{% 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: "Hunting for bumps in the margins"
date: 2022-11-18
categories: papers
---
Content generated by [gemini-2.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2022-11-18-2211.10391.txt).
Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2022-11-18-2211.10391.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': 'David Yallup'}). 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 [2211.10391](https://arxiv.org/abs/2211.10391) 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
---
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/2211.10391v2
guidislink: true
link: http://arxiv.org/abs/2211.10391v2
updated: '2023-05-22T12:28:42Z'
updated_parsed: !!python/object/apply:time.struct_time
- !!python/tuple
- 2023
- 5
- 22
- 12
- 28
- 42
- 0
- 142
- 0
- tm_zone: null
tm_gmtoff: null
published: '2022-11-18T17:48:13Z'
published_parsed: !!python/object/apply:time.struct_time
- !!python/tuple
- 2022
- 11
- 18
- 17
- 48
- 13
- 4
- 322
- 0
- tm_zone: null
tm_gmtoff: null
title: Hunting for bumps in the margins
title_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
type: text/plain
language: null
base: ''
value: Hunting for bumps in the margins
summary: 'Data driven modelling is vital to many analyses at collider experiments,
however the derived inference of physical properties becomes subject to details
of the model fitting procedure. This work brings a principled Bayesian picture,
based on the marginal likelihood, of both data modelling and signal extraction
to a common collider physics scenario. First the marginal likelihood based
method is used to propose a more principled construction of the background
process, systematically exploring a variety of candidate shapes. Second the
picture is extended to propose the marginal likelihood as a useful tool for
anomaly detection challenges in particle physics. This proposal offers insight
into both precise background model determination and demonstrates a flexible
method to extend signal determination beyond a simple bump hunt.'
summary_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
type: text/plain
language: null
base: ''
value: 'Data driven modelling is vital to many analyses at collider experiments,
however the derived inference of physical properties becomes subject to details
of the model fitting procedure. This work brings a principled Bayesian picture,
based on the marginal likelihood, of both data modelling and signal extraction
to a common collider physics scenario. First the marginal likelihood based
method is used to propose a more principled construction of the background
process, systematically exploring a variety of candidate shapes. Second the
picture is extended to propose the marginal likelihood as a useful tool for
anomaly detection challenges in particle physics. This proposal offers insight
into both precise background model determination and demonstrates a flexible
method to extend signal determination beyond a simple bump hunt.'
authors:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: David Yallup
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Will Handley
author_detail: !!python/object/new:feedparser.util.FeedParserDict
dictitems:
name: Will Handley
author: Will Handley
arxiv_doi: 10.1088/1748-0221/18/05/P05014
links:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
title: doi
href: http://dx.doi.org/10.1088/1748-0221/18/05/P05014
rel: related
type: text/html
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
href: http://arxiv.org/abs/2211.10391v2
rel: alternate
type: text/html
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
title: pdf
href: http://arxiv.org/pdf/2211.10391v2
rel: related
type: application/pdf
arxiv_comment: 10 pages, 6 figures, accepted to JINST
arxiv_journal_ref: JINST, 18, P05014 (2023)
arxiv_primary_category:
term: physics.data-an
scheme: http://arxiv.org/schemas/atom
tags:
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
term: physics.data-an
scheme: http://arxiv.org/schemas/atom
label: null
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
term: hep-ex
scheme: http://arxiv.org/schemas/atom
label: null
- !!python/object/new:feedparser.util.FeedParserDict
dictitems:
term: hep-ph
scheme: http://arxiv.org/schemas/atom
label: null
```
3. **Paper Source (TeX):**
```tex
\documentclass[reprint,preprintnumbers,nofootinbib,amsmath,amssymb,floatfix,prl]{revtex4-2}
% \bibliographystyle{apsrev4-2}
\bibliographystyle{JHEP}
\usepackage{float}
\usepackage{graphicx}
\usepackage{bm}
\usepackage{hyperref}
\usepackage{xspace}
\usepackage[capitalise]{cleveref}
\usepackage{lmodern}
\usepackage{microtype}
\usepackage[T1]{fontenc}
\usepackage{siunitx}
\usepackage{subfig}
\usepackage{booktabs}
\usepackage[algoruled,lined,linesnumbered,longend]{algorithm2e}
\renewcommand{\arraystretch}{1.5}
\hypersetup{
colorlinks=true,
allcolors=[rgb]{0.26,0.41,0.88},
}
\SetKwFor{KwIn}{initialize}{}{}{}
\sisetup{
range-phrase =\text{--}
}
\input{macros}
% comments
\usepackage[usenames]{xcolor}
%TC:macro \AF [ignore]
\newcommand{\DY}[1]{{\color{blue}\textbf{TODO DY:} \textit{#1}}}
% add back in section numbers
\setcounter{secnumdepth}{3}
% styling of treating cite as a noun
\newcommand{\refcite}{\cite}
\begin{document}
% \title{Discrete marginalised models for particle physics backgrounds}
\title{Hunting for bumps in the margins}
\author{David Yallup}
\email{dy297@cam.ac.uk}
\affiliation{Cavendish Laboratory \& Kavli Institute for Cosmology, University of Cambridge,\\ JJ Thomson Avenue, Cambridge, CB3~0HE, United Kingdom}
\author{Will Handley}
\affiliation{Cavendish Laboratory \& Kavli Institute for Cosmology, University of Cambridge,\\ JJ Thomson Avenue, Cambridge, CB3~0HE, United Kingdom}
% \date{\today}
\begin{abstract}
Data driven modelling is vital to many analyses at collider experiments, however the derived inference of physical properties becomes subject to details of the model fitting procedure. This work brings a principled Bayesian picture -- based on the marginal likelihood -- of both data modelling and signal extraction to a common collider physics scenario. First the marginal likelihood based method is used to propose a more principled construction of the background process, systematically exploring a variety of candidate shapes. Second the picture is extended to propose the marginal likelihood as a useful tool for anomaly detection challenges in particle physics. This proposal offers insight into both precise background model determination and demonstrates a flexible method to extend signal determination beyond a simple bump hunt.
\end{abstract}
\maketitle
\section{Introduction}\label{sec:intro}
The searches and measurements taking place at the energy frontiers of collider physics take place in a data rich environment. Theoretical calculations of backgrounds in these analyses are compromised by the complexity of the full theory compared to what is computationally feasible. The problem of modelling these background processes is often further complicated by experimental systematics that are accounted for in-situ from collider data~\cite{ATLAS:2022swp}. These factors motivate usage of parametric models, fit to data, to perform the background model estimation~\cite{ATLAS:2019fgd,CMS:2019gwf,CMS:2021kom,ATLAS:2020pvn}. As a result, the search for (or measurement of) rarer signal processes amongst the background becomes sensitive to this data driven fit. There will be two main sources of uncertainty on the background model in such an analysis; parametric and modelling uncertainties. Parametric uncertainty arises due to the uncertainty on the input data points that the model is fit to, leading to variance of the parameters of a given model. Modelling uncertainty arises due to the potential for model misspecification biasing the resulting inferred functional form, this is generally much harder to estimate. Typically this is achieved by independently fitting multiple models and comparing the results of the individual fits, however the resulting recipe to select the model can become rather ad-hoc. Without quantifiable justification, the uncertainty from the model selection is difficult to precisely represent.
Bayesian inference offers a natural framework to discuss both modelling and parametric uncertainties~\cite{Trotta:2008qt}. Parameter uncertainty is naturally expressed in the posterior distribution of the parameters of interest. Model uncertainty is captured in the marginal likelihood, or \emph{evidence}, of a Bayesian calculation. Comparison of evidences between models gives a concrete method of Bayesian model selection, and can be used to define Bayesian model averages which effectively ensemble the modelling uncertainty. Alternatively (but giving equivalent results) the posterior can be extended over hyperparameters governing the choice of the model itself -- for example the family of functions composing the model or the number of basis functions. Such hyperparameters can be thought of as being discrete in nature, and understanding how to deal with discrete hyperparameters is one of the core goals of this study.
Numerically marginalising over discrete parameters is a computationally difficult task as discrete model choices induce discontinuous and multimodal likelihood surfaces. Nested Sampling (NS)~\cite{skilling} is a well established method to perform numerical marginalisation over such a target likelihood function. Bayesian sparse reconstruction (BSR)~\cite{bsr} has been demonstrated to perform the marginalisation over mixtures of discrete and continuous model parameters and hyperparameters in the context of astronomical imaging.
In this analysis the framework of BSR is applied to a toy problem representing a typical case of modelling a smoothly falling background process. The resulting composition of the model is examined and contrasted with existing maximum likelihood based approaches, motivating the utility of a Bayesian model in this context. To further highlight the unique opportunities afforded by NS in tackling multimodal sampling problems, a search for generic resonances -- a bump hunt -- is considered on top of the discrete marginalised background model. Signal analysis in particle physics has been cast as a Bayesian model comparison problem in plenty of previous work (see \cite{Fowlie:2019ydo} and references therein), by extending these kinds of signal analyses over a discrete marginalised background model, a method with potential to tackle more subtle shape discrimination is proposed. The result is an analysis pipeline defining a powerful, fully Bayesian method for anomaly detection with well-calibrated background modelling uncertainties.
\section{Models for binned data}
Measuring an observable that is dominated by a single background process that has a smoothly falling rate as a function of the observable is a common scenario in many LHC physics analyses. The searches for resonances in dijet mass spectra at the ATLAS and CMS detectors~\cite{ATLAS:2019fgd,CMS:2019gwf} are flagship searches at the LHC that center on modelling the smoothly falling QCD background process. These analyses are particularly sensitive to the issue of extrapolating the fitted model into the poorly constrained tail of the spectrum. Precision measurements of Higgs properties in the diphoton decay channel with the ATLAS and CMS detectors~\cite{CMS:2021kom,ATLAS:2020pvn} provide a complementary example where the result hinges on the precision of the background model in a window as a function of the observable, surrounded and constrained by high statistics measurements of the diphoton mass spectrum. This study blends features from both these sources, examining the precise construction of a background model for a Higgs diphoton style measurement, then using this to perform an anomaly detection task reminiscent of a search analysis. In the remainder of this section the likelihood for this type of observable is defined, and the relevant basis functions to build the background model are defined.
% For the purposes of this study a simplified generic statistical model is built, incorporating common features from these two examples. No reference is made to specific details of the signal models.
\subsection{Likelihoods from histograms}
A single measurement channel consists of $M$ histogram bins in a measured property of collision events, $x$ (the dijet or diphoton mass for example). With bins indexed by $i$, data is collected in the $i$-th bin by counting events occurring in a range from the lower bin edge to upper bin edge. For simplicity the bin can instead be characterized by its mid-point, $x_i$, and the degree of freedom associated with the bin width can be effectively integrated out. Each bin records an observed event count, $n_i$, and the background model predicts this count as a function of the measured property, $b(x_i)$. The likelihood for the set of data observations $n_i$, given a background model parameterized by $\Theta$ can be taken as a product of independent Poisson distributions,
\begin{align}\label{eq:like}
\anyg{P}{n_i}{\Theta} = \prod_i^M b(x_i,\Theta)^{n_i} \frac{e^{-b(x_i,\Theta)}}{n_i!}\,.
% \anyg{P}{X}{\theta} = \prod_i \mathcal{B}_i^{n_i} \frac{e^{-\mathcal{B}_i}}{n_i!}\,,\\
% \mathcal{B} = \int^{x_i^+}_{x_i^-} f(x_i,\theta)
\end{align}
Additional constraint terms, nuisance parameters or parameters of interest on signal models can then be incorporated. This work initially explores the problem of inferring a background-only model in \cref{sec:datadrivenhiggs} to establish the challenges of surveying discrete parameters in models. In \cref{sec:bumphunt} this likelihood is then extended to consider a signal model in the guise of an anomaly detection task for generic signal models.
The smoothly falling backgrounds in this problem can be modelled as a sum of simple basis functions $\phi$,
\begin{align}
b(x_i,\Theta) = b(x_i,\theta,N,\phi) = \sum^N_k \phi_k(x_i,\theta)\,.
\end{align}
The background model is dependent on three factors in addition to the input $x_i$, redefining the background model parameters as $\Theta=\{ \theta,N,\phi\}$. These three components can be identified as -- the number of basis functions $N$, the parameters defining the basis functions $\theta$ and the family of basis functions $\phi$. The parameters $N$ and $\phi$ are identified as the discrete hyperparameters of the background model, and
% , hence the likelihood in \cref{eq:like_single} can be extended over these parameters,
% \begin{align}\label{eq:like}
% \anyg{P}{n_i}{\theta,N,\phi} = \prod_i b(x_i,\theta,N,\phi)^{n_i} \frac{e^{-b(x_i,\theta,N,\phi)}}{n_i!}\,.
% % \anyg{P}{X}{\theta} = \prod_i \mathcal{B}_i^{n_i} \frac{e^{-\mathcal{B}_i}}{n_i!}\,,\\
% % \mathcal{B} = \int^{x_i^+}_{x_i^-} f(x_i,\theta)
% \end{align}
the inference can either be extended over these parameters or they can be specified and conditioned on. A frequentist approach to deal with inference extended over nuisance parameters is to \textit{profile} them~\cite{Dauncey:2014xga}. Profiling a parameter removes the associated degree of freedom by conditioning on the maximum likelihood estimator for the parameter. Profiling discrete parameters is challenging, achieved by considering a model ensemble conditioned on the set of discrete model choices, which can be used to build an envelope of profiled curves over the continuous parameters. A method for constructing this ensemble is demonstrated in \cref{sec:mlmethod}.
% Choosing what is inferred, what is specified and conditioned on, and what is profiled define the various approaches to this problem. Profiling over discrete parameters~\cite{Dauncey:2014xga} is a commonly used approach to this type of modelling problem in particle physics in particle physics contexts.
A Bayesian interpretation of this problem instead marginalises over these discrete hyperparameters, inverting the likelihood using Bayes theorem to get a posterior distribution for the model parameters, $\Pg{\theta,N,\phi}{n_i}$. By marginalising over all parameters and hyperparameters in the model, a background model can be constructed that removes dependence on these choices by sampling directly from this full posterior distribution. An example of this marginalisation is demonstrated in \cref{sec:samplingmethod}, emphasising the role of the evidence as an alternative tool for building model averages.
Methods involving Bayesian non-parametric models have received increased attention recently~\cite{Frate:2017mai}, giving strong motivation to consider the contrasting adaptive basis function method presented in this work. Much of the attraction of Gaussian Processes derives from the fact that the expression for the marginal likelihood of such models is directly calculable as a function of the training data from simple linear operations. A marginal likelihood calculated for a set of adaptive basis functions -- as presented in this work -- retains the strengths of promoting functional diversity and embodying automatic relevance determination encapsulated by the marginal likelihood, whilst negating conceptual issues arising from using non-parametric methods in this context. When used for anomaly detection tasks, non-parametric models require the length scale over which the anomaly exhibits to be well separated from the length scale of the background process. Whilst this is generally achievable for the types of narrow width signals considered in this demonstration, posing the modelling problem as a Bayes ratio enables the data driven background models to be extended to less well separated model comparison tasks.
\subsection{Smoothly falling functions}\label{sec:prior}
In order to construct a suitable set of basis functions to examine, a weak prior belief that the background model should be both constantly decreasing and smooth is initially considered. Following similar analyses conducted at collider experiments, three distinct candidate families of basis function are considered; Exponential (exp), Power law (power), and Log Polynomial (logpoly) functions. A more complete set of such functions has been demonstrated for similar physical use cases in the context of radio astronomy~\cite{Bevins:2020jqf}, and applied to similar bump hunting style problems in the same field~\cite{deLeraAcedo:2022kiu}. The three function families chosen are by no means an exhaustive list, being an arbitrary choice inspired by a priori reasonable candidates, leaving the task of selecting and weighting these candidate models to the data. These three choices of function family can be defined in terms of the remaining background model parameters as,
\begin{align}\label{eq:bases}
\phi= \mathrm{exp}: \quad & b(x,\theta, N) = \sum_{m=1}^{N_{\mathrm{max}}} \theta_{1,m} \exp (-\theta_{2,m} x)\,, \\
\phi= \mathrm{power}: \quad & b(x,\theta, N) = \sum_{m=1}^{N_{\mathrm{max}}} \theta_{1,m} (x+1)^{-1.5\theta_{2,m}}\,, \\
\phi= \mathrm{logpoly}: \quad & b(x,\theta, N) = \sum_{m=1}^{N_{\mathrm{max}}} \theta_{1,m} \log(x+e)^{-4\theta_{2,m}}\,.
\end{align}
The data is transformed such that $x_i\in[0,1]$ and $n_i$ scaled such that it has unit variance. The factors and translations appearing in these function definitions are required to ensure that the three families to admit similarly expressive functions. The $x$ axes of all functions are shifted such that ${b(\theta_{2,m},x=0,\theta_{1,m}=1,N=1)=1}$, and the exponents scaled such that ${b(\theta_{2,m},x=1,\theta_{1,m}=1,N=1)}$ has a similar range for all families. This similarity in range is illustrated in Figure~\ref{fig:ymax}. The \emph{whitening} of the data ensures that, given the chosen prior ranges on the functions, only sensible realisations of the background function are considered. When evaluating the likelihood, the functions are transformed back to the observed counts.
\begin{figure}
\includegraphics[]{figures/intercept_max.pdf}
\caption{Value of background model at $b(\theta_{2,m},x=1,\theta_{1,m}=1,N=1)$ as a function of the exponent parameter $\theta_{2,m}$ for each discrete choice of $\phi$.\label{fig:ymax}}
\end{figure}
Each basis function in \cref{eq:bases} is characterized by two parameters; an amplitude parameter $\theta_{1,m}$ and an exponent parameter $\theta_{2,m}$. The exponent parameter is given a uniform prior on $[0,5]$. The amplitude parameter is given a \emph{sorted} (more formally, a forced identifiability prior~\cite{PhysRevD.100.084041}) uniform prior on $[-10,10]$. Allowing negative amplitudes relaxes the smooth constantly decreasing criteria in order to match the maximum likelihood implementation, however this is trivial to reinstate for a model with stronger inductive bias (as done in \cref{sec:bumphunt}).
%All $\theta$ parameters are given a Uniform prior on $[0,5]$, with the amplitude parameters having the additional constraint of being \emph{sorted} (described in Appendix A2 of~\cite{Handley:2015vkr}).
The discrete hyperparameter $N$ is given a uniform prior over integers $[1,N_\mathrm{max}]$, with $\phi$ being effectively given a uniform prior over integers $[1,3]$ indexing the family choice. The resulting sampling space is of dimension $(N_\mathrm{max}\times 2 + 2)$, with previous applications of BSR finding it sufficient to mask inactive basis parameters when $N0.1$ is used as a rough visual guide. The diversity along the $N$ axis of this table can be compensated for in the maximum likelihood method by tuning the penalty term used, the work of Dauncy et al.~\cite{Dauncey:2014xga} notes the difficulty in making general recommendations for such a tuning due to it's dependency on the studied application. It is equivalently possible to instead ``tune'' the sampling based method by introducing a sparsity promoting prior -- for example an exponential prior on $N$. There is also increased diversity in the model composition along the $\phi$ axis in the marginalising case, particularly visible for the functions with $N=3$. There is no clear mechanism to account for this effect in a discrete profiling framework. To summarize this comparison suggests that inference built from a maximum likelihood based method needs problem specific tuning and is challenged when faced with subtle shape differences.
% To some extent this can be compensated for in the maximum likelihood based method by tuning the dimensional punishment factor, with an equivalent tuning to the marginal method available by introducing a sparsity promoting prior on $N$. There is however no clear mechanism to correct the promoted diversity along the $\phi$ axis in the maximum likelihood case. Without some tuning this suggests that inference built on a maximum likelihood based method may be overconfident and weight some candidate backgrounds too highly.
These two model compositions can be visually presented by examining how the inferred background models behave as a function of the physical observable, this is shown in \cref{fig:mggfit}. The Bayesian model prediction is the result of drawing 100 samples from the posterior and deriving an ensemble of model predictions from this set of parameter values -- this forms a posterior predictive distribution for the physical observable $m_{\gamma\gamma}$. The set of all 100 function predictions are shown in red in the top panel of the figure over the considered data. The posterior predictive distribution can be used to define mean model prediction and a one standard deviation uncertainty band on this. The bottom three ratio panels show the ratio taken to this Bayesian Model mean of both the full Bayesian error band, as well as the maximum likelihood solutions for each model choice.
Examining the composition of the background in detail is a useful validation of the method, however typically the composition of the background model is understood in terms of its influence on inferred parameters of a hypothesised signal model. It is conceptually simple enough to introduce an explicit parameterized signal model into \cref{eq:like} without introducing any particularly challenging computational overhead to this method, and this is explored in \cref{sec:bumphunt}.
% The natural next step for such an analysis is to consider a signal model, the goal typically being to infer some information about hypothesised parameters of a new physics model in light of the collected data. It is conceptually simple enough to introduce an explicit parameterized signal model into \cref{eq:like} without introducing any particularly challenging computational overhead to this method, and this is explored in \cref{sec:bumphunt}.
% As a background only fit there is no consideration of signal model in this work, which would be the natural next step in such an analysis. This is historically a contentious topic in the field of particle physics, with a variety of targets and proposed solutions that require dedicated consideration in further work. It is conceptually simple enough to introduce an explicit parameterized signal model into \cref{eq:like} without introducing any particularly challenging computational overhead to this method.
% Either performing a full Bayesian credibility interval calculation, or considering the marginalisation as integrating out parameter dependency for input into familiar profile likelihood machinery are both open possibilities. The modelling of generic signals in a procedure such as \textsc{BumpHunter}~\cite{Choudalakis:2011qn} is typically performed as a compatibility test under the null hypothesis, and as such is directly interpretable with the background only uncertainty band produced in this analysis.
\begin{figure}
\includegraphics[]{figures/predict.pdf}
\caption{Prediction and data included in the inference problem for the background model in terms of the physical variables $\{n,m_{\gamma\gamma}\}$. The top panel displays the observed data and predicted spectrum resulting from 100 posterior samples of the Bayesian model in red. The three ratio panels show the ratio of maximum likelihood models to the mean prediction from the Bayesian ensemble for increasing values of $N$. The variance of the ensemble is used to predict a one standard deviation error band shown shaded in red.\label{fig:mggfit}}
\end{figure}
\section{Bump hunting with marginal likelihoods}\label{sec:bumphunt}
Performing a Bayesian analysis of typical particle physics parameterized signal models is a well examined topic, with nuanced analysis needed to navigate issues such as the impact of choice of prior on physical parameters of interest~\cite{Fowlie:2019ydo}. The key quantity needed for a Bayesian analysis of a signal model is already derived, the evidence for a hypothesis as in \cref{eq:marg_model}. A generic parametric signal model can be introduced to the existing likelihood shown in \cref{eq:like} as,
\begin{align}\label{eq:like_signal}
% H_0: \quad & \anyg{P}{X}{\theta} = \prod_i f(x_i,\theta)^{n_i} \frac{e^{-f(x_i,\theta)}}{n_i!}\,,\\
H_\psi: \quad \anyg{P}{n_i}{\psi,\Theta} = \prod_i & \big(s(x_i,\psi) + b(x_i,\Theta)\big)^{n_i} \,\\
& \cdot \frac{e^{-(s(x_i,\psi) + b(x_i,\Theta))}}{n_i!}\,,
\end{align}
where the signal model, $s(x_i,\psi)$, is a function of the input $x$ data as well as a new set of parameter(s) of interest, $\psi$. Conditioning on a choice of $\psi$ specifies the hypothesis in question, $H_\psi$, with the likelihood in \cref{eq:like} being identified as the null hypothesis, $H_0$.
Rather than further study the properties of derived intervals on signal models with strong priors -- or validating the properties of frequentist vs. Bayesian discovery metrics -- a generic anomaly detection task is considered. This scenario employs weak priors on signal parameters, and considers a likelihood with only a weak signal injected. This is a deliberately challenging scenario and is reflective of recent activity in the field trying to push the envelope of anomaly detection. By injecting a signal at around the noise threshold, a signal model that is multimodal in its parameter space is induced. This plays to a previously highlighted strength of a Nested Sampling based approach, the multimodal background model built earlier is still employed and a multimodal signal model on top of this creates a sampling problem that many common numerical sampling tools would struggle with.
For the purposes of this study a generic Gaussian signal model is considered, defining the set of $\psi=\{A,\mu,\sigma\}$, as the amplitude, mean and variance of a Gaussian distribution. In this analysis all of these parameters are included into the marginalisation process, seeking to obtain the evidence for the hypothesis $\Z(H_\psi)=\Z_\psi$. In particular the search for a generic resonance signal across the entire spectrum is sought, commonly recognised as a bump hunt in particle physics~\cite{Choudalakis:2011qn}. In the following sections the same resonance search is applied in a null dataset (\emph{i.e.}, the data used in \cref{sec:datadrivenhiggs}) and a dataset with a true signal injected. The priors on the background model parameters are the same as described in \cref{sec:prior}, apart from the amplitude prior which is given a sorted uniform prior on $[0,5]$.
% To contextualise this in the context of the collider statistical model landscape, it is helpful to make a distinction between the amplitude and the parameters governing the shape of the signal model, $\{\mu,sigma\}$. With weak priors on both amplitude and shape, the analysis can be identified as an anomaly detection exercise, weakly informing a generic signal model. The inserted signal model is partly motivated by physical considerations but
\subsection{Uncovering true positives}
First a check is performed to see if, upon injecting a ``true'' signal to the null dataset, a correct inference of the signal can be recovered. To perform both of these searches the evidence for the signal model, $\Z_\psi$, and the evidence for the null model, $\Z_0$, are calculated. To allow a generic search across the spectrum the following uniform priors are put on the components of $\psi$,
\begin{itemize}
\item $A$ - the signal amplitude is given a prior in a range $[0,500]$, noting that as the signal model has by construction an integral of 1, this corresponds to a signal model with cross section $[0,0.05]$ fb. The true injected value is $A=150$.
\item $\sigma$ - the signal variance is given a prior in a range of $[0.5,3]$. With a unit bin width this restricts the search to a narrow resonance search, confining the signal to being contained mostly within $[1,6]$ bins. The true injected value is $\sigma=1.0$.
\item $\mu$ - the signal location is given a prior range covering [$100,180$] GeV. This styles this search as a bump hunt across the full spectrum of data. The true injected value is $\mu=125$.
\end{itemize}
A summary of the results of calculating the marginal likelihood for signal and null hypotheses is shown in \cref{fig:psi_predict}. The four panels in this figure show (in descending order); the physical spectrum and background model (as previously displayed in the top panel of \cref{fig:mggfit}), the posterior predictive distribution for the physical observable under $H_\psi$, a set of 1000 candidate forms for the signal model itself sampled from the posterior, and lastly the data residuals normalised by the error. The key quantity that amortises the hunt for the signal hypothesis is the ratio of the evidences, $Z_\psi/Z_0=9.29 \pm 1.37$. This can be interpreted as ``betting odds''~\cite{Handley:2019tkm}, implying the data favouring the signal hypothesis over the null with an odds ratio of over 9 to 1. This global hypothesis odds is complemented by the calculated per-bin posterior signal model probability. By sampling a set of posterior samples for the signal model parameters, $\tilde{\psi}$, a set of candidate signal models can be constructed, $s(x,\tilde{\psi})$. The binned weighted sum of these posterior predictive distributions for $s$ gives the per-bin signal predictive posterior under $H_\psi$. In this example there is a clear preference for candidates around the true injected signal, however another potential mode for the signal model is weakly identified. The bottom panel of \cref{fig:psi_predict} shows the residuals -- the ratio of the data to the mean background model normalised to the variance of the background model. It is notable that the signal injected at $m_{\gamma\gamma}=125$ GeV consists of multiple adjacent bins where the data deviates from the model by over two standard deviations. The Bayesian significance implied by the ratio of $Z_\psi/Z_0$ is noticeably lower than one would naively expect, and the tension between this and the equivalent frequentist global significance has been described as the Bayes effect~\cite{Fowlie:2019ydo}. This tension is a significant open problem that merits further exploration building on both the NS application and discrete marginalising presented in this work.
Examining the model prediction as a function of the physical observables is a useful diagnostic, but the most natural language to discuss signal models is to calculate the posterior distributions for the parameters of the model. The corner plot for $\psi$ is shown in \cref{fig:signal_post}. In order to highlight the utility of the marginal background model as described in \cref{sec:samplingmethod}, this corner plot shows the 95\% credibility region posterior for the signal parameters under four background hypotheses. The full discrete marginalised background model is shown as a shaded contour, and comparison is drawn to three background models with fixed hyperparameters, corresponding to each of the three families considered and $N=2$. The dotted black lines show the true inserted signal parameter values. Of the three fixed background models, $\phi=\mathrm{exp}$ covers the true values well, but offers a worse resolution of the location and amplitude parameters. Both $\phi=\mathrm{power}$ and logpoly offer better resolution of the signal parameters but are close to failing to cover the true values. Marginalising over these models (and the other choices of $N$) results in a combined model that is able to retain the desirable features of the component backgrounds. In all cases the width of the signal model is poorly constrained but this conclusion is a feature of the toy problem, wherein the signal was injected at a level only slightly larger than Poisson noise. Conclusions on the various choices of fixed background hyperparameters is largely dependent on the generated pseudo-data used, but the overall method of marginalising out these choices is robust. Further investigation preferably on real data and incorporating more exhaustive comparison with results obtained via discrete profiling would be well motivated.
%The fact that marginalising over model choice as an additional uncertainty not present in the ``Fixed'' background model actually improves the resolution of the signal is perhaps counter intuitive and merits further study. It is hypothesised here that by expanding the posterior over a more diverse range of functions that the resulting inferred mean is less biased by the function choice and hence can more accurately represent the true underlying background distribution -- further investigation preferably on real data and incorporating more exhaustive comparison with results obtained via discrete profiling is needed to explore this point.
This example establishes a marginal likelihood calculation as a successful strategy to search for, and infer parameters of, anomalous bumps in smoothly falling spectra. More
in-depth study is needed to explore the threshold of signal magnitude (and variety of signal shapes) to establish
the false negative error rate for a given dataset. Examining how precisely shape deviations can be extracted is pushed to new limits with the well calibrated in-situ data driven backgrounds composed in this work, opening up an attractive prospect for an unsupervised learning algorithm built on the marginal likelihood. This can be further extended by considering signal models with shapes that go beyond a simple Gaussian~\cite{Alvarez:2022kfr}, combining the power of the functionally diverse data driven background models exhibited in this work with more exotic signal shapes.
Pushing beyond the bump hunting paradigm is a topic of renewed interest more generally in the field~\cite{Kasieczka:2021xcg}, where signal models are pushed to undetectable levels when relying on simple bump hunting paradigms. Unsupervised machine learning methods generally attempt to overcome this by extending inference over a higher dimensional input data space, however in many cases these approaches rely on some variant of bump hunting in the latent space of a larger ML architecture. Adapting this work to such a scenario is an area with strong potential.
\begin{figure}
\includegraphics[]{figures/psi_predict.pdf}
\caption{Summary of the sampled signal hypothesis, $H_\psi$ presented as a function of the physical observable, $m_{\gamma\gamma}$. This is a test to recover true positives, with a true signal model inserted to the background data.\label{fig:psi_predict}}
\end{figure}
\begin{figure}
\includegraphics[]{figures/psi_corner_compare.pdf}
\caption{Posterior distribution of signal parameters under $H_\psi$ showing the 95\% credibility regions for 3 fixed hyperparameter value background models as outlines, alongside a filled contour representing the corresponding region where marginalisation over the background hyperparameters takes place. The true values of the inserted signal model parameters are shown as dotted lines.\label{fig:signal_post}}
\end{figure}
\subsection{Checking for false positives}
Alongside exploring the ability to retrieve true positive injected signals, it is important to balance this with a consideration of the robustness in the face of false positive signal identification. This sort of false positive can be examined by running the proposed method on a spectrum with no signal injected. The summary of running a marginalisation of $H_\psi$, in the case where there is no signal injected, is shown in \cref{fig:psi_predict_null}, mirroring the structure of \cref{fig:psi_predict}. It is in examining this scenario that the strength of a Bayesian anomaly detection really shines. Whilst there are locally significant deviations, the overall evidence for the signal hypothesis disfavoured over the null -- by a probability ratio of $Z_\psi/Z_0=0.59 \pm 0.08$. The fully Bayesian method proposed in this work demonstrates the advantages of the Bayesian Occams razor end-to-end. In \cref{sec:samplingmethod} it was noted that Bayesian Occams razor optimally navigated background model complexity, punishing overly complex models based on relative lack of information gain. The principles of Occams razor are once again on display when the marginalisation is extended over signal models, allowing discrimination of signal models based on how economically they improve the description of the data. It is precisely this amortisation of models based on dimensional economy in comparison to the null, that render this a powerful yet well calibrated search method.
\begin{figure}
\includegraphics[]{figures/psi_predict_null.pdf}
\caption{Summary of signal hypothesis as in \cref{fig:psi_predict}, in the regime where no signal has been injected.\label{fig:psi_predict_null}}
\end{figure}
% \begin{itemize}
% \item set up a marginalisation with a signal model adding parameters $A \cdot \mathrm{Normal}(\mu,\sigma)$
% \item perform marginalisation on data as already shown, and again with a signal injected
% \item discuss results and point out nice features like the fact that trials factor is already included, maybe point to~\cite{Bayer:2020pva}
% \end{itemize}
\section{Conclusion}
Bayesian inference, with a particular focus on evidence calculations, gives a principled framework to handle comparison between models. By including hyperparameters governing choice of model directly into a Bayesian sampling framework we can simultaneously perform model comparison whilst sampling appropriate values of the background model parameters. The predictions can be treated simply as a single band covering all the necessary uncertainty on this background model and propagated to further inference machinery.
A toy example of inferring a data driven background model for a Higgs measurement was used to demonstrate the machinery in a familiar collider physics use case. The marginalised Bayesian model gives a principled uncertainty in terms of all the nuisance parameters -- including discrete hyperparameters -- of the model, capturing both modelling and parameter uncertainty simultaneously. Precision measurements and searches stand to benefit from the precise uncertainty and improved weighting of functionally diverse candidate models.
The benefit for search analyses was demonstrated by extending the discrete marginalised background model to include a search for a generic anomaly using a Gaussian signal model. Calculation of evidence ratios in challenging scenarios reveals a well calibrated method for anomaly detection in particle physics spectra. This constructs a prototype analysis pipeline that can handle complex signal shapes alongside diverse data driven backgrounds. This pipeline presents a challenging sampling problem, but one that is handled well by the Nested Sampling tools employed in this analysis.
More broadly, evidence calculations and subsequent evidence weighted combinations of distinct models presents a statistically justifiable method to combine discrete model choices into a single unified model. Long standing comparisons in collider phenomenology, such as comparison of hadronization models or choice of hard process scale~\cite{Yallup:2022yxe}, can be viewed as a Bayesian model comparison using the framework presented in this work.
\section*{Acknowledgements}
This work was supported by a Royal Society University Research Fellowship and enhancement award. The authors gratefully acknowledge input and comments from Mike Hobson on the construction of this manuscript.
The \texttt{anesthetic} package~\cite{anesthetic} was used to appropriately manipulate the NS chain files to construct the figures throughout this work. The \texttt{fgivenx} package~\cite{fgivenx} was used to build the function error visualisations. A prototype of the analysis code is available at \href{https://github.com/yallup/discrete_marginal}{\texttt{[github.com/yallup/discrete\_marginal]}}
% This work was performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (\url{www.csd3.cam.ac.uk}), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/T022159/1), and DiRAC funding from the Science and Technology Facilities Council (\url{www.dirac.ac.uk}).
\bibliography{references}
\end{document}
% nested sampling macros
\newcommand{\niter}{n_\text{iter}}
\newcommand{\ndynamic}{n_\text{dynamic}}
\newcommand{\pvalue}{\text{\textit{p-}value}\xspace}
\newcommand{\pvalues}{\text{\pvalue{}s}\xspace}
\newcommand{\Pvalue}{\text{\textit{P-}value}\xspace}
\newcommand{\Pvalues}{\text{\Pvalue{}s}\xspace}
\newcommand{\Z}{\ensuremath{\mathcal{Z}}\xspace}
\newcommand{\logZ}{\ensuremath{\log\Z}\xspace}
\newcommand{\like}{\ensuremath{\mathcal{L}}\xspace}
\newcommand{\post}{\ensuremath{\mathcal{P}}\xspace}
\newcommand{\prior}{\ensuremath{\Pi}\xspace}
\newcommand{\threshold}{\like^\star}
\newcommand{\pg}[2]{p\mathopen{}\left(#1\,\rvert\, #2\right)\mathclose{}}
\newcommand{\Pg}[2]{P\mathopen{}\left(#1\,\rvert\, #2\right)\mathclose{}}
\newcommand{\p}[1]{p\mathopen{}\left(#1\right)\mathclose{}}
\newcommand{\Pp}[1]{P\mathopen{}\left(#1\right)\mathclose{}}
\newcommand{\intd}{\text{d}}
\newcommand{\sampleParams}{\mathbf{x}}
\newcommand{\modelParams}{\boldsymbol{\theta}}
\newcommand{\likei}{\ensuremath{\like_\textrm{I}}\xspace}
\newcommand{\likeii}{\ensuremath{\like_\textrm{II}}\xspace}
\newcommand{\likeiii}{\ensuremath{\like_\textrm{III}}\xspace}
\newcommand{\anyg}[3]{\ensuremath{#1\mathopen{}\left(#2\,\rvert\, #3\right)\mathclose{}}\xspace}
```
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W.~Handley, \emph{{Curvature tension: evidence for a closed universe}},
\href{https://doi.org/10.1103/PhysRevD.103.L041301}{\emph{Phys. Rev. D}
{\bfseries 103} (2021) L041301}
[\href{https://arxiv.org/abs/1908.09139}{{\ttfamily 1908.09139}}].
\bibitem{Alvarez:2022kfr}
E.~Alvarez, \emph{{Bayesian inference to study a signal with two or more
decaying particles in a non-resonant background}},
\href{https://arxiv.org/abs/2210.07358}{{\ttfamily 2210.07358}}.
\bibitem{Kasieczka:2021xcg}
G.~Kasieczka et~al., \emph{{The LHC Olympics 2020 a community challenge for
anomaly detection in high energy physics}},
\href{https://doi.org/10.1088/1361-6633/ac36b9}{\emph{Rept. Prog. Phys.}
{\bfseries 84} (2021) 124201}
[\href{https://arxiv.org/abs/2101.08320}{{\ttfamily 2101.08320}}].
\bibitem{Yallup:2022yxe}
D.~Yallup, T.~Jan\ss{}en, S.~Schumann and W.~Handley, \emph{{Exploring phase
space with Nested Sampling}},
\href{https://doi.org/10.1140/epjc/s10052-022-10632-2}{\emph{Eur. Phys. J. C}
{\bfseries 82} (2022) 8} [\href{https://arxiv.org/abs/2205.02030}{{\ttfamily
2205.02030}}].
\bibitem{anesthetic}
W.~Handley, \emph{anesthetic: nested sampling visualisation},
\href{https://doi.org/10.21105/joss.01414}{\emph{The Journal of Open Source
Software} {\bfseries 4} (2019) 1414}.
\bibitem{fgivenx}
W.~Handley, \emph{fgivenx: Functional posterior plotter},
\href{https://doi.org/10.21105/joss.00849}{\emph{The Journal of Open Source
Software} {\bfseries 3} (2018) }.
\end{thebibliography}\endgroup
```
5. **Author Information:**
- Lead Author: {'name': 'David Yallup'}
- Full Authors List:
```yaml
David Yallup:
postdoc:
start: 2021-01-10
thesis: null
original_image: images/originals/david_yallup.jpg
image: /assets/group/images/david_yallup.jpg
links:
ORCiD: https://orcid.org/0000-0003-4716-5817
linkedin: https://www.linkedin.com/in/dyallup/
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
```
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 [2211.10391](https://arxiv.org/abs/2211.10391) is featured in the first sentence.
Generate only the final Markdown output that meets all these requirements.
{% endraw %}