{% 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: "Bayesian approach to radio frequency interference mitigation" date: 2022-11-28 categories: papers --- ![AI generated image](/assets/images/posts/2022-11-28-2211.15448.png) Sam LeeneyWill HandleyEloy de Lera Acedo Content generated by [gemini-1.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2022-11-28-2211.15448.txt). Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2022-11-28-2211.15448.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': 'S. A. K. Leeney'}). 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 [2211.15448](https://arxiv.org/abs/2211.15448) 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 is dedicated to advancing our understanding of the Universe through the development and application of cutting-edge artificial intelligence and Bayesian statistical inference methods. Our research spans a wide range of cosmological topics, from the very first moments of the Universe to the nature of dark matter and dark energy, with a particular focus on analyzing complex datasets from next-generation surveys. ## Research Focus Our core research revolves around developing innovative methodologies for analyzing large-scale cosmological datasets. We specialize in Simulation-Based Inference (SBI), a powerful technique that leverages our ability to simulate realistic universes to perform robust parameter inference and model comparison, even when likelihood functions are intractable ([LSBI framework](https://arxiv.org/abs/2501.03921)). This focus allows us to tackle complex astrophysical and instrumental systematics that are challenging to model analytically ([Foreground map errors](https://arxiv.org/abs/2211.10448)). A key aspect of our work is the development of next-generation SBI tools ([Gradient-guided Nested Sampling](https://arxiv.org/abs/2312.03911)), particularly those based on neural ratio estimation. These methods offer significant advantages in efficiency and scalability for high-dimensional inference problems ([NRE-based SBI](https://arxiv.org/abs/2207.11457)). We are also pioneering the application of these methods to the analysis of Cosmic Microwave Background ([CMB](https://arxiv.org/abs/1908.00906)) data, Baryon Acoustic Oscillations ([BAO](https://arxiv.org/abs/1701.08165)) from surveys like DESI and 4MOST, and gravitational wave observations. Our AI initiatives extend beyond standard density estimation to encompass a broader range of machine learning techniques, such as: * **Emulator Development:** We develop fast and accurate emulators of complex astrophysical signals ([globalemu](https://arxiv.org/abs/2104.04336)) for efficient parameter exploration and model comparison ([Neural network emulators](https://arxiv.org/abs/2503.13263)). * **Bayesian Neural Networks:** We explore the full posterior distribution of Bayesian neural networks for improved generalization and interpretability ([BNN marginalisation](https://arxiv.org/abs/2205.11151)). * **Automated Model Building:** We are developing novel techniques to automate the process of building and testing theoretical cosmological models using a combination of symbolic computation and machine learning ([Automated model building](https://arxiv.org/abs/2006.03581)). Additionally, we are active in the development and application of advanced sampling methods like nested sampling ([Nested sampling review](https://arxiv.org/abs/2205.15570)), including dynamic nested sampling ([Dynamic nested sampling](https://arxiv.org/abs/1704.03459)) and its acceleration through techniques like posterior repartitioning ([Accelerated nested sampling](https://arxiv.org/abs/2411.17663)). ## Highlight Achievements Our group has a strong publication record in high-impact journals and on the arXiv preprint server. Some key highlights include: * Development of novel AI-driven methods for analyzing the 21-cm signal from the Cosmic Dawn ([21-cm analysis](https://arxiv.org/abs/2201.11531)). * Contributing to the Planck Collaboration's analysis of CMB data ([Planck 2018](https://arxiv.org/abs/1807.06205)). * Development of the PolyChord nested sampling software ([PolyChord](https://arxiv.org/abs/1506.00171)), which is now widely used in cosmological analyses. * Contributions to the GAMBIT global fitting framework ([GAMBIT CosmoBit](https://arxiv.org/abs/2009.03286)). * Applying SBI to constrain dark matter models ([Dirac Dark Matter EFTs](https://arxiv.org/abs/2106.02056)). ## Future Directions We are committed to pushing the boundaries of cosmological analysis through our ongoing and future projects, including: * Applying SBI to test extensions of General Relativity ([Modified Gravity](https://arxiv.org/abs/2006.03581)). * Developing AI-driven tools for efficient and robust calibration of cosmological experiments ([Calibration for astrophysical experimentation](https://arxiv.org/abs/2307.00099)). * Exploring the use of transformers and large language models for automating the process of cosmological model building. * Applying our expertise to the analysis of data from next-generation surveys like Euclid, the Vera Rubin Observatory, and the Square Kilometre Array. This will allow us to probe the nature of dark energy with increased precision ([Dynamical Dark Energy](https://arxiv.org/abs/2503.08658)), search for parity violation in the large-scale structure ([Parity Violation](https://arxiv.org/abs/2410.16030)), and explore a variety of other fundamental questions. Content generated by [gemini-1.5-pro](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/2211.15448v4 guidislink: true link: http://arxiv.org/abs/2211.15448v4 updated: '2024-02-19T12:09:32Z' updated_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2024 - 2 - 19 - 12 - 9 - 32 - 0 - 50 - 0 - tm_zone: null tm_gmtoff: null published: '2022-11-28T15:43:25Z' published_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2022 - 11 - 28 - 15 - 43 - 25 - 0 - 332 - 0 - tm_zone: null tm_gmtoff: null title: Bayesian approach to radio frequency interference mitigation title_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: Bayesian approach to radio frequency interference mitigation summary: 'Interfering signals such as Radio Frequency Interference from ubiquitous satellite constellations are becoming an endemic problem in fields involving physical observations of the electromagnetic spectrum. To address this we propose a novel data cleaning methodology. Contamination is simultaneously flagged and managed at the likelihood level. It is modeled in a Bayesian fashion through a piecewise likelihood that is constrained by a Bernoulli prior distribution. The techniques described in this paper can be implemented with just a few lines of code.' summary_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: 'Interfering signals such as Radio Frequency Interference from ubiquitous satellite constellations are becoming an endemic problem in fields involving physical observations of the electromagnetic spectrum. To address this we propose a novel data cleaning methodology. Contamination is simultaneously flagged and managed at the likelihood level. It is modeled in a Bayesian fashion through a piecewise likelihood that is constrained by a Bernoulli prior distribution. The techniques described in this paper can be implemented with just a few lines of code.' authors: - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: S. A. K. Leeney - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: W. J. Handley - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: E. de Lera Acedo author_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: name: E. de Lera Acedo author: E. de Lera Acedo arxiv_doi: 10.1103/PhysRevD.108.062006 links: - !!python/object/new:feedparser.util.FeedParserDict dictitems: title: doi href: http://dx.doi.org/10.1103/PhysRevD.108.062006 rel: related type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: http://arxiv.org/abs/2211.15448v4 rel: alternate type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: title: pdf href: http://arxiv.org/pdf/2211.15448v4 rel: related type: application/pdf arxiv_comment: 6 pages, 4 figures arxiv_journal_ref: Phys. Rev. D 108, 062006, Published 29 September 2023 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 % ****** Start of file apssamp.tex ****** % % This file is part of the APS files in the REVTeX 4.2 distribution. % Version 4.2a of REVTeX, December 2014 % % Copyright (c) 2014 The American Physical Society. % % See the REVTeX 4 README file for restrictions and more information. % % TeX'ing this file requires that you have AMS-LaTeX 2.0 installed % as well as the rest of the prerequisites for REVTeX 4.2 % % See the REVTeX 4 README file % It also requires running BibTeX. The commands are as follows: % % 1) latex apssamp.tex % 2) bibtex apssamp % 3) latex apssamp.tex % 4) latex apssamp.tex % \documentclass[% reprint, %superscriptaddress, %groupedaddress, %unsortedaddress, %runinaddress, %frontmatterverose, %preprint, %preprintnumbers, %nofootinbib, %nobibnotes, %bibnotes, amsmath,amssymb, aps, %prl, %pra, prb, %rmp, %prstab, %prstper, %floatfix, ]{revtex4-1} \usepackage{graphicx}% Include figure files \usepackage{dcolumn}% Align table columns on decimal point \usepackage{bm}% bold math \usepackage{hyperref}% add hypertext capabilities \usepackage[capitalise]{cleveref} \usepackage{nameref} \newcommand{\namerefit}[1]{\textit{\nameref{#1}}} %\usepackage[mathlines]{lineno}% Enable numbering of text and display math %\linenumbers\relax % Commence numbering lines %\usepackage[showframe,%Uncomment any one of the following lines to test %%scale=0.7, marginratio={1:1, 2:3}, ignoreall,% default settings %%text={7in,10in},centering, %%margin=1.5in, %%total={6.5in,8.75in}, top=1.2in, left=0.9in, includefoot, %%height=10in,a5paper,hmargin={3cm,0.8in}, %]{geometry} \begin{document} \preprint{APS/123-QED} \title{A Bayesian approach to RFI mitigation}% Force line breaks with \\ \author{S. A. K Leeney} \altaffiliation[Also at ]{Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK}%Lines break automatically or can be forced with \\ \email{sakl2@cam.ac.uk} \author{W. J Handley }% \email{wh260@cam.ac.uk} \altaffiliation[Also at ]{Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK}%Lines break automatically or can be forced with \\ \author{E. de Lera Acedo}% \email{ed330@cam.ac.uk} \altaffiliation[Also at ]{Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK}%Lines break automatically or can be forced with \\ \affiliation{% The University of Cambridge, Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK} %\\ % This line break forced with \textbackslash\textbackslash \date{\today}% It is always \today, today, % but any date may be explicitly specified \begin{abstract} Interfering signals such as Radio Frequency Interference from ubiquitous satellite constellations are becoming an endemic problem in fields involving physical observations of the electromagnetic spectrum. To address this we propose a novel data cleaning methodology. Contamination is simultaneously flagged and managed at the likelihood level. It is modeled in a Bayesian fashion through a piecewise likelihood that is constrained by a Bernoulli prior distribution. The techniques described in this paper can be implemented with just a few lines of code. \end{abstract} %\keywords{Suggested keywords}%Use showkeys class option if keyword %display desired \maketitle %\tableofcontents \section{Introduction} Satellite constellations in low earth orbit such as SpaceX's Starlink will likely number 100,000 by 2030~\cite{venkatesan2020impact}. Described in a recent Nature article as `horrifying'~\cite{witze2022satellite}, the impact of Radio Frequency Interference (RFI) created by these `megaconstellations' on Astronomy is of significant concern. Interfering signals like RFI are problematic because they cause information in contaminated frequency channels to be lost which can lead to significant systematic error if not properly modelled. The frequency bands that these satellites and other modern telecommunications devices operate in are not protected or reserved for Astronomy. They clash with current and next generation experiments such as the ngVLA~\cite{mckinnon2019ngvla} and the SKA~\cite{bourke2015advancing}, which plan probe yet unseen epochs of the universe. Furthermore, the latest telescopes operate in large regions of the sky and across wide frequency bands, making it impossible to avoid these satellite swarms. This is pushing Astronomy to remote corners of earth and beyond, to space~\cite{sabelhaus2004overview}. However, moving projects to space is not a long term solution. There is only a short time window until the RFI-quiet dark side of the moon becomes contaminated by satellites and other devices required by projects such as LuSEE\cite{giardino2019impact} and the LCRT~\cite{bandyopadhyay2021lunar} which themselves are hoping to exploit the clear lunar skies. Furthermore space itself is filled with cosmic rays, which lead to RFI-like interference and thus have been problematic for the JWST's near infrared spectrometer~\cite{giardino2019impact}. Consequentially, RFI is becoming the fundamental bottleneck in modern Astronomy and beyond; with modern observations being at the forefront of many fields of fundamental Physics, such as dark matter detection. New statistical techniques are therefore urgently required to address this rapidly growing problem. Furthermore, projects such as the Event Horizon Telescope~\cite{event2019first}, Planck~\cite{ade2014planck} and BICEP~\cite{ade2021bicep} all used Bayesian statistics in their data analysis pipelines, but there is currently no way to manage RFI in a Bayesian sense forcing astronomers to patch traditional RFI mitigation algorithms into their Bayesian systems. Modern projects implement Singular Value Decomposition~\cite{offringa2010post} watershed segmentation~\cite{kerrigan2019optimizing}, Deep Learning methods~\cite{akeret2017radio};~\cite{vafaei2020deep};~\cite{sun2022robust} and more recently Gibbs sampling~\cite{2022arXiv221105088K}. Scientists are constantly searching for more elusive and faint signals, thus data cleaning is becoming increasingly important. The Laser Interferometer Gravitational-wave Observatory (LIGO)~\cite{abbott2009ligo}, for example, is sensitive enough to detect a car starting miles away and thus is extremely sensitive to data corruption~\cite{ormiston2020noise}. The next generation aLIGO~\cite{harry2010advanced} will be even more so. As such there is a need for novel data cleaning methodologies beyond fields involving measurements of the electromagnetic spectrum. In this paper we propose a data cleaning methodology that takes a Bayesian approach, where contamination is both flagged and managed at the likelihood level. Detailed benchmarking of this technique will be provided when it is applied on data from a next generation low frequency global experiment, REACH~\cite{de2022reach}, which will see first light this year. A usage example of the methodologies described in this work can be found at \href{https://github.com/samleeney/Publications}{github.com/samleeney/publications}. \section{Theory} \subsection{Bayesian Inference} \label{sec:bayesianinftheory} Bayesian methods can be used to perform parameter estimates and model comparison. A model $\mathcal{M}$ uses data $\mathcal{D}$ to infer its free parameters $\theta$. Using Bayes Theorem, \begin{align} P(\mathcal{D}|\theta) \times P(\theta) &= P(\theta|\mathcal{D}) \times P(\mathcal{D}), \\ \mathcal{L} \times \pi &= \mathcal{P} \times \mathcal{Z}, \end{align} the prior $\pi$ is updated onto the posterior $\mathcal{P}$ in light of the likelihood $\mathcal{L}$ and furthermore the Bayesian Evidence $\mathcal{Z}$ can be inferred by computing the integral \begin{equation} \mathcal{Z} = \int \mathcal{L}(\theta) \times \pi(\theta) \; d\theta. \end{equation} In practice, $\mathcal{P}$ and $\mathcal{Z}$ can be determined simultaneously using a Bayesian numerical solver. We use the Nested Sampling algorithm \textsc{polychord}~\cite{handley2015polychord}; where a series of points generated within $\pi$ are updated such that they sequentially contract around the peak(s) of the likelihood, forming the posterior which can be used to generate parameter estimates. The artefacts of this process can then be used to compute $\mathcal{Z}$, which is used for model comparison. For a more detailed description of Bayesian Inference and Nested Sampling see~\cite{mackay2003information}~\cite{sivia2006data}. \subsection{Data cleaning likelihood}\label{sec:rficorrtheory} A sufficiently contaminated data point can be considered corrupted. Any information relevant to the model is lost and furthermore it cannot be modelled as Gaussian noise. Assuming $\mathcal{D}$ is uncorrelated, the likelihood \begin{equation} \begin{aligned} \mathcal{L} &= P(\mathcal{D}|\theta) = \prod_{i} \mathcal{L}_{i}(\theta),\label{l1} = \prod_{i} P(\mathcal{D}_i|\theta) \end{aligned} \end{equation} where $i$ represents the $i$'th data point, is insufficient to model such contaminated data. It is therefore necessary to model the likelihood that each data point is corrupted. Thus, we introduce a piecewise likelihood including the possibility of corruption of data \begin{equation} P(\mathcal{D}_i|\theta) = \begin{cases} \mathcal{L}_i(\theta) &: \text{uncorrupted}\\ \Delta^{-1}[ 0<\mathcal{D}_i<\Delta] &: \text{corrupted}.\\ \end{cases} \end{equation} Corruption is modelled as the data becoming completely unreliable and therefore being distributed uniformly within some range $\Delta$ (which, as a scale of corruption, has the same dimensions as the data). An efficient way to write this likelihood is \begin{equation} P(\mathcal{D}|\theta, \varepsilon) = \prod_{i} \mathcal{L}_{i}^{\varepsilon_{i}} \Delta^{\varepsilon_i-1} \label{eq:li2} \end{equation} where the Boolean mask vector $\varepsilon$ has a $i$th component which takes the value $1$ if the datum $i$ is uncorrupted and value $0$ if corrupted. We do not know before the data arrive whether or not they are corrupted. We may infer this in a Bayesian fashion, by ascribing a Bernoulli prior probability $p_i$ of corruption (which has dimensions of probability) i.e: \begin{equation} P(\varepsilon_i) = p_i^{(1-\varepsilon_i)}(1-p_i)^{\varepsilon_i}.\label{eq:pei} \end{equation} Both $\Delta$ and $p_i$ are required for a dimensionally consistent analysis. It should be noted that above we assume the a-priori probability that each bin is corrupted is uncorrelated, i.e $P(\varepsilon)=\prod _i P(\varepsilon_i)$, which in practice will almost certainly not be true. We will discuss later the extent to which this assumption can be considered valid. Multiplying \cref{eq:pei,eq:li2} yields \begin{equation} P(\mathcal{D},\varepsilon|\theta) = \prod_{i} \left[\mathcal{L}_{i}(1-p_i)\right]^{\varepsilon_{i}} \left[p_i/\Delta\right]^{(1-\varepsilon_i)} \label{eqn:likelihood_eps} \end{equation} and to recover a likelihood independent of $\varepsilon$ we formally can marginalise out: \begin{align} P(\mathcal{D}|\theta) &=\sum_{\varepsilon \in \{ 0, 1 \} ^N}P(\mathcal{D},\varepsilon|\theta) \\ &= \sum_{\varepsilon \in \{ 0, 1 \} ^N} \prod_{i} \left[\mathcal{L}_{i}(1-p_i)\right]^{\varepsilon_{i}} \left[p_i/\Delta_i\right]^{(1-\varepsilon_i)}. \label{eq:pdtheta2} \end{align} This would require the computation of the all $2^N$ terms in \cref{eq:pdtheta2}. For realistic values of $N$, this computation becomes impractical. However, if it is assumed that the most likely model (i.e the maximum term in \cref{eq:pdtheta2}) dominates over the next to leading order terms, we can make the approximation \begin{equation} P(\mathcal{D},\varepsilon|\theta) \approx \delta_{\varepsilon \varepsilon^\mathrm{max}} \times P(\mathcal{D},\varepsilon^{\mathrm{max}}|\theta)\label{eqn:postierioreps} \end{equation} where $\delta_{ij}$ is the usual Kroneker delta function, and $\varepsilon^\mathrm{max}$ is the mask vector which maximises the likelihood $P(D,\varepsilon|\theta)$, namely: \begin{equation} \varepsilon^{\mathrm{max}}_{i}= \begin{cases} 1, & \mathcal{L}_i(1-p_i) > p_i/\Delta_i \\ 0, & \text{otherwise}. \end{cases} \label{eqn:emax} \end{equation} Under this approximation we find that the sum in \cref{eq:pdtheta2} becomes \begin{equation} P(\mathcal{D}|\theta) \approx P(\mathcal{D},\varepsilon^{\mathrm{max}}|\theta).\label{eq:approx} \end{equation} In practice the approximation in \cref{eq:approx} is only valid if the next to leading order term is much smaller, such that \begin{equation} P(\mathcal{D}|\theta, \varepsilon_{\mathrm{max}}) \gg \mathrm{max}_j P(\mathcal{D}|\theta,\varepsilon^{(j)})\label{eq:nlo}, \end{equation} where $\varepsilon^{(j)}$ is $\varepsilon^\mathrm{max}$ with its $j$th bit flipped: \begin{equation} \varepsilon^{(j)}_k = \begin{cases} 1-\varepsilon^{\mathrm{max}}_k & k=j \\ \varepsilon^{\mathrm{max}}_k & k\ne j \\ \end{cases} \end{equation} and we may use \cref{eq:nlo} as a consistency check. To summarise, we can correct for contamination under these approximations by replacing the original likelihood $\mathcal{L} = \prod_i\mathcal{L}_i$ in \cref{l1} with \begin{equation} P(\mathcal{D}|\theta) = \prod_i\left[\mathcal{L}_{i}(1-p_i)\right]^{\varepsilon^{\mathrm{max}}_{i}} \left[p_i/\Delta\right]^{(1-\varepsilon^\mathrm{max}_i)} \label{eqn:likelihood} \end{equation} where $\varepsilon^{\mathrm{max}}$ is defined by \cref{eqn:emax}. \subsection{Computing the posterior} The posterior and evidence are computed from \cref{eqn:likelihood} via Nested Sampling (although any numerical Bayesian sampling method could be used). Taking logs for convenience gives \begin{equation} \begin{aligned} \log{P(\mathcal{D}|\theta)} = \sum_{i} &[\log{\mathcal{L}_i}+\log({1-p_i})]\varepsilon^{\mathrm{max}}\\ &+ [\log{p}_i - \log{\Delta}](1 - \varepsilon^\mathrm{max}_i), \label{eq:loglikelihood} \end{aligned} \end{equation} yielding a masked chi squared like term which can be used to distinguish whether there is a statistically significant difference between the classes of data, i.e corrupted or non corrupted. Furthermore, the second term in \cref{eq:loglikelihood} introduces an Occam penalty. Each time a data point is predicted to be contaminated its likelihood is replaced with the penalty rather than being completely removed. Without this term, the likelihood where all data points are flagged would be larger and thus `more likely' than all other possibilities. Therefore, flagging all datum would always be preferable. We compute this by imposing the condition in \cref{eqn:emax} on \cref{eq:loglikelihood} as follows, \begin{equation} \log{P(\mathcal{D}|\theta)} = \begin{cases} \log \mathcal{L}_i + \log (1-p_i), & \begin{aligned} &[\log{\mathcal{L}_i} + \log({1-p_i}) \\ &> \log p_i - \log \Delta] \end{aligned}\\ \log p_i - \log \Delta, & \text{otherwise}. \end{cases} \label{eqn:loglcompute} \end{equation} The corrected likelihood is then updated iteratively via the selected Bayesian sampling method, compressing the prior onto the posterior while simultaneously correcting for conamination. One may also notice that the condition $\log \mathcal{L}_i + \log(1-p_i) > \log p_i - \log \Delta$ in~\cref{eqn:loglcompute} relates to a Logit function, such that \begin{equation} \log \mathcal{L}_i + \log \Delta > \textsc{logit(p)}. \end{equation} Logit functions are used routinely as an activation function in binary classification tasks, hinting at the potential of a future extension of this work using machine learning. \section{Toy example}\label{sec:toyex} We will initially test this approach on a simple toy model with a basic contaminant signal injected. We then move onto a more realistic and complex case in \cref{sec:reachex}. \subsection{Initial Testing}\label{sec:initialsetup} Two simple datasets are generated for comparison consisting of a line with $m=1$ and $c=1$ with $\sigma=5$ order Gaussian noise. One is contaminated by an RFI like signal and the other (the ground truth) is not. They are fit in a Bayesian sense, attempting to recover the two free parameters $m$ and $c$ using the correcting likelihood in \cref{eqn:loglcompute} with \begin{equation} \mathcal{L}_i = -\frac{\log(2\pi \sigma^{2})}{2} - \frac{[y_{i} - y_{\mathcal{S}}(x_i;m, c)]^2}{2\sigma^2}, \end{equation} where $\theta = m, c, \sigma$, $y_i$ is the simulated data and $y_\mathcal{S}(x_i; m, c)$ are the parameter estimates at the $i$'th sampling iteration are used to compute the model $y_i=mx_i + c$. We set $\Delta = \mathcal{D}_\textsc{max}$, to encapsulate the full range of possible data values. $\Delta$ could likely be fit as a free parameter, as will be investigated further in future works. Evaluating the posterior on $\varepsilon$ we can assess how frequently across the entire sampling run each datum was believed to fit (non corrupted) or not fit (corrupted) the model. Contaminated points should make up a near zero fraction of the posterior. Conversely, points that are not contaminated would often fit the model and as such contribute significantly to the final posterior distribution. There can also be some points that lie somewhere in between, which the model is less confident are uncontaminated. This may occur with datum that deviate the most from the true signal due to higher order Gaussian noise, for example. It should be emphasised that although $\varepsilon_i$ is constrained to binary values, the subsequent mask on $\varepsilon$ is not. Unlike traditional RFI flagging algorithms, points are not classified in a binary manor. The mask takes the weighted mean across the posterior. Thus, points more likely to contain RFI will have less `impact' on the final posterior distribution than points believed to be uncontaminated. The mask could be thought of as being slightly opaque to these data points, accounting for the models uncertainty. Incorporating the models confidence in its correction directly into the subsequent parameter estimates makes this approach unique in comparison with its counterparts. \subsection{Model Evaluation} The two aforementioned data sets are evaluated when fit using the likelihood capable of correcting for contamination and also when using a standard likelihood, which cannot natively account for contamination. This generates a total of four posterior distributions for comparison. Of these, all but the contaminated, uncorrected case would be expected to perform similarly if RFI has been effectively mitigated. From a Bayesian standpoint, the simplest model will always be preferable. Thus, for the clean dataset it would be expected that the standard likelihood would be preferred slightly over the correcting likelihood. Fig.~\ref{fig:anesthetic} shows the parameter distributions inferred from the data. The results are as expected; parameters in all but the `RFI No correction' are inferred to within $1\sigma$ of their true value. As seen in~\cref{fig:fgx}, the model that does not correct for RFI is slightly preferred for uncontaminated data. Conversely the correcting model is strongly preferred on the contaminated data indicating that the correction is working as predicted. \begin{figure*} \includegraphics[width=\textwidth]{f_4pane_samples2.pdf} \caption{Showing the parameter distributions inferred from the dataset described in~\cref{sec:initialsetup}. The top left to bottom right panes show probability distribution functions for $m$, $c$ and $\sigma$, respectively. Plots generated using posterior plotting tool \textsc{anesthetic}~\protect\cite{anesthetic}.} \label{fig:anesthetic} \end{figure*} \begin{figure} \includegraphics[width=\columnwidth]{f_4pane_toy_sidebar.pdf} \caption{Showing the inferred parameter estimates in a contour plot, where darker tones indicate higher $\sigma$ confidence in the parameter estimates. Generated from the dataset described in Section~\ref{sec:initialsetup}. The Bayes factor is $-1.4\pm0.3$ for the no RFI case and $25.0\pm0.4$ for the RFI case. The plots are generated using the functional posterior plotter \textsc{fgivenx}~\protect\cite{fgivenx}.} \label{fig:fgx} \end{figure} Viewing the posterior plots of $P(y|x, \mathcal{D})$ in \cref{fig:fgx} it is clear that when RFI is not corrected, the true parameter values are outside the $1\sigma$ and sometimes $2\sigma$ confidence bounds. Conversely the other three cases fit almost entirely within the $1\sigma$ bounds, indicating that the RFI has been mitigated. \subsection{Evaluating the $\log p$ dependence}\label{sec:logpdependence} Proper selection of the probability thresholding term $\log p$ is essential. From a Bayesian standpoint it should be set to represent our prior degree of belief in there being RFI in each datum. We assess the $\log p$ dependence while varying $\log p$ as a function of the RMSE on the fit generated from the parameter estimates, the $\log$ Bayesian Evidence and the mean number of points flagged across all samples. \begin{figure} \centering \includegraphics[width=\columnwidth]{f_approx_current_sig5_2.pdf} \caption{Assessing how various methods of model evaluation vary as function of $\log p$. From top to bottom: the RMSE, the $\log$ of the Bayesian Evidence, then the weighted average number of points flagged and finally the radio between $P_{\textsc{max}}$ and $P_{\textsc{NLO}}$. All dependant variables (excluding $\log \mathcal{Z}$) are averaged over the weighted posterior samples. The noise observable in these plots is sampling noise; the noise in the simulated data is seeded.} \label{fig:4pane} \end{figure} For high $\log p$, the RMSE is high and we observe in \cref{fig:4pane} that the model generates less accurate parameter estimates. Here the threshold is so high that the model is more confident that any of the points are RFI than non RFI. This matches the corresponding low evidence. The RMSE drops as $\log p$ decreases to near its minimum. The model incorrectly flags $\approx 5$ data points while the RMSE is low, showing that the model is able to generate accurate parameter estimates whilst over flagging, indicating it is insensitive to false positives. As $\log p$ decreases further, the model is better able to distinguish between higher order Gaussian noise and as such the average number of points predicted to be RFI approaches the true value. As this happens the evidence also reaches its maximum, which indicates that the Bayesian Evidence is appropriately showing how well each of the many models created by different $\log p$ values fit the data. \subsection{To what extent is $P(\mathcal{D}|\theta) \approx P(\mathcal{D}|\theta, \varepsilon_{\mathrm{max}})$ valid?} A key assumption is made in \cref{eq:approx} is that the leading order term, \cref{eq:loglikelihood}, is considerably larger than all the other possible terms for $\varepsilon \in (0, 1)^N$. It is necessary to test the validity of this approximation by computing \cref{eqn:loglcompute} and comparing the result ($P_{\textsc{max}}$) with the next leading order term ($P_{\textsc{NLO}}$) as calculated by \cref{eq:nlo}. For $-5 < \log p < -0.1$. These results are displayed in the bottom pane of \cref{fig:4pane}. $P_{\textsc{max}}$ is 18 times larger than $P_{\textsc{NLO}}$ at peak $\log \mathcal{Z}$ and increases linearly for $\log p$ below this. Depending on the $\log p$ selection strategy, $P_{\textsc{max}}$ is at least 11 times more likely than the next leading order term. Assuming an appropriate $\log p$ selection strategy, the ratio would be higher, indicating that $P(\mathcal{D}|\theta) \approx P(\mathcal{D}|\theta \varepsilon_{\mathrm{max}})$ is valid. \subsection{Selection Strategy for $\log p$} Various selection strategies could be taken to select the optimal $\log p$ value. For each $\log p$, the model changes. As such, selecting the $\log p$ that maximises the evidence seems to be the most obvious selection strategy. In the case of the toy model, the peak $\log \mathcal{Z}$ occurs where $\log p = -2.7$. Here, $P_{\textsc{max}}$ is 18 times larger than $P_{\textsc{NLO}}$. Another possible strategy could be to select $\log p$ where the number of points flagged is at its minimum. It is also possible to ascribe a prior to $\log p$, fitting it as a free parameter thus fully automating the approach. This will be examined further in future works. \section{REACH Example}\label{sec:reachex} \begin{figure} \includegraphics[width=\columnwidth]{f_4pane_reach_sidebar.pdf} \caption{Showing the results when the RFI correction is applied on simulated data in the REACH data analysis pipeline. The Bayes factor for the no RFI case is $-0.6\pm0.6$ and the Bayes factor for the RFI case is $9e7\pm0.6$.}\label{fig:reach_dual_plot} \end{figure} Finally, we examine a real use case for this method. The REACH~\cite{de2022reach} radio telescope is designed to detect the 21cm signal from the Cosmic Dawn. We select REACH as a testing ground for our methodology because it operates in the same low frequency ranges as the next generation of Astronomical experiments, such as the nvGLA the SKA and takes a Bayesian approach to data analysis~\cite{anstey2021general}. The 21cm signal is expected to take the shape of an inverted Gaussian, so the model takes the form \begin{equation} f(x) = A \exp{\bigg( -\frac{(x-\mu)^{2}}{2 \sigma^{2}}\bigg)} \end{equation} with center frequency $\mu$, standard deviation $\sigma$ and magnitude $A$ all free parameters. The four cases discussed in \cref{sec:toyex} are then examined, but this time on a simulated sky data set containing a 21cm signal with two RFI spikes injected. The No RFI Correction and ground truth cases are very similar with the simpler (ground truth) case marginally preferred as expected. The RFI Corrected case is again similar with a slightly lower evidence due to the penalties incurred during the corrections. The above is also evident when viewing \cref{fig:reach_dual_plot}. The reconstructed signal is within $2\sigma$ of the true signal for all but the RFI No Correction case. \section{Conclusions}\label{sec:conclusions} In this paper, which serves as a proof-of-concept, we show that contamination can be both cleaned and corrected in a fully Bayesian sense at the likelihood level. We demonstrate our general approach in the context of signal processing for Astronomy, but these methods will likely be useful beyond. Forthcoming results from current state of the art low frequency Astronomy experiments~\cite{de2019reach} (where these methods will be benchmarked in the coming months) and extensive testing of this approach as a general Bayesian data cleaning methodology will be used to provide a more detailed analysis in the future. % The \nocite command causes all entries in a bibliography to be printed out % whether or not they are actually referenced in the text. This is appropriate % for the sample file to show the different styles of references, but authors % most likely will not want to use it. \nocite{*} \bibliography{ref}% Produces the bibliography via BibTeX. \end{document} % % ****** End of file apssamp.tex ****** %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End: ``` 4. **Bibliographic Information:** ```bbl %merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked %Control: key (0) %Control: author (8) initials jnrlst %Control: editor formatted (1) identically to author %Control: production of article title (-1) disabled %Control: page (0) single %Control: year (1) truncated %Control: production of eprint (0) enabled \begin{thebibliography}{43}% \makeatletter \providecommand \@ifxundefined [1]{% \@ifx{#1\undefined} }% \providecommand \@ifnum [1]{% \ifnum #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi }% \providecommand \@ifx [1]{% \ifx #1\expandafter \@firstoftwo \else \expandafter \@secondoftwo \fi 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{Wang}}, \bibinfo {author} {\bibfnamefont {Y.}~\bibnamefont {Mei}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Xu}}, \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont {Smirnov}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Deng}}, \ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Wei}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Monthly Notices of the Royal Astronomical Society}\ }\textbf {\bibinfo {volume} {512}},\ \bibinfo {pages} {2025} (\bibinfo {year} {2022})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {{Kennedy}}\ \emph {et~al.}(2022)\citenamefont {{Kennedy}}, \citenamefont {{Bull}}, \citenamefont {{Wilensky}},\ and\ \citenamefont {{Choudhuri}}}]{2022arXiv221105088K}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {F.}~\bibnamefont {{Kennedy}}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {{Bull}}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {{Wilensky}}}, \ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {{Choudhuri}}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv e-prints}\ ,\ \bibinfo {eid} {arXiv:2211.05088}} (\bibinfo {year} {2022})},\ \Eprint {http://arxiv.org/abs/2211.05088} {arXiv:2211.05088 [astro-ph.CO]} \BibitemShut {NoStop}% \bibitem [{\citenamefont {Abbott}\ \emph {et~al.}(2009)\citenamefont {Abbott}, \citenamefont {Abbott}, \citenamefont {Adhikari}, \citenamefont {Ajith}, \citenamefont {Allen}, \citenamefont {Allen}, \citenamefont {Amin}, \citenamefont {Anderson}, \citenamefont {Anderson}, \citenamefont {Arain} \emph {et~al.}}]{abbott2009ligo}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Abbott}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Abbott}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Adhikari}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Ajith}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Allen}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont {Allen}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Amin}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Anderson}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Anderson}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Arain}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Reports on Progress in Physics}\ }\textbf {\bibinfo {volume} {72}},\ \bibinfo {pages} {076901} (\bibinfo {year} {2009})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Ormiston}\ \emph {et~al.}(2020)\citenamefont {Ormiston}, \citenamefont {Nguyen}, \citenamefont {Coughlin}, \citenamefont {Adhikari},\ and\ \citenamefont {Katsavounidis}}]{ormiston2020noise}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Ormiston}}, \bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Nguyen}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Coughlin}}, \bibinfo {author} {\bibfnamefont {R.~X.}\ \bibnamefont {Adhikari}}, \ and\ \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Katsavounidis}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physical Review Research}\ }\textbf {\bibinfo {volume} {2}},\ \bibinfo {pages} {033066} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Harry}\ \emph {et~al.}(2010)\citenamefont {Harry}, \citenamefont {forthe LIGO Scientific~Collaboration} \emph {et~al.}}]{harry2010advanced}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {G.~M.}\ \bibnamefont {Harry}}, \bibinfo {author} {\bibnamefont {forthe LIGO Scientific~Collaboration}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Classical and Quantum Gravity}\ }\textbf {\bibinfo {volume} {27}},\ \bibinfo {pages} {084006} (\bibinfo {year} {2010})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {de~Lera~Acedo}\ \emph {et~al.}(2022)\citenamefont {de~Lera~Acedo}, \citenamefont {de~Villiers}, \citenamefont {Razavi-Ghods}, \citenamefont {Handley}, \citenamefont {Fialkov}, \citenamefont {Magro}, \citenamefont {Anstey}, \citenamefont {Bevins}, \citenamefont {Chiello}, \citenamefont {Cumner} \emph {et~al.}}]{de2022reach}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {de~Lera~Acedo}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {de~Villiers}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Razavi-Ghods}}, \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Handley}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Fialkov}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Magro}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Anstey}}, \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Bevins}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Chiello}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Cumner}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Nature Astronomy}\ ,\ \bibinfo {pages} {1}} (\bibinfo {year} {2022})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Handley}\ \emph {et~al.}(2015)\citenamefont {Handley}, \citenamefont {Hobson},\ and\ \citenamefont {Lasenby}}]{handley2015polychord}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Handley}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Hobson}}, \ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Lasenby}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Monthly Notices of the Royal Astronomical Society}\ }\textbf {\bibinfo {volume} {453}},\ \bibinfo {pages} {4384} (\bibinfo {year} {2015})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {MacKay}(2003)}]{mackay2003information}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont {MacKay}},\ }\href@noop {} {\emph {\bibinfo {title} {Information theory, inference and learning algorithms}}}\ (\bibinfo {publisher} {Cambridge university press},\ \bibinfo {year} {2003})\BibitemShut {NoStop}% \bibitem [{\citenamefont {Sivia}\ and\ \citenamefont {Skilling}(2006)}]{sivia2006data}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Sivia}}\ and\ \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Skilling}},\ }\href@noop {} {\emph {\bibinfo {title} {Data analysis: a Bayesian tutorial}}}\ (\bibinfo {publisher} {OUP Oxford},\ \bibinfo {year} {2006})\BibitemShut {NoStop}% \bibitem [{\citenamefont {Handley}(2019)}]{anesthetic}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Handley}},\ }\href {\doibase 10.21105/joss.01414} {\bibfield {journal} {\bibinfo {journal} {The Journal of Open Source Software}\ }\textbf {\bibinfo {volume} {4}},\ \bibinfo {pages} {1414} (\bibinfo {year} {2019})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Handley}(2018)}]{fgivenx}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Handley}},\ }\href {\doibase 10.21105/joss.00849} {\bibfield {journal} {\bibinfo {journal} {The Journal of Open Source Software}\ }\textbf {\bibinfo {volume} {3}} (\bibinfo {year} {2018}),\ 10.21105/joss.00849}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Anstey}\ \emph {et~al.}(2021)\citenamefont {Anstey}, \citenamefont {de~Lera~Acedo},\ and\ \citenamefont {Handley}}]{anstey2021general}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Anstey}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {de~Lera~Acedo}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Handley}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Monthly Notices of the Royal Astronomical Society}\ }\textbf {\bibinfo {volume} {506}},\ \bibinfo {pages} {2041} (\bibinfo {year} {2021})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {de~Lera~Acedo}(2019)}]{de2019reach}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {de~Lera~Acedo}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {2019 International Conference on Electromagnetics in Advanced Applications (ICEAA)}}}\ (\bibinfo {organization} {IEEE},\ \bibinfo {year} {2019})\ pp.\ \bibinfo {pages} {0626--0629}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Blanchard}\ \emph {et~al.}(2021)\citenamefont {Blanchard}, \citenamefont {Higham},\ and\ \citenamefont {Higham}}]{blanchard2021accurately}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Blanchard}}, \bibinfo {author} {\bibfnamefont {D.~J.}\ \bibnamefont {Higham}}, \ and\ \bibinfo {author} {\bibfnamefont {N.~J.}\ \bibnamefont {Higham}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {IMA Journal of Numerical Analysis}\ }\textbf {\bibinfo {volume} {41}},\ \bibinfo {pages} {2311} (\bibinfo {year} {2021})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Johnston}\ \emph {et~al.}(2008)\citenamefont {Johnston}, \citenamefont {Taylor}, \citenamefont {Bailes}, \citenamefont {Bartel}, \citenamefont {Baugh}, \citenamefont {Bietenholz}, \citenamefont {Blake}, \citenamefont {Braun}, \citenamefont {Brown}, \citenamefont {Chatterjee} \emph {et~al.}}]{johnston2008science}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Johnston}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Taylor}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Bailes}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Bartel}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Baugh}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Bietenholz}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Blake}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Braun}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Brown}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Chatterjee}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Experimental astronomy}\ }\textbf {\bibinfo {volume} {22}},\ \bibinfo {pages} {151} (\bibinfo {year} {2008})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Offringa}(2010)}]{offringa2010aoflagger}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Offringa}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Astrophysics Source Code Library}\ ,\ \bibinfo {pages} {ascl}} (\bibinfo {year} {2010})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Singh}\ \emph {et~al.}(2017)\citenamefont {Singh}, \citenamefont {Subrahmanyan}, \citenamefont {Shankar}, \citenamefont {Rao}, \citenamefont {Fialkov}, \citenamefont {Cohen}, \citenamefont {Barkana}, \citenamefont {Girish}, \citenamefont {Raghunathan}, \citenamefont {Somashekar} \emph {et~al.}}]{singh2017first}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Singh}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Subrahmanyan}}, \bibinfo {author} {\bibfnamefont {N.~U.}\ \bibnamefont {Shankar}}, \bibinfo {author} {\bibfnamefont {M.~S.}\ \bibnamefont {Rao}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Fialkov}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Cohen}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Barkana}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Girish}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Raghunathan}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Somashekar}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {The Astrophysical Journal Letters}\ }\textbf {\bibinfo {volume} {845}},\ \bibinfo {pages} {L12} (\bibinfo {year} {2017})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Oslick}\ \emph {et~al.}(1998)\citenamefont {Oslick}, \citenamefont {Linscott}, \citenamefont {Maslakovic},\ and\ \citenamefont {Twicken}}]{oslick1998general}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Oslick}}, \bibinfo {author} {\bibfnamefont {I.~R.}\ \bibnamefont {Linscott}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Maslakovic}}, \ and\ \bibinfo {author} {\bibfnamefont {J.~D.}\ \bibnamefont {Twicken}},\ }in\ \href@noop {} {\emph {\bibinfo {booktitle} {Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP'98 (Cat. No. 98CH36181)}}},\ Vol.~\bibinfo {volume} {3}\ (\bibinfo {organization} {IEEE},\ \bibinfo {year} {1998})\ pp.\ \bibinfo {pages} {1537--1540}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Baan}\ \emph {et~al.}(2004)\citenamefont {Baan}, \citenamefont {Fridman},\ and\ \citenamefont {Millenaar}}]{baan2004radio}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Baan}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Fridman}}, \ and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Millenaar}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {The Astronomical Journal}\ }\textbf {\bibinfo {volume} {128}},\ \bibinfo {pages} {933} (\bibinfo {year} {2004})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Bowman}\ \emph {et~al.}(2018)\citenamefont {Bowman}, \citenamefont {Rogers}, \citenamefont {Monsalve}, \citenamefont {Mozdzen},\ and\ \citenamefont {Mahesh}}]{bowman2018absorption}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~D.}\ \bibnamefont {Bowman}}, \bibinfo {author} {\bibfnamefont {A.~E.}\ \bibnamefont {Rogers}}, \bibinfo {author} {\bibfnamefont {R.~A.}\ \bibnamefont {Monsalve}}, \bibinfo {author} {\bibfnamefont {T.~J.}\ \bibnamefont {Mozdzen}}, \ and\ \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Mahesh}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Nature}\ }\textbf {\bibinfo {volume} {555}},\ \bibinfo {pages} {67} (\bibinfo {year} {2018})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Ellingson}\ and\ \citenamefont {Lewis}(2006)}]{ellingson2006rfi}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Ellingson}}\ and\ \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Lewis}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {An SKA Engineering Overview}\ ,\ \bibinfo {pages} {116}} (\bibinfo {year} {2006})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Wilensky}(2021)}]{wilensky2021improving}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Wilensky}},\ }\emph {\bibinfo {title} {Improving 21-cm Epoch of Reionization Power Spectrum Limits by Characterizing and Mitigating Radio Frequency Interference}},\ \href@noop {} {Ph.D. thesis},\ \bibinfo {school} {University of Washington} (\bibinfo {year} {2021})\BibitemShut {NoStop}% \bibitem [{\citenamefont {Anstey}\ \emph {et~al.}(2022)\citenamefont {Anstey}, \citenamefont {Acedo},\ and\ \citenamefont {Handley}}]{anstey2022use}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Anstey}}, \bibinfo {author} {\bibfnamefont {E.~d.~L.}\ \bibnamefont {Acedo}}, \ and\ \bibinfo {author} {\bibfnamefont {W.}~\bibnamefont {Handley}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv preprint arXiv:2210.04707}\ } (\bibinfo {year} {2022})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Takahashi}\ \emph {et~al.}(2010)\citenamefont {Takahashi}, \citenamefont {Ade}, \citenamefont {Barkats}, \citenamefont {Battle}, \citenamefont {Bierman}, \citenamefont {Bock}, \citenamefont {Chiang}, \citenamefont {Dowell}, \citenamefont {Duband}, \citenamefont {Hivon} \emph {et~al.}}]{takahashi2010characterization}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {Y.~D.}\ \bibnamefont {Takahashi}}, \bibinfo {author} {\bibfnamefont {P.~A.}\ \bibnamefont {Ade}}, \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Barkats}}, \bibinfo {author} {\bibfnamefont {J.~O.}\ \bibnamefont {Battle}}, \bibinfo {author} {\bibfnamefont {E.~M.}\ \bibnamefont {Bierman}}, \bibinfo {author} {\bibfnamefont {J.~J.}\ \bibnamefont {Bock}}, \bibinfo {author} {\bibfnamefont {H.~C.}\ \bibnamefont {Chiang}}, \bibinfo {author} {\bibfnamefont {C.~D.}\ \bibnamefont {Dowell}}, \bibinfo {author} {\bibfnamefont {L.}~\bibnamefont {Duband}}, \bibinfo {author} {\bibfnamefont {E.~F.}\ \bibnamefont {Hivon}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {The Astrophysical Journal}\ }\textbf {\bibinfo {volume} {711}},\ \bibinfo {pages} {1141} (\bibinfo {year} {2010})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Darling}(2020)}]{PhysRevLett.125.121103}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Darling}},\ }\href {\doibase 10.1103/PhysRevLett.125.121103} {\bibfield {journal} {\bibinfo {journal} {Phys. Rev. Lett.}\ }\textbf {\bibinfo {volume} {125}},\ \bibinfo {pages} {121103} (\bibinfo {year} {2020})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {di~Serego~Alighieri}\ \emph {et~al.}(2008)\citenamefont {di~Serego~Alighieri}, \citenamefont {Kurk}, \citenamefont {Ciardi}, \citenamefont {Cimatti}, \citenamefont {Daddi},\ and\ \citenamefont {Ferrara}}]{di2008search}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {di~Serego~Alighieri}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Kurk}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Ciardi}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Cimatti}}, \bibinfo {author} {\bibfnamefont {E.}~\bibnamefont {Daddi}}, \ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Ferrara}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Proceedings of the International Astronomical Union}\ }\textbf {\bibinfo {volume} {4}},\ \bibinfo {pages} {75} (\bibinfo {year} {2008})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Linde}(1983)}]{linde1983chaotic}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {A.~D.}\ \bibnamefont {Linde}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physics Letters B}\ }\textbf {\bibinfo {volume} {129}},\ \bibinfo {pages} {177} (\bibinfo {year} {1983})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Bale}\ \emph {et~al.}(2023)\citenamefont {Bale}, \citenamefont {Bassett}, \citenamefont {Burns}, \citenamefont {Jones}, \citenamefont {Goetz}, \citenamefont {Hellum-Bye}, \citenamefont {Hermann}, \citenamefont {Hibbard}, \citenamefont {Maksimovic}, \citenamefont {McLean} \emph {et~al.}}]{bale2023lusee}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~D.}\ \bibnamefont {Bale}}, \bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Bassett}}, \bibinfo {author} {\bibfnamefont {J.~O.}\ \bibnamefont {Burns}}, \bibinfo {author} {\bibfnamefont {J.~D.}\ \bibnamefont {Jones}}, \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Goetz}}, \bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Hellum-Bye}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Hermann}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Hibbard}}, \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Maksimovic}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {McLean}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {arXiv preprint arXiv:2301.10345}\ } (\bibinfo {year} {2023})}\BibitemShut {NoStop}% \bibitem [{\citenamefont {Bernhardt}\ \emph {et~al.}(2021)\citenamefont {Bernhardt}, \citenamefont {de~Castro}, \citenamefont {Tanno}, \citenamefont {Schwaighofer}, \citenamefont {Tezcan}, \citenamefont {Monteiro}, \citenamefont {Bannur}, \citenamefont {Lungren}, \citenamefont {Nori}, \citenamefont {Glocker}, \citenamefont {Alvarez-Valle},\ and\ \citenamefont {Oktay}}]{Bernhardt2021ActiveLC}% \BibitemOpen \bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Bernhardt}}, \bibinfo {author} {\bibfnamefont {D.~C.}\ \bibnamefont {de~Castro}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {Tanno}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Schwaighofer}}, \bibinfo {author} {\bibfnamefont {K.~C.}\ \bibnamefont {Tezcan}}, \bibinfo {author} {\bibfnamefont {M.~A.~B.}\ \bibnamefont {Monteiro}}, \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Bannur}}, \bibinfo {author} {\bibfnamefont {M.~P.}\ \bibnamefont {Lungren}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Nori}}, \bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Glocker}}, \bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Alvarez-Valle}}, \ and\ \bibinfo {author} {\bibfnamefont {O.}~\bibnamefont {Oktay}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Nature Communications}\ }\textbf {\bibinfo {volume} {13}} (\bibinfo {year} {2021})}\BibitemShut {NoStop}% \end{thebibliography}% ``` 5. **Author Information:** - Lead Author: {'name': 'S. A. K. Leeney'} - Full Authors List: ```yaml 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/ 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 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 [2211.15448](https://arxiv.org/abs/2211.15448) is featured in the first sentence. Generate only the final Markdown output that meets all these requirements. {% endraw %}