampling the polynomial orders as parameters. The third method introduces a physically motivated noise model to the marginalised polynomial method. Running these methods on a suite of simulated datasets based on the REACH receiver design and a lab dataset, we found that our methods produced a calibration solution which is equally as or more accurate than the existing conjugate priors method when compared with an estimate of the calibrator's noise. We find in the case of the measured lab dataset the conjugate priors method is biased heavily by the large noise on the shorted load calibrator, resulting in incorrect noise wave parameter fits. This is mitigated by the methods introduced in this paper which calibrate the validation source spectra to within 5\% of the noise floor. \end{abstract} % Select between one and six entries from the list of approved keywords. % Don't make up new ones. \begin{keywords} instrumentation: interferometers -- methods: data analysis -- cosmology: dark ages, reionization, first stars -- cosmology: early Universe \end{keywords} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% BODY OF PAPER %%%%%%%%%%%%%%%%%% \section{Introduction} The aim of 21-cm cosmology experiments is to measure the emission from neutral hydrogen (HI) in the periods of the early universe known as the Cosmic Dark Ages, Cosmic Dawn and the Epoch of Reionisation. They detect this emission using the hyperfine transition at a rest frequency of $\nu \approx 1420$ MHz or wavelength $\lambda \approx 21$ cm produced by the HI gas and define a statistical `spin temperature' that is measured relative to the temperature of cosmic microwave background \citep{furlanettoCosmologyLowFrequencies2006}. Using this measurement we can infer the properties of the first galaxies and dark matter in the cosmic dawn \citep{monsalveResultsEDGESHighband2018,monsalveResultsEDGESHighBand2019, bevinsAstrophysicalConstraintsSARAS2022}. There are several experiments which are designed to measure the global sky-averaged 21-cm signal such as Experiment to Detect the Global EoR Signature (EDGES) \citep{bowmanEmpiricalConstraintsGlobal2008}, Shaped Antenna measurement of the background Radio Spectrum (SARAS) \citep{singhSARASSpectralRadiometer2018,singhDetectionCosmicDawn2022}, Large Aperture Experiment to Detect the Dark Ages (LEDA) \citep{priceDesignCharacterizationLargeaperture2018}, Probing Radio Intensity at high-Z from Marion (PRIZM) \citep{philipProbingRadioIntensity2019}, Mapper of the IGM Spin Temperature (MIST) \citep{monsalveMapperIGMSpin2023} and Radio Experiment for the Analysis of Cosmic Hydrogen (REACH) \citep{deleraacedoREACHRadiometerDetecting2022a}. These experiments use low-frequency radio antennae to detect the 21-cm signal from neutral hydrogen in the Cosmic Dawn and Epoch of Reionisation to place constraints on the physics of the early universe. The first claimed detection of the global 21-cm signal was made by the EDGES team \citep{bowmanAbsorptionProfileCentred2018}. EDGES found the best fitting profile to be a $0.5^{+0.5}_{-0.2}$ K deep flattened Gaussian centred at $78\pm1$ MHz. This detection raised concerns because of its unusually deep absorption trough and unphysical galactic foreground model \citep{hillsConcernsModellingEDGES2018}, requiring exotic physics to explain the detection \citep{barkanaPossibleInteractionBaryons2018, fengEnhancedGlobalSignal2018}. Another 21-cm global signal experiment, SARAS \citep{nambissanSARASCDEoR2021}, recently placed constraints on the global 21-cm signal, rejecting the EDGES detection with 95.3\% confidence \citep{singhDetectionCosmicDawn2022}. An alternative explanation for the EDGES signal is uncorrected for systematics in the data \citep{hillsConcernsModellingEDGES2018,singhRedshifted21Cm2019,simsTestingCalibrationSystematics2020}. \cite{hillsConcernsModellingEDGES2018} found evidence of a sinusoidal systematic in the EDGES data, finding the goodness-of-fit improving when a 12.5 MHz sinusoid is removed. This result is consistent with the results of \cite{simsTestingCalibrationSystematics2020}. In this work we will focus on the REACH experiment \citep{deleraacedoREACHRadiometerDetecting2022a}, a single antenna experiment which is designed to verify the EDGES detection. To do so requires careful characterisation and calibration of systematics to guarantee the sensitivity required to detect the global 21-cm signal, for example \cite{roqueBayesianNoiseWave2021}, \cite{shenBayesianDataAnalysis2022}, \cite{scheutwinkelBayesianEvidencedrivenDiagnosis2022a}, \cite{pattisonModellingHotHorizon2023}, \cite{kirkhamBayesianMethodMitigate2024}, and \cite{cumnerEffectsAntennaPower2024}. In this paper we will present four methods of calibration for 21-cm global experiments using Bayesian methods to characterise the response of the receiver. We will introduce three methods which are similar to the approach used in \cite{roqueBayesianNoiseWave2021}, but with modifications to consider the variation of noise of the calibrated temperatures. In section \ref{s:methods} we will introduce the REACH receiver system and present the calibration methods. In section \ref{s:method_comparison} we will introduce the techniques used to generate a set of simulated data and present a set of lab data, and use both to demonstrate and compare our calibration methods. Finally in section \ref{s:conclusions} we will present our conclusions. \section{Methods} \label{s:methods} \subsection{REACH Receiver Calibration} \begin{figure*} \centering \includegraphics[width=0.8\linewidth]{figs/dicke_switching_2.pdf} \caption{Diagram of the REACH calibration setup using a Dicke switch. Adapted from \protect\cite{roqueBayesianNoiseWave2021}.} \label{fig:dicke-switching} \end{figure*} To calibrate the instrument the REACH system uses the method of `Dicke switching' which was outlined in \cite{dickeMeasurementThermalRadiation1946} and adapted for global 21-cm experiments in \cite{rogersAbsoluteCalibrationWideband2012}. In order to do this, we measure two reference sources -- a resistive load and a noise source. To account for impedence mismatches in the system we also have twelve calibration sources \citep{meysWaveApproachNoise1978} which were chosen with the aim of maximising the Smith chart coverage, allowing us to measure an unknown source such as the antenna \citep{roqueReceiverDesignREACH2025}. A schematic of this calibration method can be seen in figure \ref{fig:dicke-switching}. The twelve calibration sources in the REACH receiver are listed below. % \begin{itemize} \item An ambient 50 $\Omega$ `cold' load \item Ambient 25 $\Omega$ and 100 $\Omega$ loads \item A 50 $\Omega$ heated `hot' load at 370K which is connected to a 4 inch cable \item 27 $\Omega$, 36 $\Omega$, 69 $\Omega$ and 91 $\Omega$ ambient loads which are connected to a `short' 2 metre cable \item 10 $\Omega$, 250 $\Omega$, open and short loads which are connected to a `long' 10 metre cable \end{itemize} % The antenna is a hexagonal dipole \citep{cumnerRadioAntennaDesign2022} which is connected to a cable approximately one metre in length. For each calibrator we measure the power spectral density (PSD) of the calibration source, $P_\text{s}$, the PSD of the reference load, $P_\text{L}$, and the PSD of the noise source, $P_\text{NS}$, using a spectrometer after amplification by a low noise amplifier (LNA). We also measure the reflection coefficient of the calibration source connected to the receiver input, $\Gamma_\text{s}$, and the reflection coefficient of the receiver itself, $\Gamma_\text{r}$, by measuring their $S_{11}$s with a vector network analyser (VNA). Using the \cite{meysWaveApproachNoise1978} noise wave approach of modelling the LNA response we can write the calibrated source temperature, $T_\text{s}$, in terms of the PSDs and reflection coefficients as \citep{monsalveCALIBRATIONEDGESHIGHBAND2017,roqueBayesianNoiseWave2021}, % \begin{align} \label{e:full_calibration_eqn} \begin{split} T_\text{NS} \left(\frac{P_\text{s} - P_\text{L}}{P_\text{NS} - P_\text{L}}\right) + T_\text{L} = &T_\text{s} \left[ \frac{1 - |\Gamma_\text{s}|^2}{|1 - \Gamma_\text{s} \Gamma_\text{r}|^2} \right]\\ &+ T_\text{unc} \left[ \frac{|\Gamma_\text{s}|^2}{|1 - \Gamma_\text{s} \Gamma_\text{r}|^2} \right]\\ &+ T_\text{cos} \left[ \frac{\mathrm{Re}\left( \frac{\Gamma_\text{s}}{1 - \Gamma_\text{s}\Gamma_\text{r}} \right)}{\sqrt{1 - |\Gamma_\text{r}|^2}} \right]\\ &+ T_\text{sin} \left[ \frac{\mathrm{Im}\left( \frac{\Gamma_\text{s}}{1 - \Gamma_\text{s}\Gamma_\text{r}} \right)}{\sqrt{1 - |\Gamma_\text{r}|^2}} \right], \end{split} \end{align} % where $T_\text{unc}$, $T_\text{cos}$ and $T_\text{sin}$ are functions of frequency known as the noise wave parameters (NWPs) which are fitted for \citep{meysWaveApproachNoise1978}. We also fit for an `effective' load temperature, $T_\text{L}$, and `effective' noise temperature of the internal noise source, $T_\text{NS}$. For brevity we will hereafter refer to all five functions as the `noise wave parameters'. In this work we will be fitting these five functions as polynomials. Following \cite{roqueBayesianNoiseWave2021}, we can simplify equation \ref{e:full_calibration_eqn} by defining the following terms, % \begin{equation} \label{e:x_matrices} X_\text{unc} = - \frac{|\Gamma_\text{s}|^2}{1 - |\Gamma_\text{s}|^2}, \end{equation} \begin{equation} \label{e:X_load} X_\text{L} = \frac{|1 - \Gamma_\text{s} \Gamma_\text{r}|^2}{1 - |\Gamma_\text{s}|^2}, \end{equation} \begin{equation} X_\text{cos} = -\mathrm{Re}\left(\frac{\Gamma_\text{s} }{1 - \Gamma_\text{s}\Gamma_\text{r}} \times \frac{X_\text{L}}{\sqrt{1 - |\Gamma_\text{r}|^2}}\right), \end{equation} \begin{equation} X_\text{sin} = -\mathrm{Im}\left(\frac{\Gamma_\text{s} }{1 - \Gamma_\text{s}\Gamma_\text{r}} \times \frac{X_\text{L}}{\sqrt{1 - |\Gamma_\text{r}|^2}}\right), \end{equation} \begin{equation} \label{e:X_noise_source} X_\text{NS} = \left(\frac{P_\text{s} - P_\text{L}}{P_\text{NS} - P_\text{L}}\right) X_\text{L}, \end{equation} % which contain all of the measured quantities of our receiver system. We can hence rearrange equation \ref{e:full_calibration_eqn} to get the simple linear calibration equation % \begin{equation} \label{e:simplified_calibration_eqn} T_s(\nu) = X_\mathrm{unc} T_\mathrm{unc} + X_\mathrm{cos} T_\mathrm{cos} + X_\mathrm{sin} T_\mathrm{sin} + X_\mathrm{NS} T_\mathrm{NS} + X_\mathrm{L} T_\mathrm{L}. \end{equation} Finally we can define the matrices % \begin{equation} \bld{X} = \begin{pmatrix}X_\text{unc} & X_\text{cos} & X_\text{sin} & X_\text{NS} & X_\text{L}\end{pmatrix}, \end{equation} \begin{equation} \bld{\Theta} = \begin{pmatrix}T_\text{unc} & T_\text{cos} & T_\text{sin} & T_\text{NS} & T_\text{L}\end{pmatrix}^T, \end{equation} % which reduces the calibration equation down to the linear equation, % \begin{equation} \label{e:calibration_equation} \bld{T}_\text{s} = \bld{X}\bld{\Theta}, \end{equation} % where $\bld{T}_\text{s}$ is a vector of calibrated source temperatures over frequency. \subsection{Noise Estimation} In order to have a benchmark by which to test our calibration methods, we estimate the noise on the calibrated temperature by propagating the PSD noise through equation \ref{e:full_calibration_eqn}. Since the S11 noise is orders of magnitude smaller than the PSD noise we make the assumption that the reflection coefficient measurements are noiseless. This means that the only contribution to noise in the final calibrated temperature comes from the measurements of $P_\text{s}$, $P_\text{L}$ and $P_\text{NS}$ in $X_\text{NS}$, equation \ref{e:X_noise_source}. The noise estimate is as follows, % \begin{align} \label{e:noise_estimate} \begin{split} (\sigma^T_s)^2 = \left(\frac{T^\text{fit}_\text{NS}X_\text{L}}{P_\text{NS} - P_\text{L}}\right)^2& \Bigg(\sigma_{P_\text{s}}^2 + \sigma_{P_\text{L}}^2 - 2\sigma_{P_\text{s}P_\text{L}}\\ &+ \left(\frac{P_\text{s} - P_\text{L}}{P_\text{NS} - P_\text{L}}\right)^2 (\sigma^2_{P_\text{L}} + \sigma^2_{P_\text{NS}}\\ &- 2\sigma_{P_\text{L}P_\text{NS}}) - 2\frac{P_\text{s} - P_\text{L}}{P_\text{NS} - P_\text{L}}\sigma_{AB} \Bigg), \end{split} \end{align} % where $A = P_\text{s} - P_\text{L}$ and $B = P_\text{NS} - P_\text{L}$. $\sigma_{IJ}$ denotes the covariance of $I$ and $J$. Note that $T^\text{fit}_\text{NS}$ is the fitted valued of $T_\text{NS}$, and that the noise is a frequency dependent quantity. The noise on the PSDs is determined by smoothing the PSD and subtracting the smooth function, leaving the noise as the residual. In order to estimate the noise before fitting, we approximate $T_\text{NS}^\text{fit}$ using the excess noise ratio (ENR) of the noise diode used in the receiver system. In the REACH system we use a noise diode with an ENR of around 6 dB. \subsection{Bayesian Inference} To estimate both the noise wave parameters and the parameters describing the noise of each of the calibrated temperatures, we will apply Bayesian inference, a statistical method that reverses conditional probabilities through Bayes' theorem: % \begin{equation} P(\bld \theta|\bld D, \mathcal M) = \frac{P(\bld D|\bld \theta, \mathcal M) \cdot P(\bld \theta | \mathcal M)}{P(\bld D | \mathcal M)} = \frac{\mathcal{L}(\theta)\cdot\pi(\theta)}{\mathcal{Z}}, \end{equation} % Here, $\bld \theta$ represents the parameters of the model, $\mathcal M$, that we aim to fit, and $\bld D$ denotes the vector of data points \citep{siviaDataAnalysisBayesian2006}. The term $P(\bld \theta | \mathcal M)$, or $\pi (\bld \theta)$, is the `prior distribution,' which reflects our prior knowledge of the parameter's probability distribution. The `likelihood', $P(\bld D|\bld \theta, \mathcal M)$, or $\mathcal L (\bld \theta)$, is the probability of observing the data given the assumed model and parameters. The `posterior distribution,' $P(\bld \theta|\bld D, \mathcal M)$, or $\mathcal P (\bld \theta)$, represents the probability of the parameters given the data and model, inferred by combining the prior and likelihood distributions. Lastly, $P(\bld D|\mathcal M)$, often denoted as $\mathcal Z$ or the `Bayesian evidence,' can be used to compare different models. We use a Gaussian likelihood for calibration \citep{roqueBayesianNoiseWave2021} of the form, % \begin{equation} \label{e:gaussian_likelihood} \mathcal L = \frac{1}{\sqrt{|2\pi \bld{C}|}}\exp\left\{-\frac{1}{2} (\bld{T}_s - \bld{X\Theta})^T \bld{C}^{-1} (\bld{T}_s - \bld{X\Theta}) \right\}, \end{equation} % where $\bld{C}$ is the covariance matrix which incorporates our calibrator noise parameters. \subsection{$\Gamma$-Weighted Conjugate Priors} \cite{roqueBayesianNoiseWave2021} uses a conjugate prior method to quickly evaluate the posterior probability by using a multivariate normal inverse gamma distribution as the prior on the polynomial coefficients. As the likelihood is Gaussian this results in a posterior which is also a normal inverse gamma distribution which can be evaluated analytically from the prior and the likelihood. This is a computationally efficient method for calibration but it makes the assumption that all of the calibration sources have the same calibrator noise parameter, $\sigma$. This assumption is problematic as the noise in the calculated calibrator temperature is proportional to $1 / (1 - |\Gamma_\text{s}|^2)$ (from equations \ref{e:X_load} and \ref{e:noise_estimate}) which can theoretically vary between 1 and $\infty$ for the different calibrators, thereby inflating the radiometric noise in the PSD measurement, $P_\text{s}$. We also found a serious issue for all inference methods when using an open or short load as in equation \ref{e:x_matrices} all of the $\bld{X}$ matrices are proportional to $1 / (1 - |\Gamma_\text{s}|^2)$, resulting in a singularity in the likelihood for loads with $|\Gamma_\text{s}| \approx 1$. This was highlighted as an issue in \cite{sutinjoMeasureWellSpreadPoints2020} and \cite{priceMeasuringNoiseParameters2023} who show that it is possible to develop methods to mitigate the issues caused by singularities. In this paper we will use a similar method to \cite{sutinjoMeasureWellSpreadPoints2020} to resolve the issue of singularities due to the $1/ (1 - |\Gamma_\text{s}|^2)$ factor in the $\bld{X}$ matrices by rewriting equation \ref{e:simplified_calibration_eqn} as % \begin{equation} \label{e:gamma_weighted_calibration_eqn} \left(1 - |\Gamma_\text{s}|^2\right) \bld{T}_\text{s} = \left(1 - |\Gamma_\text{s}|^2\right) \bld{X\Theta}, \end{equation} % and writing a new calibration equation, % \begin{equation} \bld{T}'_\text{s} = \bld{X}'\bld{\Theta}, \end{equation} % where primed quantities represent the new singularity-free quantities, ${\bld{T}'_\text{s} = (1 - |\Gamma_\text{s}|^2) \bld{T}_\text{s}}$, which are finite for all $\Gamma_\text{s}$. The polynomial coefficients, $\bld{\Theta}$, are left unchanged in this new formalisation and we can continue to use the conjugate priors method. Note that in order to find the final calibration solution it is still necessary to calculate the calibrator source temperature, % \begin{equation} \bld{T}_\text{s} = \frac{1}{1 - |\Gamma_\text{s}|^2} \bld{T}'_\text{s}. \end{equation} % As a secondary effect we found that this resulted in all of the calibrated source temperatures exhibiting almost equal RMSE noise levels, allowing the conjugate priors method to fit a single $\sigma$ parameter without biasing the final result. As the noise in the system is radiometric the noise will still vary slightly between calibrators due to temperature variance, but to an extent that it does not significantly bias the final result. In practice this is equivalent to weighting the calibrators by the factor $(1-|\Gamma_\text{s}|^2)$ in the likelihood. We want to highlight the similarities of this method to the EDGES Bayesian calibration framework \citep{murrayBayesianCalibrationFramework2022} where the authors found it necessary to downweight their open and short cables but noted that their approach is not self-consistent. We believe that this $\Gamma$-weighting method is a physically motivated way of downweighting cables, as open and short cables will have a weight of $(1 - |\Gamma_\text{s}|^2) \sim 0$ while a matched 50 $\Omega$ load will have a weight of $(1 - |\Gamma_\text{s}|^2) \sim 1$. This may suggest that these the open and short cables are not required to calibrate with this method. \subsection{Marginalised Polynomial Method} \label{s:marg_poly_method} As an alternative to the conjugate priors method we can fit for each calibrator noise parameter separately using a numerical sampler such as \textsc{PolyChord} \citep{handleyPolychordNestedSampling2015, handleyPolychordNextgenerationNested2015}. In \cite{roqueBayesianNoiseWave2021}, the conjugate priors method used a gradient ascent method to determine the polynomial orders with the highest Bayesian evidence -- a method which is unfeasible with a slow numerical fit and can get stuck in local minima. In this work we will outline a new method to determine the optimal NWP polynomial order by sampling the polynomial orders as fit parameters. To do so we first introduce two parameter vectors, % \begin{equation} \bld{n} = \begin{pmatrix}n_\text{unc} & n_\text{cos} & n_\text{sin} & n_\text{NS} & n_\text{L}\end{pmatrix}, \end{equation} % the vector of polynomial orders for each of the noise wave parameters, and % \begin{equation} \bld{\eta} = \begin{pmatrix}\sigma_0& \sigma_1& \dots\end{pmatrix}, \end{equation} % the vector of calibrator noise parameters for each of the twelve calibrators. In this case we are fitting a single monochromatic noise parameter for each calibrator. We can then write the covariance matrix in terms of the calibrator noise parameters as, % \begin{center} \begin{equation} \bld{C} = \mathrm{diag} \begin{pmatrix}\bovermat{N_\nu}{\sigma^2_0& \sigma^2_0& \dots}& \bovermat{N_\nu}{\sigma^2_1& \sigma^2_1& \dots}\end{pmatrix}, \end{equation} \end{center} % where $N_\nu$ is the number of frequency channels in the data. We can now say that the likelihood is a function of the three parameter vectors $\mathcal L (\eta, \bld n, \bld \Theta)$. Similar to approaches taken by EDGES and HERA \citep{simsBayesianPowerSpectrum2017, monsalveResultsEDGESHighband2018, tauscherGlobal21Cm2021, murrayBayesianCalibrationFramework2022}, we can exploit the linearity of the likelihood in equation \ref{e:gaussian_likelihood} and analytically marginalise over the polynomial parameter vector, $\bld \Theta$, in order to speed up the sampling. This also simplifies the likelihood by removing the need for transdimensional sampling methods \citep{heeBayesianModelSelection2016, kroupaKernelMeanNoisemarginalised2023} when sampling the NWP polynomial orders. We do this by setting a Gaussian prior on the polynomial coefficients, % \begin{equation} \label{e:gaussian_prior} \pi(\bld{\Theta}) = \frac{1}{\sqrt{|2\pi \bld{\Sigma}_\pi|}}\exp\left\{-\frac{1}{2} (\bld{\Theta} - \bld{\mu}_\pi)^T \bld{\Sigma}_\pi^{-1} (\bld{\Theta} - \bld{\mu}_\pi) \right\}, \end{equation} % where we have the mean, $\bld{\mu}_\pi$, and the prior covariance matrix, ${\bld{\Sigma}_\pi = \sigma_\pi^2 \bld{I}}$. Using this prior to marginalise over the polynomial parameters we hence get the marginal log likelihood, % %\begin{equation} \label{e:marginal_likelihood} % \mathcal{L}(\bld{\eta},\bld{n}) = \sqrt{\frac{1}{|2\pi \bld{C}||\bld{\Sigma}_\pi||\bld{\Sigma}^{-1}|}} \exp\left\{\frac{1}{2} \bld{\mu}^T \bld{\Sigma}^{-1} \bld{\mu} -\frac{1}{2} \bld{T}^T \bld{C}^{-1} \bld{T}\right\}, %\end{equation} \begin{align} \begin{split}\label{e:marginal_likelihood} \log \mathcal{L}(\bld{\eta},\bld{n}) = &\frac{1}{2} \log\left|\frac{\bld{\Sigma}_P}{\bld{\Sigma}_\pi}\right| - \frac{1}{2} (\bld{\mu}_P - \bld{\mu}_\pi)^T \bld{\Sigma}_\pi^{-1} (\bld{\mu}_P - \bld{\mu}_\pi) \\ &- \frac{1}{2}\log |2\pi\bld{C}| - \frac{1}{2}(\bld{T} - \bld{X}\bld{\mu}_P)^T \bld{C}^{-1}(\bld{T} - \bld{X}\bld{\mu}_P), \end{split} \end{align} % with % \begin{align} \begin{split} \bld{\Sigma}_P^{-1} &= \frac{1}{\sigma_\pi^2} \bld{I} + \sum_i \frac{1}{\sigma_i^2} \bld{X}_\mathrm{i}^T \bld{X}_\mathrm{i}, \\ \bld{\mu}_P &= \frac{1}{\sigma_\pi^2}\bld{\Sigma}_P\bld{\mu}_\pi + \sum_i \frac{1}{\sigma_i^2} \bld{\Sigma}_P \bld{X}_\mathrm{i}^T \bld{T}_\mathrm{s,i}, \end{split} \end{align} % where $i$ denotes the $i$th calibration source. Here, $\bld{C}$ is a function of $\bld{\eta}$ and the sizes of $\bld{\Sigma}_P$ and $\bld{\mu}_P$ depend on $\bld{n}$. Once the posteriors of $\bld \eta$ and $\bld n$ have been sampled with \textsc{PolyChord} we can then find the posterior on the polynomial coefficients as draws from a multivariate normal distribution, % \begin{equation} \bld \Theta \sim \mathcal{N} (\bld \mu_P, \bld \Sigma_P), \end{equation} % as per equation \ref{e:gaussian_prior}. We set an exponential prior, $n \sim \mathrm{Exp}(5)$, on the polynomial orders and a log uniform prior, $\sigma_i \sim \mathrm{LogUniform}(10^{-4},0.1)$ K on the calibrator noise parameters. The width of the Gaussian prior on the polynomial coefficients is set to $\sigma_\pi = 10$ K, with the prior mean for the zeroth polynomial coefficients informed by the least squares solution \citep{roqueReceiverDesignREACH2025}. For this work we set the zeroth order coefficient means to 270 K, 175 K, 40 K, 1120 K and 300 K for $T_\text{unc}$, $T_\text{cos}$, $T_\text{sin}$, $T_\text{NS}$ and $T_\text{L}$ respectively. These priors are motivated based on a visual inspection of a preliminary least squares fit to the noise wave parameters \citep{roqueReceiverDesignREACH2025} and provide a good estimate of the true values. Higher order polynomial coefficients have a prior mean of 0 K. We can also use this method with the $\Gamma$-weighted calibration equation from equation \ref{e:gamma_weighted_calibration_eqn} with no changes to the method. This can be useful if singularities are causing numerical instabilities when fitting. \subsection{Marginalised Polynomial Method with Noise Model} One major benefit of the marginalised polynomial method is that it allows us to use arbitrary models for the noise in our likelihood by modifying the covariance matrix, $\bld C$. In this work we will construct the covariance matrix using the frequency-dependent estimated noise from equation \ref{e:noise_estimate}, defining this new matrix as $\tilde{\bld C}$. Since the noise estimate has a $T_\text{NS}$ term, care must be taken with the likelihood to preserve the linearity of the likelihood necessary to marginalise over the polynomial coefficients. In this work we will assume $T_\text{NS}$ to be a zeroth order polynomial, motivated by inspection of the fits from other polynomial methods. In order to marginalise over the other noise wave parameters, we rearrange equation \ref{e:full_calibration_eqn}, and define the following terms, % \begin{equation} \tilde{T}_\text{s} = T_\text{s} - X_\text{NS}T_\text{NS}, \end{equation} % \begin{equation} \tilde{\bld{X}} = \begin{pmatrix}X_\text{unc} & X_\text{cos} & X_\text{sin} & X_\text{L}\end{pmatrix}, \end{equation} % \begin{equation} \tilde{\bld{\Theta}} = \begin{pmatrix}T_\text{unc} & T_\text{cos} & T_\text{sin} & T_\text{L}\end{pmatrix}^T, \end{equation} % % \begin{equation} \tilde{\bld{n}} = \begin{pmatrix}n_\text{unc} & n_\text{cos} & n_\text{sin} & n_\text{L}\end{pmatrix}. \end{equation} % The likelihood is hence % \begin{equation} \label{e:gaussian_likelihood_noise_model} \mathcal L = \frac{1}{\sqrt{|2\pi \tilde{\bld{C}}|}}\exp\left\{-\frac{1}{2} (\tilde{\bld{T}}_s - \bld{\tilde{X}\tilde{\Theta}})^T \tilde{\bld{C}}^{-1} (\tilde{\bld{T}}_s - \bld{\tilde{X}\tilde{\Theta}}) \right\}, \end{equation} % which can be marginalised as before, resulting in the marginal likelihood $\mathcal{L}(\tilde{\bld{n}}, T_\text{NS})$, as in equation \ref{e:marginal_likelihood}. Note that we no longer sample over the calibrator noise parameters as this has been calculated using our fit value of $T_\text{NS}$ and equation \ref{e:noise_estimate}, thereby reducing the total number of parameters that we must numerically sample over to just five. \subsection{Cable Corrections} The set of REACH calibrators include nine sources which are connected to the LNA with cables. In a real system we would expect the cables to have a different temperature, $T_\text{c}$, to the calibrator source temperature which will introduce losses and reflected power from the cable. To correct for this, we calculate the realised gain of the cable, % \begin{equation} G_\text{c} = \frac{|S_{21}|^2 (1 - |\Gamma_\text{r}|^2)}{|1 - S_{11} \Gamma_\text{r}|^2(1 - |\Gamma_\text{s}|^2)}, \end{equation} % where $S_{11}$ and $S_{21}$ are the forward S-parameters of the cable \citep{monsalveCALIBRATIONEDGESHIGHBAND2017,roqueReceiverDesignREACH2025}. This can then be used to calculate an effective source temperature, % \begin{equation} T^\text{eff}_\text{s} = G_\text{c} T_\text{s} + (1-G_\text{c})T_\text{c}, \end{equation} % which is used to calibrate the source and fit the noise wave parameters. Once the source has been calibrated we can rearrange this expression to get the final source temperature, % \begin{equation} T_\text{s}^\text{final} = \frac{1}{G_\text{c}} (T^\text{fit}_\text{s} + (G_\text{c}-1)T_\text{c}). \end{equation} % For $\Gamma$-weighted calibration methods we must also weight the cable temperature as $T_\text{c}' = (1 - |\Gamma_\text{s}|^2) T_\text{c}$ when correcting $T_\text{s}'$. \section{Method Comparison} \label{s:method_comparison} \subsection{Testing with Simulated Mock Data} \label{s:mock_data} In order to test this method we will generate five sets of mock data intended to simulate the REACH receiver system to varying levels of complexity following the methods used in \cite{sunCalibrationError21centimeter2024}. These mock datasets are produced using lab $S_{11}$ and $S_{21}$ measurements of the LNA and simulated noise parameters \citep{priceNewTechniqueMeasure2023}. The five datasets are as follows: \begin{itemize} \item \textbf{Dataset 1:} The antenna is simulated as a flat 5000 K spectrum with the system gain, $g=1$. The temperatures of the cables are set to the temperature of the sources. \item \textbf{Dataset 2:} The antenna is simulated as a flat 5000 K spectrum with the system gain, $g=1$. The temperatures of the cables are set to 317 K. \item \textbf{Dataset 3:} The antenna is simulated as a flat 5000 K spectrum with a realistic value of the system gain, $g=|S_{21}|^2$. The temperatures of the cables are set to 317 K. \item \textbf{Dataset 4:} The antenna is simulated as a power law spectrum with a realistic value of the system gain, $g=|S_{21}|^2$. The temperatures of the cables are set to the temperature of the sources. \item \textbf{Dataset 5:} The antenna is simulated as a power law spectrum with a realistic value of the system gain, $g=|S_{21}|^2$. The temperatures of the cables are set to 317 K. \end{itemize} In all datasets we do not model the antenna as being connected to a cable. We set the temperature of the hot load to 370 K, the cold load to 320 K and all other sources are randomly selected from a Gaussian distribution centred around 320 K with a standard deviation of 20 K. The power law spectrum in datasets 4 and 5 is generated using the EDGES log-polynomial fit to the galactic foreground power, \citep{bowmanAbsorptionProfileCentred2018} % \begin{align} \begin{split} T_\text{ant} = &1801.4\left(\frac{\nu}{\nu_0}\right)^{-2.5} + 116.7\left(\frac{\nu}{\nu_0}\right)^{-1.5} -289.8\left(\frac{\nu}{\nu_0}\right)^{-0.5} \\&+ 144.8 \left(\frac{\nu}{\nu_0}\right)^{0.5} - 22.9\left(\frac{\nu}{\nu_0}\right)^{1.5}, \end{split} \end{align} % where $\nu_0 = 75$ MHz, generated over the range $110 \leq \nu \leq 140$ MHz to match the band of the lab dataset in section \ref{s:real_data}. We do not consider antenna chromaticity effects, assuming an achromatic antenna gain. For each mock dataset we will then compare the marginalised polynomial method introduced in this paper with the original conjugate priors method \citep{roqueBayesianNoiseWave2021} and the $\Gamma$-weighted conjugate priors method. \subsection{Testing with Lab Data} \label{s:real_data} Finally we will test the method on lab measurements of a real system which is designed to be a simplified version of REACH system. The data was measured with an integration time of 5 minutes and we used the data in the range $110 \leq \nu \leq 140$ MHz. This system is made up of seven sources, listed below. \begin{itemize} \item An ambient 50 $\Omega$ `cold' load \item An ambient 90 $\Omega$ load at the end of a 0.37 m cable \item An ambient 100 $\Omega$ load \item A 50 $\Omega$ heated `hot' load heated to approximately 360 K \item An ambient short load \item 0.3 m and 2 m open-terminated cables \end{itemize} In order to validate the calibration and simulate an antenna we leave the 90 $\Omega$ ambient load out of the calibration and compare the calibrated temperature with the measured temperature of the load. This will be referred to as the `validation source'. In this case all sources are plugged directly into the VNA and LNA, avoiding the need to use switches which will cause signal loss. We do not perform any cable temperature gradient corrections and treat the loads and cables as sources. \subsection{Results and Discussion} \begin{figure*} \centering \includegraphics[width=0.8\linewidth]{figs/rmse_vs_true_bar_graph_v3.pdf} \caption{Summary of the three calibration methods tested on the five mock datasets and the lab dataset. Here we plot the fractional difference between the calculated noise of the calibrated validation source solution and the expected noise of the validation source. Mock datasets highlighted in red indicate datasets which include cable temperature gradients. It can be seen that while the methods perform equally for the mock datasets, we see the methods presented in this paper outperform the existing conjugate priors method on a lab dataset.} \label{fig:rmse_mock_data} \end{figure*} The results of running the four calibration methods on the five mock datasets and the lab dataset can be seen in figure \ref{fig:rmse_mock_data}. We use the fractional error between the RMSE of the residuals of the calibrated validation source temperature and the estimated RMSE value derived from equation \ref{e:noise_estimate} using the true value of $T_\text{NS}$ as a figure of merit. The plot is separated into two, with the datasets which have a flat 5000 K antenna on the left and datasets with a EDGES foreground spectrum on the right. For all mock datasets we see that all four methods perform comparably, giving antenna solutions whose RMSEs are within 25\% of the noise level. In general we see that the flat spectrum datasets have a higher error, since the absolute calibration error scales with the antenna temperature. Since the flat antenna is 5000 K across the frequency range we see this larger error at all frequencies. In comparison, the EDGES foreground datasets which ranges from 650 K to 350 K across the band have a much lower absolute error overall. Since the largest reflection coefficient in the mock datasets is of order $|\Gamma_\text{s}| \sim 0.8$, this is far enough from $1$ that the conjugate priors method does not get significantly affected by the noise biasing issues discussed. This produces calibrated temperatures which are very similar to those of the other methods. However, we find negative values of $T_\text{unc}$ which are unphysical as this is the magnitude of the reflected receiver noise. As this results in a good solution for the antenna, we believe this is a result of degeneracies in the noise wave parameter solutions that the three other methods are able to break, but this requires further investigation. %For all datasets the marginalised polynomial method performs better than the conjugate priors method which we attribute to the reduction in residual structure resulting in a lower RMSE value. Fitting for each calibrator's noise parameter separately allows the noise wave parameters to be fitted more accurately. We also find that since the marginalised polynomial method's polynomial order posteriors peak well below the upper edge of the priors -- as is the case for mock dataset 5 in figure \ref{fig:poly_order_triangle_plot} -- we can be sure we are not overfitting the noise wave parameters. This is mitigated by the conjugate priors method's use of the Bayesian evidence to determine the optimal polynomial orders \citep{hergtBayesianEvidenceTensortoscalar2021}. \begin{figure*} \centering \includegraphics[width=0.8\linewidth]{figs/poly_order_triangle_plot.pdf} \caption{Corner plot of the marginalised polynomial order posterior probabilities for the lab dataset. As the higher orders are disfavoured, we can conclude that the noise wave parameters are not being overfitted by the marginalised polynomial method.} \label{fig:poly_order_triangle_plot} \end{figure*} In contrast to the mock datasets we see that the three new methods presented in this paper greatly outperform the conjugate priors method for the lab dataset. We note that these methods perform better on the lab data than the mock data because shorter integration times result in higher noise on the lab data PSDs, hiding some residual structure. A corner plot showing the posterior distribution of the polynomial orders can be seen in figure \ref{fig:poly_order_triangle_plot}. All of the posteriors peak at low order polynomials, demonstrating that the methods are avoiding overfitting over the noise wave polynomials. The degree to which high order polynomials are disfavoured can be tuned by adjusting the parameter of the exponential prior. \begin{figure*} \centering \includegraphics[width=0.9\linewidth]{figs/TNS_method_comparison_simple_real_2.pdf} \caption{Comparison of the fitted polynomial to $T_\text{NS}$ for each calibration method for the lab dataset. The existing conjugate priors method greatly underestimates the noise source temperature while the methods presented in this paper are closer to the true value. Despite the improvement, the marginalised polynomial methods still slightly underestimate the value of $T_\text{NS}$.} \label{fig:TNS_method_comparison} \end{figure*} We also find that the $\Gamma$-weighted conjugate priors method performs comparably well to the marginalised polynomial method. This is due to the weighting factor normalising the noise in the calibrated temperature reducing the bias resulting from the same calibrator noise level assumption. In addition we find that the weighting factor increases the numerical stability of the method when fitting the lab data as the short load has a reflection coefficient close to unity. The reason the conjugate priors method struggles to calibrate the lab data is that the singularities and noise differences bias the fit to the noise source temperature, $T_\text{NS}$. This can be seen in the functional posteriors for $T_\text{NS}$ in figure \ref{fig:TNS_method_comparison}, where the conjugate priors method has a much lower $T_\text{NS}$ value than the other three methods. As a result, as shown in equation \ref{e:noise_estimate}, the noise on the calibrated temperature is much lower than the expected value. This is a key point we wish to highlight -- the smallest RMSE noise value possible is not the goal of calibration, as if it is smaller than the estimated value then it is likely that you have fitted your noise wave parameters incorrectly. Since in equation \ref{e:full_calibration_eqn} $P_\text{s}$ is multiplied by $T_\text{NS}$, incorrectly fitting the noise source temperature will result in the 21-cm signal amplitude being scaled incorrectly, just as the noise is scaled. For example, an overestimate of $T_\text{NS}$ will result in a global 21-cm signal depth which is larger than reality. In addition, incorrectly fitting the other noise wave parameters will introduce residual structure in the final calibration solution. The three new methods presented in this work recover a more realistic $T_\text{NS}$ value of approximately 1100 K for the lab dataset, however we note that the marginalised polynomial methods still slightly underestimate its value. The noise diode used for the lab measurements has an ENR of 6 dB or 800 K, which when added to the ambient temperature, $T_\text{L}$, of approximately 300 K results in the expected $T_\text{NS}$ of around 1100 K. We cannot verify the other noise wave parameters, $T_\text{unc}$, $T_\text{sin}$, or $T_\text{cos}$ independently but, as discussed, we require $T_\text{unc} > 0$. \begin{figure*} \centering \includegraphics[width=0.9\linewidth]{figs/ant_res_method_comparison_simple_real_2.pdf} \caption{Comparison of the difference between the calibrated validation source spectrum and the measured validation source temperature for each calibration method for the lab dataset. While there is little residual structure in all of the spectra, the absolute calibration is best for the $\Gamma$-weighted conjugate priors method, indicative of the method's ability to constrain $T_\text{NS}$ accurately. The noise level of the residuals here are close to what we expect for the short integration time of the lab dataset.} \label{fig:ant_res_method_comparison} \end{figure*} We can see the residuals on the validation source in figure \ref{fig:ant_res_method_comparison}. While the methods do not show significant structure we can see there is a visible difference in the absolute offset which can be attributed to the inaccurate fits to $T_\text{NS}$ as demonstrated in figure \ref{fig:TNS_method_comparison}. We also note that the noise level of the residuals are close to what we expect, taking into account the short five minute integration. In order to get the noise down to science-ready noise we would require further integration. Finally, we wish to highlight that the marginalised polynomial methods and $\Gamma$-weighted conjugate priors method successfully calibrate lab data to within 5\% of the theoretical noise limit, demonstrating the power of these Bayesian methods. In particular we have shown that these methods are more robust than the previous conjugate priors method, even when faced with loads which introduce singularities into the system of equations. Further improvements could be made by improving the fitting of the noise model used in the marginalised polynomial which, for example, assumes that the S11 measurements are noiseless. \section{Conclusions} \label{s:conclusions} In this paper we introduced three new Bayesian methods for receiver calibration of global 21-cm experiments and compare them with the conjugate priors method introduced in \cite{roqueBayesianNoiseWave2021} using a suite of simulated REACH datasets and a lab measured dataset. We benchmarked all three methods by finding the difference between the measured RMSE noise of the validation source residuals and the estimated value of the noise. The lab dataset tests showed the $\Gamma$-weighted conjugate priors method outperformed the original conjugate priors method owing to the noise normalisation and increased numerical stability due to the mitigation of the effects of singularities. We also find that the marginalised polynomial method introduced in this paper also outperforms the conjugate priors method since the ability to fit separate noise parameters for the calibrators adds extra degrees of freedom and prevents the calibration result from being biased by high-noise sources such as a shorted cable. Furthermore, the marginalised polynomial method introduces the freedom to choose an arbitrary noise model which further improves the fit to the lab dataset when a physically motivated covariance matrix is used. We also highlight that the goal of calibration is not just to have the smallest RMSE -- although lower RMSE does imply less residual structure -- but rather that the RMSE should be as close to the noise estimate as possible. A measured RMSE which is lower than the estimate likely implies that you have incorrectly fitted your noise wave parameters. \section*{Acknowledgements} CJK was supported by Science and Technology Facilities Council grant number ST/V506606/1. WJH was supported by a Royal Society University Research Fellowship. SAKL was supported by the European Research Council and the UKRI. HTJB acknowledges support from the Kavli Institute for Cosmology Cambridge and Kavli Foundation. DJA was supported by Science and Technology Facilities Council grant number ST/X00239X/1. EdLA was supported by Science and Technology Facilities Council grant number ST/V004425/1. We would also like to thank the Kavli Foundation for their support of REACH. The authors thank the Science and Technology Facilities Council grant number EP/Y02916X/1 for supporting the REACH project. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Data Availability} The data that supported the findings of this article will be shared on reasonable request to the corresponding author. %%%%%%%%%%%%%%%%%%%% REFERENCES %%%%%%%%%%%%%%%%%% % The best way to enter references is to use BibTeX: \bibliographystyle{mnras} \bibliography{21cm_Cosmology} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%% APPENDICES %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Don't change these lines \bsp % typesetting comment \label{lastpage} \end{document} % End of mnras_template.tex ``` 4. **Bibliographic Information:** ```bbl \begin{thebibliography}{} \makeatletter \relax \def\mn@urlcharsother{\let\do\@makeother \do\$\do\&\do\#\do\^\do\_\do\%\do\~} \def\mn@doi{\begingroup\mn@urlcharsother \@ifnextchar [ {\mn@doi@} {\mn@doi@[]}} \def\mn@doi@[#1]#2{\def\@tempa{#1}\ifx\@tempa\@empty \href {http://dx.doi.org/#2} {doi:#2}\else \href {http://dx.doi.org/#2} {#1}\fi \endgroup} \def\mn@eprint#1#2{\mn@eprint@#1:#2::\@nil} \def\mn@eprint@arXiv#1{\href {http://arxiv.org/abs/#1} {{\tt arXiv:#1}}} \def\mn@eprint@dblp#1{\href {http://dblp.uni-trier.de/rec/bibtex/#1.xml} {dblp:#1}} \def\mn@eprint@#1:#2:#3:#4\@nil{\def\@tempa {#1}\def\@tempb {#2}\def\@tempc {#3}\ifx \@tempc \@empty \let \@tempc \@tempb \let \@tempb \@tempa \fi \ifx \@tempb \@empty \def\@tempb {arXiv}\fi \@ifundefined {mn@eprint@\@tempb}{\@tempb:\@tempc}{\expandafter \expandafter \csname mn@eprint@\@tempb\endcsname \expandafter{\@tempc}}} \bibitem[\protect\citeauthoryear{Barkana}{Barkana}{2018}]{barkanaPossibleInteractionBaryons2018} Barkana R., 2018, \mn@doi [Nature] {10.1038/nature25791}, 555, 71 \bibitem[\protect\citeauthoryear{Bevins, Fialkov, {de Lera Acedo}, Handley, Singh, Subrahmanyan \& Barkana}{Bevins et~al.}{2022}]{bevinsAstrophysicalConstraintsSARAS2022} Bevins H. T.~J., Fialkov A., {de Lera Acedo} E., Handley W.~J., Singh S., Subrahmanyan R., Barkana R., 2022, \mn@doi [Nat Astron] {10.1038/s41550-022-01825-6}, 6, 1473 \bibitem[\protect\citeauthoryear{Bowman, Rogers \& Hewitt}{Bowman et~al.}{2008}]{bowmanEmpiricalConstraintsGlobal2008} Bowman J.~D., Rogers A. E.~E., Hewitt J.~N., 2008, \mn@doi [The Astrophysical Journal] {10.1086/528675}, 676, 1 \bibitem[\protect\citeauthoryear{Bowman, Rogers, Monsalve, Mozdzen \& Mahesh}{Bowman et~al.}{2018}]{bowmanAbsorptionProfileCentred2018} Bowman J.~D., Rogers A. E.~E., Monsalve R.~A., Mozdzen T.~J., Mahesh N., 2018, \mn@doi [Nature] {10.1038/nature25792}, 555, 67 \bibitem[\protect\citeauthoryear{Cumner et~al.,}{Cumner et~al.}{2022}]{cumnerRadioAntennaDesign2022} Cumner J., et~al., 2022, \mn@doi [J. Astron. Instrum.] {10.1142/S2251171722500015}, 11, 2250001 \bibitem[\protect\citeauthoryear{Cumner, Pieterse, {de Villiers} \& {de Lera Acedo}}{Cumner et~al.}{2024}]{cumnerEffectsAntennaPower2024} Cumner J., Pieterse C., {de Villiers} D., {de Lera Acedo} E., 2024, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stae1475}, 531, 4734 \bibitem[\protect\citeauthoryear{Dicke}{Dicke}{1946}]{dickeMeasurementThermalRadiation1946} Dicke R.~H., 1946, \mn@doi [Review of Scientific Instruments] {10.1063/1.1770483}, 17, 268 \bibitem[\protect\citeauthoryear{Feng \& Holder}{Feng \& Holder}{2018}]{fengEnhancedGlobalSignal2018} Feng C., Holder G., 2018, \mn@doi [ApJL] {10.3847/2041-8213/aac0fe}, 858, L17 \bibitem[\protect\citeauthoryear{Furlanetto, Peng~Oh \& Briggs}{Furlanetto et~al.}{2006}]{furlanettoCosmologyLowFrequencies2006} Furlanetto S.~R., Peng~Oh S., Briggs F.~H., 2006, \mn@doi [Physics Reports] {10.1016/j.physrep.2006.08.002}, 433, 181 \bibitem[\protect\citeauthoryear{Handley, Hobson \& Lasenby}{Handley et~al.}{2015a}]{handleyPolychordNestedSampling2015} Handley W.~J., Hobson M.~P., Lasenby A.~N., 2015a, \mn@doi [Monthly Notices of the Royal Astronomical Society: Letters] {10.1093/mnrasl/slv047}, 450, L61 \bibitem[\protect\citeauthoryear{Handley, Hobson \& Lasenby}{Handley et~al.}{2015b}]{handleyPolychordNextgenerationNested2015} Handley W.~J., Hobson M.~P., Lasenby A.~N., 2015b, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stv1911}, 453, 4384 \bibitem[\protect\citeauthoryear{Hee, Handley, Hobson \& Lasenby}{Hee et~al.}{2016}]{heeBayesianModelSelection2016} Hee S., Handley W.~J., Hobson M.~P., Lasenby A.~N., 2016, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stv2217}, 455, 2461 \bibitem[\protect\citeauthoryear{Hills, Kulkarni, Meerburg \& Puchwein}{Hills et~al.}{2018}]{hillsConcernsModellingEDGES2018} Hills R., Kulkarni G., Meerburg P.~D., Puchwein E., 2018, \mn@doi [Nature] {10.1038/s41586-018-0796-5}, 564, E32 \bibitem[\protect\citeauthoryear{Kirkham, Anstey \& {de Lera Acedo}}{Kirkham et~al.}{2024}]{kirkhamBayesianMethodMitigate2024} Kirkham C.~J., Anstey D.~J., {de Lera Acedo} E., 2024, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stad3725}, 527, 8305 \bibitem[\protect\citeauthoryear{Kroupa, Yallup, Handley \& Hobson}{Kroupa et~al.}{2023}]{kroupaKernelMeanNoisemarginalised2023} Kroupa N., Yallup D., Handley W., Hobson M., 2023, Kernel-, Mean- and Noise-Marginalised {{Gaussian}} Processes for Exoplanet Transits and \${{H}}\_0\$ Inference (\mn@eprint {arXiv} {2311.04153}), \mn@doi{10.48550/arXiv.2311.04153} \bibitem[\protect\citeauthoryear{Meys}{Meys}{1978}]{meysWaveApproachNoise1978} Meys R., 1978, \mn@doi [IEEE Transactions on Microwave Theory and Techniques] {10.1109/TMTT.1978.1129303}, 26, 34 \bibitem[\protect\citeauthoryear{Monsalve, Rogers, Bowman \& Mozdzen}{Monsalve et~al.}{2017}]{monsalveCALIBRATIONEDGESHIGHBAND2017} Monsalve R.~A., Rogers A. E.~E., Bowman J.~D., Mozdzen T.~J., 2017, \mn@doi [ApJ] {10.3847/1538-4357/835/1/49}, 835, 49 \bibitem[\protect\citeauthoryear{Monsalve, Greig, Bowman, Mesinger, Rogers, Mozdzen, Kern \& Mahesh}{Monsalve et~al.}{2018}]{monsalveResultsEDGESHighband2018} Monsalve R.~A., Greig B., Bowman J.~D., Mesinger A., Rogers A. E.~E., Mozdzen T.~J., Kern N.~S., Mahesh N., 2018, \mn@doi [ApJ] {10.3847/1538-4357/aace54}, 863, 11 \bibitem[\protect\citeauthoryear{Monsalve, Fialkov, Bowman, Rogers, Mozdzen, Cohen, Barkana \& Mahesh}{Monsalve et~al.}{2019}]{monsalveResultsEDGESHighBand2019} Monsalve R.~A., Fialkov A., Bowman J.~D., Rogers A. E.~E., Mozdzen T.~J., Cohen A., Barkana R., Mahesh N., 2019, \mn@doi [The Astrophysical Journal] {10.3847/1538-4357/ab07be}, 875, 67 \bibitem[\protect\citeauthoryear{Monsalve et~al.,}{Monsalve et~al.}{2023}]{monsalveMapperIGMSpin2023} Monsalve R.~A., et~al., 2023, Mapper of the {{IGM Spin Temperature}} ({{MIST}}): {{Instrument Overview}} (\mn@eprint {arXiv} {2309.02996}), \mn@doi{10.48550/arXiv.2309.02996} \bibitem[\protect\citeauthoryear{Murray, Bowman, Sims, Mahesh, Rogers, Monsalve, Samson \& Vydula}{Murray et~al.}{2022}]{murrayBayesianCalibrationFramework2022} Murray S.~G., Bowman J.~D., Sims P.~H., Mahesh N., Rogers A. E.~E., Monsalve R.~A., Samson T., Vydula A.~K., 2022, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stac2600}, 517, 2264 \bibitem[\protect\citeauthoryear{Nambissan et~al.,}{Nambissan et~al.}{2021}]{nambissanSARASCDEoR2021} Nambissan T.~J., et~al., 2021, \mn@doi [Exp Astron] {10.1007/s10686-020-09697-2}, 51, 193 \bibitem[\protect\citeauthoryear{Pattison, Anstey \& {de Lera Acedo}}{Pattison et~al.}{2023}]{pattisonModellingHotHorizon2023} Pattison J. H.~N., Anstey D.~J., {de Lera Acedo} E., 2023, Modelling a {{Hot Horizon}} in {{Global}} 21 Cm {{Experimental Foregrounds}}, \mn@doi{10.48550/arXiv.2307.02908} \bibitem[\protect\citeauthoryear{Philip et~al.,}{Philip et~al.}{2019}]{philipProbingRadioIntensity2019} Philip L., et~al., 2019, \mn@doi [J. Astron. Instrum.] {10.1142/S2251171719500041}, 08, 1950004 \bibitem[\protect\citeauthoryear{Price et~al.,}{Price et~al.}{2018}]{priceDesignCharacterizationLargeaperture2018} Price D.~C., et~al., 2018, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/sty1244}, 478, 4193 \bibitem[\protect\citeauthoryear{Price, Tong, Sutinjo, Patra \& Greenhill}{Price et~al.}{2023a}]{priceNewTechniqueMeasure2023} Price D.~C., Tong C.-Y.~E., Sutinjo A.~T., Patra N., Greenhill L.~J., 2023a, A New Technique to Measure Noise Parameters for Global 21-Cm Experiments, \mn@doi{10.48550/arXiv.2305.11479} \bibitem[\protect\citeauthoryear{Price, Tong, Sutinjo, Greenhill \& Patra}{Price et~al.}{2023b}]{priceMeasuringNoiseParameters2023} Price D.~C., Tong C.-Y.~E., Sutinjo A.~T., Greenhill L.~J., Patra N., 2023b, \mn@doi [IEEE Transactions on Microwave Theory Techniques] {10.1109/TMTT.2022.3225317}, 71, 1102 \bibitem[\protect\citeauthoryear{Rogers \& Bowman}{Rogers \& Bowman}{2012}]{rogersAbsoluteCalibrationWideband2012} Rogers A. E.~E., Bowman J.~D., 2012, \mn@doi [Radio Science] {10.1029/2011RS004962}, 47 \bibitem[\protect\citeauthoryear{Roque, Handley \& {Razavi-Ghods}}{Roque et~al.}{2021}]{roqueBayesianNoiseWave2021} Roque I. L.~V., Handley W.~J., {Razavi-Ghods} N., 2021, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stab1453}, 505, 2638 \bibitem[\protect\citeauthoryear{Roque et~al.,}{Roque et~al.}{2025}]{roqueReceiverDesignREACH2025} Roque I. L.~V., et~al., 2025, \mn@doi [Exp Astron] {10.1007/s10686-024-09975-3}, 59, 7 \bibitem[\protect\citeauthoryear{Scheutwinkel, {de Lera Acedo} \& Handley}{Scheutwinkel et~al.}{2022}]{scheutwinkelBayesianEvidencedrivenDiagnosis2022a} Scheutwinkel K.~H., {de Lera Acedo} E., Handley W., 2022, \mn@doi [Publ. Astron. Soc. Aust.] {10.1017/pasa.2022.49}, 39, e052 \bibitem[\protect\citeauthoryear{Shen, Anstey, {de~Lera~Acedo} \& Fialkov}{Shen et~al.}{2022}]{shenBayesianDataAnalysis2022} Shen E., Anstey D., {de~Lera~Acedo} E., Fialkov A., 2022, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stac1900}, 515, 4565 \bibitem[\protect\citeauthoryear{Sims \& Pober}{Sims \& Pober}{2020}]{simsTestingCalibrationSystematics2020} Sims P.~H., Pober J.~C., 2020, \mn@doi [Monthly Notices of the Royal Astronomical Society] {10.1093/mnras/stz3388}, 492, 22 \bibitem[\protect\citeauthoryear{Sims, Lentati, Pober, Carilli, Hobson, Alexander \& Sutter}{Sims et~al.}{2017}]{simsBayesianPowerSpectrum2017} Sims P.~H., Lentati L., Pober J.~C., Carilli C., Hobson M.~P., Alexander P., Sutter P., 2017, Bayesian Power Spectrum Estimation at the {{Epoch}} of {{Reionization}}, \mn@doi{10.48550/arXiv.1701.03384} \bibitem[\protect\citeauthoryear{Singh \& Subrahmanyan}{Singh \& Subrahmanyan}{2019}]{singhRedshifted21Cm2019} Singh S., Subrahmanyan R., 2019, \mn@doi [ApJ] {10.3847/1538-4357/ab2879}, 880, 26 \bibitem[\protect\citeauthoryear{Singh, Subrahmanyan, Shankar, Rao, Girish, Raghunathan, Somashekar \& Srivani}{Singh et~al.}{2018}]{singhSARASSpectralRadiometer2018} Singh S., Subrahmanyan R., Shankar N.~U., Rao M.~S., Girish B.~S., Raghunathan A., Somashekar R., Srivani K.~S., 2018, \mn@doi [Exp Astron] {10.1007/s10686-018-9584-3}, 45, 269 \bibitem[\protect\citeauthoryear{Singh et~al.,}{Singh et~al.}{2022}]{singhDetectionCosmicDawn2022} Singh S., et~al., 2022, \mn@doi [Nat Astron] {10.1038/s41550-022-01610-5}, 6, 607 \bibitem[\protect\citeauthoryear{Sivia}{Sivia}{2006}]{siviaDataAnalysisBayesian2006} Sivia D.~S., 2006, Data Analysis: A {{Bayesian}} Tutorial., 2nd ed. / d.s. sivia with j. skilling. edn. Oxford Science Publications, University Press, Oxford \bibitem[\protect\citeauthoryear{Sun, Acedo, Wu, Yue, Zhu \& Chen}{Sun et~al.}{2024}]{sunCalibrationError21centimeter2024} Sun S., Acedo E. d.~L., Wu F., Yue B., Zhu J., Chen X., 2024, Calibration {{Error}} in 21-Centimeter {{Global Spectrum Experiments}} (\mn@eprint {arXiv} {2405.17742}) \bibitem[\protect\citeauthoryear{Sutinjo, Belostotski, Juswardy \& Ung}{Sutinjo et~al.}{2020}]{sutinjoMeasureWellSpreadPoints2020} Sutinjo A.~T., Belostotski L., Juswardy B., Ung D. X.~C., 2020, \mn@doi [IEEE Transactions on Microwave Theory Techniques] {10.1109/TMTT.2020.2977287}, 68, 1783 \bibitem[\protect\citeauthoryear{Tauscher et~al.,}{Tauscher et~al.}{2021}]{tauscherGlobal21Cm2021} Tauscher K., et~al., 2021, \mn@doi [The Astrophysical Journal] {10.3847/1538-4357/ac00af}, 915, 66 \bibitem[\protect\citeauthoryear{{de Lera Acedo} et~al.,}{{de Lera Acedo} et~al.}{2022}]{deleraacedoREACHRadiometerDetecting2022a} {de Lera Acedo} E., et~al., 2022, \mn@doi [Nat Astron] {10.1038/s41550-022-01709-9}, 6, 984 \makeatother \end{thebibliography} ``` 5. **Author Information:** - Lead Author: {'name': 'Christian J. Kirkham'} - Full Authors List: ```yaml Christian J. Kirkham: {} Will Handley: pi: start: 2020-10-01 thesis: null postdoc: start: 2016-10-01 end: 2020-10-01 thesis: null phd: start: 2012-10-01 end: 2016-09-30 supervisors: - Anthony Lasenby - Mike Hobson thesis: 'Kinetic initial conditions for inflation: theory, observation and methods' original_image: images/originals/will_handley.jpeg image: /assets/group/images/will_handley.jpg links: Webpage: https://willhandley.co.uk Jiacong Zhu: {} Kaan Artuc: {} Ian Roque: phd: start: 2019-10-01 end: 2023-12-31 supervisors: - Will Handley - Nima Razavi-Ghods thesis: 'EXCALIBRATE: Calibration for astrophysical experimentation' mphil: start: 2018-10-01 end: 2019-09-15 supervisors: - Nima Razavi-Ghods - Will Handley thesis: Bayesian Techniques for the Calibration of 21 cm Global Experiments original_image: images/originals/ian_roque.png image: /assets/group/images/ian_roque.jpg links: Webpage: https://www.astro.phy.cam.ac.uk/directory/ian-roque destination: 2024-08-01: Stanford SLAC staff engineer Sam Leeney: phd: start: 2023-10-01 supervisors: - Eloy de Lera Acedo - Harry Bevins - Will Handley thesis: null mphil: start: 2022-04-11 end: 2022-12-30 supervisors: - Eloy de Lera Acedo thesis: 'Data science in early universe Cosmology: a novel Bayesian RFI mitigation approach using numerical sampling techniques' original_image: images/originals/sam_leeney.jpeg image: /assets/group/images/sam_leeney.jpg links: Webpage: https://github.com/samleeney Group Webpage: https://www.cavendishradiocosmology.com/ Harry Bevins: coi: start: 2023-10-01 thesis: null phd: start: 2019-10-01 end: 2023-03-31 supervisors: - Will Handley - Eloy de Lera Acedo - Anastasia Fialkov thesis: A Machine Learning-enhanced Toolbox for Bayesian 21-cm Data Analysis and Constraints on the Astrophysics of the Early Universe original_image: images/originals/harry_bevins.jpeg image: /assets/group/images/harry_bevins.jpg links: Webpage: https://htjb.github.io/ GitHub: https://github.com/htjb ADS: https://ui.adsabs.harvard.edu/search/q=author%3A%22Bevins%2C%20H.%20T.%20J.%22&sort=date%20desc%2C%20bibcode%20desc&p_=0 Publons: https://publons.com/researcher/5239833/harry-bevins/ destination: 2023-04-01: Postdoc in Cambridge (Eloy) 2023-10-01: Cambridge Kavli Fellowship Dominic Anstey: phd: start: 2018-10-01 end: 2022-09-30 supervisors: - Eloy de Lera Acedo - Will Handley thesis: 'Data Analysis in Global 21cm Experiments: Physically Motivated Bayesian Modelling Techniques' original_image: images/originals/dominic_anstey.jpg image: /assets/group/images/dominic_anstey.jpg destination: 2022-10-01: Postdoc at Cambridge (UK) Eloy de Lera Acedo: coi: start: 2018-10-01 thesis: null image: https://www.astro.phy.cam.ac.uk/sites/default/files/styles/inline/public/images/profile/headshotlow.jpg?itok=RMrJ4zTa links: Department webpage: https://www.phy.cam.ac.uk/directory/dr-eloy-de-lera-acedo ``` This YAML file provides a concise snapshot of an academic research group. It lists members by name along with their academic roles—ranging from Part III and summer projects to MPhil, PhD, and postdoctoral positions—with corresponding dates, thesis topics, and supervisor details. Supplementary metadata includes image paths and links to personal or departmental webpages. A dedicated "coi" section profiles senior researchers, highlighting the group’s collaborative mentoring network and career trajectories in cosmology, astrophysics, and Bayesian data analysis. ==================================================================================== Final Output Instructions ==================================================================================== - Combine all data sources to create a seamless, engaging narrative. - Follow the exact Markdown output format provided at the top. - Do not include any extra explanation, commentary, or wrapping beyond the specified Markdown. - Validate that every bibliographic reference with a DOI or arXiv identifier is converted into a Markdown link as per the examples. - Validate that every Markdown author link corresponds to a link in the author information block. - Before finalizing, confirm that no LaTeX citation commands or other undesired formatting remain. - Before finalizing, confirm that the link to the paper itself [2412.14023](https://arxiv.org/abs/2412.14023) is featured in the first sentence. Generate only the final Markdown output that meets all these requirements. {% endraw %}