{% 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: "Marginal Bayesian Statistics Using Masked Autoregressive Flows and Kernel Density Estimators with Examples in Cosmology" date: 2022-07-23 categories: papers --- ![AI generated image](/assets/images/posts/2022-07-23-2207.11457.png) Harry BevinsWill HandleyEloy de Lera AcedoAnastasia Fialkov Content generated by [gemini-2.5-pro](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/2022-07-23-2207.11457.txt). Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/2022-07-23-2207.11457.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': 'Harry Bevins'}). 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 [2207.11457](https://arxiv.org/abs/2207.11457) is included in the first sentence. 5. **Final Formatting Requirements:** - The output must be plain Markdown; do not wrap it in Markdown code fences. - Preserve the YAML front matter exactly as provided. ==================================================================================== Section 2: Provided Data for Integration ==================================================================================== 1. **Homepage Content (Tone and Style Reference):** ```markdown --- layout: home --- ![AI generated image](/assets/images/index.png) The Handley Research Group stands at the forefront of cosmological exploration, pioneering novel approaches that fuse fundamental physics with the transformative power of artificial intelligence. We are a dynamic team of researchers, including PhD students, postdoctoral fellows, and project students, based at the University of Cambridge. Our mission is to unravel the mysteries of the Universe, from its earliest moments to its present-day structure and ultimate fate. We tackle fundamental questions in cosmology and astrophysics, with a particular focus on leveraging advanced Bayesian statistical methods and AI to push the frontiers of scientific discovery. Our research spans a wide array of topics, including the [primordial Universe](https://arxiv.org/abs/1907.08524), [inflation](https://arxiv.org/abs/1807.06211), the nature of [dark energy](https://arxiv.org/abs/2503.08658) and [dark matter](https://arxiv.org/abs/2405.17548), [21-cm cosmology](https://arxiv.org/abs/2210.07409), the [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1807.06209), and [gravitational wave astrophysics](https://arxiv.org/abs/2411.17663). ### Our Research Approach: Innovation at the Intersection of Physics and AI At The Handley Research Group, we develop and apply cutting-edge computational techniques to analyze complex astronomical datasets. Our work is characterized by a deep commitment to principled [Bayesian inference](https://arxiv.org/abs/2205.15570) and the innovative application of [artificial intelligence (AI) and machine learning (ML)](https://arxiv.org/abs/2504.10230). **Key Research Themes:** * **Cosmology:** We investigate the early Universe, including [quantum initial conditions for inflation](https://arxiv.org/abs/2002.07042) and the generation of [primordial power spectra](https://arxiv.org/abs/2112.07547). We explore the enigmatic nature of [dark energy, using methods like non-parametric reconstructions](https://arxiv.org/abs/2503.08658), and search for new insights into [dark matter](https://arxiv.org/abs/2405.17548). A significant portion of our efforts is dedicated to [21-cm cosmology](https://arxiv.org/abs/2104.04336), aiming to detect faint signals from the Cosmic Dawn and the Epoch of Reionization. * **Gravitational Wave Astrophysics:** We develop methods for [analyzing gravitational wave signals](https://arxiv.org/abs/2411.17663), extracting information about extreme astrophysical events and fundamental physics. * **Bayesian Methods & AI for Physical Sciences:** A core component of our research is the development of novel statistical and AI-driven methodologies. This includes advancing [nested sampling techniques](https://arxiv.org/abs/1506.00171) (e.g., [PolyChord](https://arxiv.org/abs/1506.00171), [dynamic nested sampling](https://arxiv.org/abs/1704.03459), and [accelerated nested sampling with $\beta$-flows](https://arxiv.org/abs/2411.17663)), creating powerful [simulation-based inference (SBI) frameworks](https://arxiv.org/abs/2504.10230), and employing [machine learning for tasks such as radiometer calibration](https://arxiv.org/abs/2504.16791), [cosmological emulation](https://arxiv.org/abs/2503.13263), and [mitigating radio frequency interference](https://arxiv.org/abs/2211.15448). We also explore the potential of [foundation models for scientific discovery](https://arxiv.org/abs/2401.00096). **Technical Contributions:** Our group has a strong track record of developing widely-used scientific software. Notable examples include: * [**PolyChord**](https://arxiv.org/abs/1506.00171): A next-generation nested sampling algorithm for Bayesian computation. * [**anesthetic**](https://arxiv.org/abs/1905.04768): A Python package for processing and visualizing nested sampling runs. * [**GLOBALEMU**](https://arxiv.org/abs/2104.04336): An emulator for the sky-averaged 21-cm signal. * [**maxsmooth**](https://arxiv.org/abs/2007.14970): A tool for rapid maximally smooth function fitting. * [**margarine**](https://arxiv.org/abs/2205.12841): For marginal Bayesian statistics using normalizing flows and KDEs. * [**fgivenx**](https://arxiv.org/abs/1908.01711): A package for functional posterior plotting. * [**nestcheck**](https://arxiv.org/abs/1804.06406): Diagnostic tests for nested sampling calculations. ### Impact and Discoveries Our research has led to significant advancements in cosmological data analysis and yielded new insights into the Universe. Key achievements include: * Pioneering the development and application of advanced Bayesian inference tools, such as [PolyChord](https://arxiv.org/abs/1506.00171), which has become a cornerstone for cosmological parameter estimation and model comparison globally. * Making significant contributions to the analysis of major cosmological datasets, including the [Planck mission](https://arxiv.org/abs/1807.06209), providing some of the tightest constraints on cosmological parameters and models of [inflation](https://arxiv.org/abs/1807.06211). * Developing novel AI-driven approaches for astrophysical challenges, such as using [machine learning for radiometer calibration in 21-cm experiments](https://arxiv.org/abs/2504.16791) and [simulation-based inference for extracting cosmological information from galaxy clusters](https://arxiv.org/abs/2504.10230). * Probing the nature of dark energy through innovative [non-parametric reconstructions of its equation of state](https://arxiv.org/abs/2503.08658) from combined datasets. * Advancing our understanding of the early Universe through detailed studies of [21-cm signals from the Cosmic Dawn and Epoch of Reionization](https://arxiv.org/abs/2301.03298), including the development of sophisticated foreground modelling techniques and emulators like [GLOBALEMU](https://arxiv.org/abs/2104.04336). * Developing new statistical methods for quantifying tensions between cosmological datasets ([Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio](https://arxiv.org/abs/1902.04029)) and for robust Bayesian model selection ([Bayesian model selection without evidences: application to the dark energy equation-of-state](https://arxiv.org/abs/1506.09024)). * Exploring fundamental physics questions such as potential [parity violation in the Large-Scale Structure using machine learning](https://arxiv.org/abs/2410.16030). ### Charting the Future: AI-Powered Cosmological Discovery The Handley Research Group is poised to lead a new era of cosmological analysis, driven by the explosive growth in data from next-generation observatories and transformative advances in artificial intelligence. Our future ambitions are centred on harnessing these capabilities to address the most pressing questions in fundamental physics. **Strategic Research Pillars:** * **Next-Generation Simulation-Based Inference (SBI):** We are developing advanced SBI frameworks to move beyond traditional likelihood-based analyses. This involves creating sophisticated codes for simulating [Cosmic Microwave Background (CMB)](https://arxiv.org/abs/1908.00906) and [Baryon Acoustic Oscillation (BAO)](https://arxiv.org/abs/1607.00270) datasets from surveys like DESI and 4MOST, incorporating realistic astrophysical effects and systematic uncertainties. Our AI initiatives in this area focus on developing and implementing cutting-edge SBI algorithms, particularly [neural ratio estimation (NRE) methods](https://arxiv.org/abs/2407.15478), to enable robust and scalable inference from these complex simulations. * **Probing Fundamental Physics:** Our enhanced analytical toolkit will be deployed to test the standard cosmological model ($\Lambda$CDM) with unprecedented precision and to explore [extensions to Einstein's General Relativity](https://arxiv.org/abs/2006.03581). We aim to constrain a wide range of theoretical models, from modified gravity to the nature of [dark matter](https://arxiv.org/abs/2106.02056) and [dark energy](https://arxiv.org/abs/1701.08165). This includes leveraging data from upcoming [gravitational wave observatories](https://arxiv.org/abs/1803.10210) like LISA, alongside CMB and large-scale structure surveys from facilities such as Euclid and JWST. * **Synergies with Particle Physics:** We will continue to strengthen the connection between cosmology and particle physics by expanding the [GAMBIT framework](https://arxiv.org/abs/2009.03286) to interface with our new SBI tools. This will facilitate joint analyses of cosmological and particle physics data, providing a holistic approach to understanding the Universe's fundamental constituents. * **AI-Driven Theoretical Exploration:** We are pioneering the use of AI, including [large language models and symbolic computation](https://arxiv.org/abs/2401.00096), to automate and accelerate the process of theoretical model building and testing. This innovative approach will allow us to explore a broader landscape of physical theories and derive new constraints from diverse astrophysical datasets, such as those from GAIA. Our overarching goal is to remain at the forefront of scientific discovery by integrating the latest AI advancements into every stage of our research, from theoretical modeling to data analysis and interpretation. We are excited by the prospect of using these powerful new tools to unlock the secrets of the cosmos. Content generated by [gemini-2.5-pro-preview-05-06](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/content/index.txt). Image generated by [imagen-3.0-generate-002](https://deepmind.google/technologies/gemini/) using [this prompt](/prompts/images/index.txt). ``` 2. **Paper Metadata:** ```yaml !!python/object/new:feedparser.util.FeedParserDict dictitems: id: http://arxiv.org/abs/2207.11457v3 guidislink: true link: http://arxiv.org/abs/2207.11457v3 updated: '2022-11-24T14:33:11Z' updated_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2022 - 11 - 24 - 14 - 33 - 11 - 3 - 328 - 0 - tm_zone: null tm_gmtoff: null published: '2022-07-23T08:24:50Z' published_parsed: !!python/object/apply:time.struct_time - !!python/tuple - 2022 - 7 - 23 - 8 - 24 - 50 - 5 - 204 - 0 - tm_zone: null tm_gmtoff: null title: "Marginal Bayesian Statistics Using Masked Autoregressive Flows and\n Kernel\ \ Density Estimators with Examples in Cosmology" title_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: "Marginal Bayesian Statistics Using Masked Autoregressive Flows and\n\ \ Kernel Density Estimators with Examples in Cosmology" summary: 'Cosmological experiments often employ Bayesian workflows to derive constraints on cosmological and astrophysical parameters from their data. It has been shown that these constraints can be combined across different probes such as Planck and the Dark Energy Survey and that this can be a valuable exercise to improve our understanding of the universe and quantify tension between multiple experiments. However, these experiments are typically plagued by differing systematics, instrumental effects and contaminating signals, which we collectively refer to as `nuisance'' components, that have to be modelled alongside target signals of interest. This leads to high dimensional parameter spaces, especially when combining data sets, with > 20 dimensions of which only around 5 correspond to key physical quantities. We present a means by which to combine constraints from different data sets in a computationally efficient manner by generating rapid, reusable and reliable marginal probability density estimators, giving us access to nuisance-free likelihoods. This is possible through the unique combination of nested sampling, which gives us access to Bayesian evidences, and the marginal Bayesian statistics code MARGARINE. Our method is lossless in the signal parameters, resulting in the same posterior distributions as would be found from a full nested sampling run over all nuisance parameters, and typically quicker than evaluating full likelihoods. We demonstrate our approach by applying it to the combination of posteriors from the Dark Energy Survey and Planck.' summary_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: type: text/plain language: null base: '' value: 'Cosmological experiments often employ Bayesian workflows to derive constraints on cosmological and astrophysical parameters from their data. It has been shown that these constraints can be combined across different probes such as Planck and the Dark Energy Survey and that this can be a valuable exercise to improve our understanding of the universe and quantify tension between multiple experiments. However, these experiments are typically plagued by differing systematics, instrumental effects and contaminating signals, which we collectively refer to as `nuisance'' components, that have to be modelled alongside target signals of interest. This leads to high dimensional parameter spaces, especially when combining data sets, with > 20 dimensions of which only around 5 correspond to key physical quantities. We present a means by which to combine constraints from different data sets in a computationally efficient manner by generating rapid, reusable and reliable marginal probability density estimators, giving us access to nuisance-free likelihoods. This is possible through the unique combination of nested sampling, which gives us access to Bayesian evidences, and the marginal Bayesian statistics code MARGARINE. Our method is lossless in the signal parameters, resulting in the same posterior distributions as would be found from a full nested sampling run over all nuisance parameters, and typically quicker than evaluating full likelihoods. We demonstrate our approach by applying it to the combination of posteriors from the Dark Energy Survey and Planck.' authors: - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Harry Bevins - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Will Handley - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Pablo Lemos - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Peter Sims - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Eloy de Lera Acedo - !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Anastasia Fialkov author_detail: !!python/object/new:feedparser.util.FeedParserDict dictitems: name: Anastasia Fialkov author: Anastasia Fialkov arxiv_doi: 10.3390/psf2022005001 links: - !!python/object/new:feedparser.util.FeedParserDict dictitems: title: doi href: http://dx.doi.org/10.3390/psf2022005001 rel: related type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: href: http://arxiv.org/abs/2207.11457v3 rel: alternate type: text/html - !!python/object/new:feedparser.util.FeedParserDict dictitems: title: pdf href: http://arxiv.org/pdf/2207.11457v3 rel: related type: application/pdf arxiv_comment: "Published in Phys. Sci. Forum 2022, 5(1), 1;\n https://doi.org/10.3390/psf2022005001" arxiv_primary_category: term: astro-ph.CO scheme: http://arxiv.org/schemas/atom tags: - !!python/object/new:feedparser.util.FeedParserDict dictitems: term: astro-ph.CO scheme: http://arxiv.org/schemas/atom label: null - !!python/object/new:feedparser.util.FeedParserDict dictitems: term: astro-ph.IM scheme: http://arxiv.org/schemas/atom label: null ``` 3. **Paper Source (TeX):** ```tex \DeclareOption{acoustics}{ \gdef\@journal{acoustics} \gdef\@journalshort{Acoustics} \gdef\@journalfull{Acoustics} \gdef\@doiabbr{acoustics} \gdef\@ISSN{2624-599X} } \DeclareOption{actuators}{ \gdef\@journal{actuators} \gdef\@journalshort{Actuators} \gdef\@journalfull{Actuators} \gdef\@doiabbr{act} \gdef\@ISSN{2076-0825} } \DeclareOption{addictions}{ \gdef\@journal{addictions} \gdef\@journalshort{Addictions} \gdef\@journalfull{Addictions} \gdef\@doiabbr{} \gdef\@ISSN{0006-0006} } \DeclareOption{admsci}{ \gdef\@journal{admsci} \gdef\@journalshort{Adm. Sci.} \gdef\@journalfull{Administrative Sciences} \gdef\@doiabbr{admsci} \gdef\@ISSN{2076-3387} } \DeclareOption{aerospace}{ \gdef\@journal{aerospace} \gdef\@journalshort{Aerospace} \gdef\@journalfull{Aerospace} \gdef\@doiabbr{aerospace} \gdef\@ISSN{2226-4310} } \DeclareOption{agriculture}{ \gdef\@journal{agriculture} \gdef\@journalshort{Agriculture} \gdef\@journalfull{Agriculture} \gdef\@doiabbr{agriculture} \gdef\@ISSN{2077-0472} } \DeclareOption{agriengineering}{ \gdef\@journal{agriengineering} \gdef\@journalshort{AgriEngineering} \gdef\@journalfull{AgriEngineering} \gdef\@doiabbr{agriengineering} \gdef\@ISSN{2624-7402} } \DeclareOption{agronomy}{ \gdef\@journal{agronomy} \gdef\@journalshort{Agronomy} \gdef\@journalfull{Agronomy} \gdef\@doiabbr{agronomy} \gdef\@ISSN{2073-4395} } \DeclareOption{algorithms}{ \gdef\@journal{algorithms} \gdef\@journalshort{Algorithms} \gdef\@journalfull{Algorithms} \gdef\@doiabbr{a} \gdef\@ISSN{1999-4893} } \DeclareOption{animals}{ \gdef\@journal{animals} \gdef\@journalshort{Animals} \gdef\@journalfull{Animals} \gdef\@doiabbr{ani} \gdef\@ISSN{2076-2615} } \DeclareOption{antibiotics}{ \gdef\@journal{antibiotics} \gdef\@journalshort{Antibiotics} \gdef\@journalfull{Antibiotics} \gdef\@doiabbr{antibiotics} \gdef\@ISSN{2079-6382} } \DeclareOption{antibodies}{ \gdef\@journal{antibodies} \gdef\@journalshort{Antibodies} \gdef\@journalfull{Antibodies} \gdef\@doiabbr{antib} \gdef\@ISSN{2073-4468} } \DeclareOption{antioxidants}{ \gdef\@journal{antioxidants} \gdef\@journalshort{Antioxidants} \gdef\@journalfull{Antioxidants} \gdef\@doiabbr{antiox} \gdef\@ISSN{2076-3921} } \DeclareOption{applsci}{ \gdef\@journal{applsci} \gdef\@journalshort{Appl. Sci.} \gdef\@journalfull{Applied Sciences} \gdef\@doiabbr{app} \gdef\@ISSN{2076-3417} } \DeclareOption{arts}{ \gdef\@journal{arts} \gdef\@journalshort{Arts} \gdef\@journalfull{Arts} \gdef\@doiabbr{arts} \gdef\@ISSN{2076-0752} } \DeclareOption{asc}{ \gdef\@journal{asc} \gdef\@journalshort{Autom. Syst. Control} \gdef\@journalfull{Automatic Systems and Control} \gdef\@doiabbr{} \gdef\@ISSN{} } \DeclareOption{asi}{ \gdef\@journal{asi} \gdef\@journalshort{Appl. Syst. Innov.} \gdef\@journalfull{Applied System Innovation} \gdef\@doiabbr{asi} \gdef\@ISSN{2571-5577} } \DeclareOption{atmosphere}{ \gdef\@journal{atmosphere} \gdef\@journalshort{Atmosphere} \gdef\@journalfull{Atmosphere} \gdef\@doiabbr{atmos} \gdef\@ISSN{2073-4433} } \DeclareOption{atoms}{ \gdef\@journal{atoms} \gdef\@journalshort{Atoms} \gdef\@journalfull{Atoms} \gdef\@doiabbr{atoms} \gdef\@ISSN{2218-2004} } \DeclareOption{axioms}{ \gdef\@journal{axioms} \gdef\@journalshort{Axioms} \gdef\@journalfull{Axioms} \gdef\@doiabbr{axioms} \gdef\@ISSN{2075-1680} } \DeclareOption{batteries}{ \gdef\@journal{batteries} \gdef\@journalshort{Batteries} \gdef\@journalfull{Batteries} \gdef\@doiabbr{batteries} \gdef\@ISSN{2313-0105} } \DeclareOption{bdcc}{ \gdef\@journal{bdcc} \gdef\@journalshort{Big Data Cogn. Comput.} \gdef\@journalfull{Big Data and Cognitive Computing} \gdef\@doiabbr{bdcc} \gdef\@ISSN{2504-2289} } \DeclareOption{behavsci}{ \gdef\@journal{behavsci} \gdef\@journalshort{Behav. Sci.} \gdef\@journalfull{Behavioral Sciences} \gdef\@doiabbr{bs} \gdef\@ISSN{2076-328X} } \DeclareOption{beverages}{ \gdef\@journal{beverages} \gdef\@journalshort{Beverages} \gdef\@journalfull{Beverages} \gdef\@doiabbr{beverages} \gdef\@ISSN{2306-5710} } \DeclareOption{bioengineering}{ \gdef\@journal{bioengineering} \gdef\@journalshort{Bioengineering} \gdef\@journalfull{Bioengineering} \gdef\@doiabbr{bioengineering} \gdef\@ISSN{2306-5354} } \DeclareOption{biology}{ \gdef\@journal{biology} \gdef\@journalshort{Biology} \gdef\@journalfull{Biology} \gdef\@doiabbr{biology} \gdef\@ISSN{2079-7737} } \DeclareOption{biomedicines}{ \gdef\@journal{biomedicines} \gdef\@journalshort{Biomedicines} \gdef\@journalfull{Biomedicines} \gdef\@doiabbr{biomedicines} \gdef\@ISSN{2227-9059} } \DeclareOption{biomimetics}{ \gdef\@journal{biomimetics} \gdef\@journalshort{Biomimetics} \gdef\@journalfull{Biomimetics} \gdef\@doiabbr{biomimetics} \gdef\@ISSN{2313-7673} } \DeclareOption{biomolecules}{ \gdef\@journal{biomolecules} \gdef\@journalshort{Biomolecules} \gdef\@journalfull{Biomolecules} \gdef\@doiabbr{biom} \gdef\@ISSN{2218-273X} } \DeclareOption{biosensors}{ \gdef\@journal{biosensors} \gdef\@journalshort{Biosensors} \gdef\@journalfull{Biosensors} \gdef\@doiabbr{bios} \gdef\@ISSN{2079-6374} } \DeclareOption{brainsci}{ \gdef\@journal{brainsci} \gdef\@journalshort{Brain Sci.} \gdef\@journalfull{Brain Sciences} \gdef\@doiabbr{brainsci} \gdef\@ISSN{2076-3425} } \DeclareOption{buildings}{ \gdef\@journal{buildings} \gdef\@journalshort{Buildings} \gdef\@journalfull{Buildings} \gdef\@doiabbr{buildings} \gdef\@ISSN{2075-5309} } \DeclareOption{cancers}{ \gdef\@journal{cancers} \gdef\@journalshort{Cancers} \gdef\@journalfull{Cancers} \gdef\@doiabbr{cancers} \gdef\@ISSN{2072-6694} } \DeclareOption{carbon}{ \gdef\@journal{carbon} \gdef\@journalshort{C} \gdef\@journalfull{C} \gdef\@doiabbr{c} \gdef\@ISSN{2311-5629} } \DeclareOption{catalysts}{ \gdef\@journal{catalysts} \gdef\@journalshort{Catalysts} \gdef\@journalfull{Catalysts} \gdef\@doiabbr{catal} \gdef\@ISSN{2073-4344} } \DeclareOption{cells}{ \gdef\@journal{cells} \gdef\@journalshort{Cells} \gdef\@journalfull{Cells} \gdef\@doiabbr{cells} \gdef\@ISSN{2073-4409} } \DeclareOption{ceramics}{ \gdef\@journal{ceramics} \gdef\@journalshort{Ceramics} \gdef\@journalfull{Ceramics} \gdef\@doiabbr{ceramics} \gdef\@ISSN{2571-6131} } \DeclareOption{challenges}{ \gdef\@journal{challenges} \gdef\@journalshort{Challenges} \gdef\@journalfull{Challenges} \gdef\@doiabbr{challe} \gdef\@ISSN{2078-1547} } \DeclareOption{chemengineering}{ \gdef\@journal{chemengineering} \gdef\@journalshort{ChemEngineering} \gdef\@journalfull{ChemEngineering} \gdef\@doiabbr{chemengineering} \gdef\@ISSN{2305-7084} } \DeclareOption{chemistry}{ \gdef\@journal{chemistry} \gdef\@journalshort{Chemistry} \gdef\@journalfull{Chemistry} \gdef\@doiabbr{chemistry} \gdef\@ISSN{2624-8549} } \DeclareOption{chemosensors}{ \gdef\@journal{chemosensors} \gdef\@journalshort{Chemosensors} \gdef\@journalfull{Chemosensors} \gdef\@doiabbr{chemosensors} \gdef\@ISSN{2227-9040} } \DeclareOption{children}{ \gdef\@journal{children} \gdef\@journalshort{Children} \gdef\@journalfull{Children} \gdef\@doiabbr{children} \gdef\@ISSN{2227-9067} } \DeclareOption{cleantechnol}{ \gdef\@journal{cleantechnol} \gdef\@journalshort{Clean Technol.} \gdef\@journalfull{Clean Technologies} \gdef\@doiabbr{cleantechnol} \gdef\@ISSN{2571-8797} } \DeclareOption{climate}{ \gdef\@journal{climate} \gdef\@journalshort{Climate} \gdef\@journalfull{Climate} \gdef\@doiabbr{cli} \gdef\@ISSN{2225-1154} } \DeclareOption{clockssleep}{ \gdef\@journal{clockssleep} \gdef\@journalshort{Clocks\&Sleep} \gdef\@journalfull{Clocks \& Sleep} \gdef\@doiabbr{clockssleep} \gdef\@ISSN{2624-5175} } \DeclareOption{cmd}{ \gdef\@journal{cmd} \gdef\@journalshort{Corros. Mater. Degrad.} \gdef\@journalfull{Corrosion and Materials Degradation} \gdef\@doiabbr{cmd} \gdef\@ISSN{2624-5558} } \DeclareOption{coatings}{ \gdef\@journal{coatings} \gdef\@journalshort{Coatings} \gdef\@journalfull{Coatings} \gdef\@doiabbr{coatings} \gdef\@ISSN{2079-6412} } \DeclareOption{colloids}{ \gdef\@journal{colloids} \gdef\@journalshort{Colloids Interfaces} \gdef\@journalfull{Colloids Interfaces} \gdef\@doiabbr{colloids} \gdef\@ISSN{2504-5377} } \DeclareOption{computation}{ \gdef\@journal{computation} \gdef\@journalshort{Computation} \gdef\@journalfull{Computation} \gdef\@doiabbr{computation} \gdef\@ISSN{2079-3197} } \DeclareOption{computers}{ \gdef\@journal{computers} \gdef\@journalshort{Computers} \gdef\@journalfull{Computers} \gdef\@doiabbr{computers} \gdef\@ISSN{2073-431X} } \DeclareOption{condensedmatter}{ \gdef\@journal{condensedmatter} \gdef\@journalshort{Condens. Matter} \gdef\@journalfull{Condensed Matter} \gdef\@doiabbr{condmat} \gdef\@ISSN{2410-3896} } \DeclareOption{cosmetics}{ \gdef\@journal{cosmetics} \gdef\@journalshort{Cosmetics} \gdef\@journalfull{Cosmetics} \gdef\@doiabbr{cosmetics} \gdef\@ISSN{2079-9284} } \DeclareOption{cryptography}{ \gdef\@journal{cryptography} \gdef\@journalshort{Cryptography} \gdef\@journalfull{Cryptography} \gdef\@doiabbr{cryptography} \gdef\@ISSN{2410-387X} } \DeclareOption{crystals}{ \gdef\@journal{crystals} \gdef\@journalshort{Crystals} \gdef\@journalfull{Crystals} \gdef\@doiabbr{cryst} \gdef\@ISSN{2073-4352} } \DeclareOption{dairy}{ \gdef\@journal{dairy} \gdef\@journalshort{Dairy} \gdef\@journalfull{Dairy} \gdef\@doiabbr{dairy} \gdef\@ISSN{2624-862X} } \DeclareOption{data}{ \gdef\@journal{data} \gdef\@journalshort{Data} \gdef\@journalfull{Data} \gdef\@doiabbr{data} \gdef\@ISSN{2306-5729} } \DeclareOption{dentistry}{ \gdef\@journal{dentistry} \gdef\@journalshort{Dent. J.} \gdef\@journalfull{Dentistry Journal} \gdef\@doiabbr{dj} \gdef\@ISSN{2304-6767} } \DeclareOption{designs}{ \gdef\@journal{designs} \gdef\@journalshort{Designs} \gdef\@journalfull{Designs} \gdef\@doiabbr{designs} \gdef\@ISSN{2411-9660} } \DeclareOption{diagnostics}{ \gdef\@journal{diagnostics} \gdef\@journalshort{Diagnostics} \gdef\@journalfull{Diagnostics} \gdef\@doiabbr{diagnostics} \gdef\@ISSN{2075-4418} } \DeclareOption{diseases}{ \gdef\@journal{diseases} \gdef\@journalshort{Diseases} \gdef\@journalfull{Diseases} \gdef\@doiabbr{diseases} \gdef\@ISSN{2079-9721} } \DeclareOption{diversity}{ \gdef\@journal{diversity} \gdef\@journalshort{Diversity} \gdef\@journalfull{Diversity} \gdef\@doiabbr{d} \gdef\@ISSN{1424-2818} } \DeclareOption{drones}{ \gdef\@journal{drones} \gdef\@journalshort{Drones} \gdef\@journalfull{Drones} \gdef\@doiabbr{drones} \gdef\@ISSN{2504-446X} } \DeclareOption{econometrics}{ \gdef\@journal{econometrics} \gdef\@journalshort{Econometrics} \gdef\@journalfull{Econometrics} \gdef\@doiabbr{econometrics} \gdef\@ISSN{2225-1146} } \DeclareOption{economies}{ \gdef\@journal{economies} \gdef\@journalshort{Economies} \gdef\@journalfull{Economies} \gdef\@doiabbr{economies} \gdef\@ISSN{2227-7099} } \DeclareOption{education}{ \gdef\@journal{education} \gdef\@journalshort{Educ. Sci.} \gdef\@journalfull{Education Sciences} \gdef\@doiabbr{educsci} \gdef\@ISSN{2227-7102} } \DeclareOption{ejihpe}{ \gdef\@journal{ejihpe} \gdef\@journalshort{Eur. J. Investig. Health Psychol. Educ.} \gdef\@journalfull{European Journal of Investigation in Health, Psychology and Education} \gdef\@doiabbr{ejihpe} \gdef\@ISSN{2254-9625} } \DeclareOption{electrochem}{ \gdef\@journal{electrochem} \gdef\@journalshort{Electrochem} \gdef\@journalfull{Electrochem} \gdef\@doiabbr{electrochem} \gdef\@ISSN{} } \DeclareOption{electronics}{ \gdef\@journal{electronics} \gdef\@journalshort{Electronics} \gdef\@journalfull{Electronics} \gdef\@doiabbr{electronics} \gdef\@ISSN{2079-9292} } \DeclareOption{energies}{ \gdef\@journal{energies} \gdef\@journalshort{Energies} \gdef\@journalfull{Energies} \gdef\@doiabbr{en} \gdef\@ISSN{1996-1073} } \DeclareOption{entropy}{ \gdef\@journal{entropy} \gdef\@journalshort{Entropy} \gdef\@journalfull{Entropy} \gdef\@doiabbr{e} \gdef\@ISSN{1099-4300} } \DeclareOption{environments}{ \gdef\@journal{environments} \gdef\@journalshort{Environments} \gdef\@journalfull{Environments} \gdef\@doiabbr{environments} \gdef\@ISSN{2076-3298} } \DeclareOption{epigenomes}{ \gdef\@journal{epigenomes} \gdef\@journalshort{Epigenomes} \gdef\@journalfull{Epigenomes} \gdef\@doiabbr{epigenomes} \gdef\@ISSN{2075-4655} } \DeclareOption{est}{ \gdef\@journal{est} \gdef\@journalshort{Electrochem. Sci. Technol.} \gdef\@journalfull{Electrochemical Science and Technology} \gdef\@doiabbr{} \gdef\@ISSN{} } \DeclareOption{fermentation}{ \gdef\@journal{fermentation} \gdef\@journalshort{Fermentation} \gdef\@journalfull{Fermentation} \gdef\@doiabbr{fermentation} \gdef\@ISSN{2311-5637} } \DeclareOption{fibers}{ \gdef\@journal{fibers} \gdef\@journalshort{Fibers} \gdef\@journalfull{Fibers} \gdef\@doiabbr{fib} \gdef\@ISSN{2079-6439} } \DeclareOption{fire}{ \gdef\@journal{fire} \gdef\@journalshort{Fire} \gdef\@journalfull{Fire} \gdef\@doiabbr{fire} \gdef\@ISSN{2571-6255} } \DeclareOption{fishes}{ \gdef\@journal{fishes} \gdef\@journalshort{Fishes} \gdef\@journalfull{Fishes} \gdef\@doiabbr{fishes} \gdef\@ISSN{2410-3888} } \DeclareOption{fluids}{ \gdef\@journal{fluids} \gdef\@journalshort{Fluids} \gdef\@journalfull{Fluids} \gdef\@doiabbr{fluids} \gdef\@ISSN{2311-5521} } \DeclareOption{foods}{ \gdef\@journal{foods} \gdef\@journalshort{Foods} \gdef\@journalfull{Foods} \gdef\@doiabbr{foods} \gdef\@ISSN{2304-8158} } \DeclareOption{forecasting}{ \gdef\@journal{forecasting} \gdef\@journalshort{Forecasting} \gdef\@journalfull{Forecasting} \gdef\@doiabbr{forecast} \gdef\@ISSN{2571-9394} } \DeclareOption{forests}{ \gdef\@journal{forests} \gdef\@journalshort{Forests} \gdef\@journalfull{Forests} \gdef\@doiabbr{f} \gdef\@ISSN{1999-4907} } \DeclareOption{fractalfract}{ \gdef\@journal{fractalfract} \gdef\@journalshort{Fractal Fract.} \gdef\@journalfull{Fractal and Fractional} \gdef\@doiabbr{fractalfract} \gdef\@ISSN{2504-3110} } \DeclareOption{futureinternet}{ \gdef\@journal{futureinternet} \gdef\@journalshort{Future Internet} \gdef\@journalfull{Future Internet} \gdef\@doiabbr{fi} \gdef\@ISSN{1999-5903} } \DeclareOption{futurephys}{ \gdef\@journal{futurephys} \gdef\@journalshort{Future Phys.} \gdef\@journalfull{Future Physics} \gdef\@doiabbr{futurephys} \gdef\@ISSN{2624-6503} } \DeclareOption{galaxies}{ \gdef\@journal{galaxies} \gdef\@journalshort{Galaxies} \gdef\@journalfull{Galaxies} \gdef\@doiabbr{galaxies} \gdef\@ISSN{2075-4434} } \DeclareOption{games}{ \gdef\@journal{games} \gdef\@journalshort{Games} \gdef\@journalfull{Games} \gdef\@doiabbr{g} \gdef\@ISSN{2073-4336} } \DeclareOption{gastrointestdisord}{ \gdef\@journal{gastrointestdisord} \gdef\@journalshort{Gastrointest. Disord.} \gdef\@journalfull{Gastrointestinal Disorders} \gdef\@doiabbr{gidisord} \gdef\@ISSN{2624-5647} } \DeclareOption{gels}{ \gdef\@journal{gels} \gdef\@journalshort{Gels} \gdef\@journalfull{Gels} \gdef\@doiabbr{gels} \gdef\@ISSN{2310-2861} } \DeclareOption{genealogy}{ \gdef\@journal{genealogy} \gdef\@journalshort{Genealogy} \gdef\@journalfull{Genealogy} \gdef\@doiabbr{genealogy} \gdef\@ISSN{2313-5778} } \DeclareOption{genes}{ \gdef\@journal{genes} \gdef\@journalshort{Genes} \gdef\@journalfull{Genes} \gdef\@doiabbr{genes} \gdef\@ISSN{2073-4425} } \DeclareOption{geohazards}{ \gdef\@journal{geohazards} \gdef\@journalshort{GeoHazards} \gdef\@journalfull{GeoHazards} \gdef\@doiabbr{geohazards} \gdef\@ISSN{2624-795X} } \DeclareOption{geosciences}{ \gdef\@journal{geosciences} \gdef\@journalshort{Geosciences} \gdef\@journalfull{Geosciences} \gdef\@doiabbr{geosciences} \gdef\@ISSN{2076-3263} } \DeclareOption{geriatrics}{ \gdef\@journal{geriatrics} \gdef\@journalshort{Geriatrics} \gdef\@journalfull{Geriatrics} \gdef\@doiabbr{geriatrics} \gdef\@ISSN{2308-3417} } \DeclareOption{hazardousmatters}{ \gdef\@journal{hazardousmatters} \gdef\@journalshort{Hazard. Matters} \gdef\@journalfull{Hazardous Matters} \gdef\@doiabbr{} \gdef\@ISSN{0014-0014} } \DeclareOption{healthcare}{ \gdef\@journal{healthcare} \gdef\@journalshort{Healthcare} \gdef\@journalfull{Healthcare} \gdef\@doiabbr{healthcare} \gdef\@ISSN{2227-9032} } \DeclareOption{heritage}{ \gdef\@journal{heritage} \gdef\@journalshort{Heritage} \gdef\@journalfull{Heritage} \gdef\@doiabbr{heritage} \gdef\@ISSN{2571-9408} } \DeclareOption{highthroughput}{ \gdef\@journal{highthroughput} \gdef\@journalshort{High-Throughput} \gdef\@journalfull{High-Throughput} \gdef\@doiabbr{ht} \gdef\@ISSN{2571-5135} } \DeclareOption{horticulturae}{ \gdef\@journal{horticulturae} \gdef\@journalshort{Horticulturae} \gdef\@journalfull{Horticulturae} \gdef\@doiabbr{horticulturae} \gdef\@ISSN{2311-7524} } \DeclareOption{humanities}{ \gdef\@journal{humanities} \gdef\@journalshort{Humanities} \gdef\@journalfull{Humanities} \gdef\@doiabbr{h} \gdef\@ISSN{2076-0787} } \DeclareOption{hydrology}{ \gdef\@journal{hydrology} \gdef\@journalshort{Hydrology} \gdef\@journalfull{Hydrology} \gdef\@doiabbr{hydrology} \gdef\@ISSN{2306-5338} } \DeclareOption{ijerph}{ \gdef\@journal{ijerph} \gdef\@journalshort{Int. J. Environ. Res. Public Health} \gdef\@journalfull{International Journal of Environmental Research and Public Health} \gdef\@doiabbr{ijerph} \gdef\@ISSN{1660-4601} } \DeclareOption{ijfs}{ \gdef\@journal{ijfs} \gdef\@journalshort{Int. J. Financial Stud.} \gdef\@journalfull{International Journal of Financial Studies} \gdef\@doiabbr{ijfs} \gdef\@ISSN{2227-7072} } \DeclareOption{ijgi}{ \gdef\@journal{ijgi} \gdef\@journalshort{ISPRS Int. J. Geo-Inf.} \gdef\@journalfull{ISPRS International Journal of Geo-Information} \gdef\@doiabbr{ijgi} \gdef\@ISSN{2220-9964} } \DeclareOption{ijms}{ \gdef\@journal{ijms} \gdef\@journalshort{Int. J. Mol. Sci.} \gdef\@journalfull{International Journal of Molecular Sciences} \gdef\@doiabbr{ijms} \gdef\@ISSN{1422-0067} } \DeclareOption{ijtpp}{ \gdef\@journal{ijtpp} \gdef\@journalshort{Int. J. Turbomach. Propuls. Power} \gdef\@journalfull{International Journal of Turbomachinery, Propulsion and Power} \gdef\@doiabbr{ijtpp} \gdef\@ISSN{2504-186X} } \DeclareOption{informatics}{ \gdef\@journal{informatics} \gdef\@journalshort{Informatics} \gdef\@journalfull{Informatics} \gdef\@doiabbr{informatics} \gdef\@ISSN{2227-9709} } \DeclareOption{information}{ \gdef\@journal{information} \gdef\@journalshort{Information} \gdef\@journalfull{Information} \gdef\@doiabbr{info} \gdef\@ISSN{2078-2489} } \DeclareOption{infrastructures}{ \gdef\@journal{infrastructures} \gdef\@journalshort{Infrastructures} \gdef\@journalfull{Infrastructures} \gdef\@doiabbr{infrastructures} \gdef\@ISSN{2412-3811} } \DeclareOption{inorganics}{ \gdef\@journal{inorganics} \gdef\@journalshort{Inorganics} \gdef\@journalfull{Inorganics} \gdef\@doiabbr{inorganics} \gdef\@ISSN{2304-6740} } \DeclareOption{insects}{ \gdef\@journal{insects} \gdef\@journalshort{Insects} \gdef\@journalfull{Insects} \gdef\@doiabbr{insects} \gdef\@ISSN{2075-4450} } \DeclareOption{instruments}{ \gdef\@journal{instruments} \gdef\@journalshort{Instruments} \gdef\@journalfull{Instruments} \gdef\@doiabbr{instruments} \gdef\@ISSN{2410-390X} } \DeclareOption{inventions}{ \gdef\@journal{inventions} \gdef\@journalshort{Inventions} \gdef\@journalfull{Inventions} \gdef\@doiabbr{inventions} \gdef\@ISSN{2411-5134} } \DeclareOption{iot}{ \gdef\@journal{iot} \gdef\@journalshort{IoT} \gdef\@journalfull{IoT} \gdef\@doiabbr{iot} \gdef\@ISSN{2624-831X} } \DeclareOption{j}{ \gdef\@journal{j} \gdef\@journalshort{J} \gdef\@journalfull{J} \gdef\@doiabbr{j} \gdef\@ISSN{2571-8800} } \DeclareOption{jcdd}{ \gdef\@journal{jcdd} \gdef\@journalshort{J. Cardiovasc. Dev. Dis.} \gdef\@journalfull{Journal of Cardiovascular Development and Disease} \gdef\@doiabbr{jcdd} \gdef\@ISSN{2308-3425} } \DeclareOption{jcm}{ \gdef\@journal{jcm} \gdef\@journalshort{J. Clin. Med.} \gdef\@journalfull{Journal of Clinical Medicine} \gdef\@doiabbr{jcm} \gdef\@ISSN{2077-0383} } \DeclareOption{jcp}{ \gdef\@journal{jcp} \gdef\@journalshort{J. Cybersecur. Priv.} \gdef\@journalfull{Journal of Cybersecurity and Privacy} \gdef\@doiabbr{jcp} \gdef\@ISSN{2624-800X} } \DeclareOption{jcs}{ \gdef\@journal{jcs} \gdef\@journalshort{J. Compos. Sci.} \gdef\@journalfull{Journal of Composites Science} \gdef\@doiabbr{jcs} \gdef\@ISSN{2504-477X} } \DeclareOption{jdb}{ \gdef\@journal{jdb} \gdef\@journalshort{J. Dev. Biol.} \gdef\@journalfull{Journal of Developmental Biology} \gdef\@doiabbr{jdb} \gdef\@ISSN{2221-3759} } \DeclareOption{jfb}{ \gdef\@journal{jfb} \gdef\@journalshort{J. Funct. Biomater.} \gdef\@journalfull{Journal of Functional Biomaterials} \gdef\@doiabbr{jfb} \gdef\@ISSN{2079-4983} } \DeclareOption{jfmk}{ \gdef\@journal{jfmk} \gdef\@journalshort{J. Funct. Morphol. Kinesiol.} \gdef\@journalfull{Journal of Functional Morphology and Kinesiology} \gdef\@doiabbr{jfmk} \gdef\@ISSN{2411-5142} } \DeclareOption{jimaging}{ \gdef\@journal{jimaging} \gdef\@journalshort{J. Imaging} \gdef\@journalfull{Journal of Imaging} \gdef\@doiabbr{jimaging} \gdef\@ISSN{2313-433X} } \DeclareOption{jintelligence}{ \gdef\@journal{jintelligence} \gdef\@journalshort{J. Intell.} \gdef\@journalfull{Journal of Intelligence} \gdef\@doiabbr{jintelligence} \gdef\@ISSN{2079-3200} } \DeclareOption{jlpea}{ \gdef\@journal{jlpea} \gdef\@journalshort{J. Low Power Electron. Appl.} \gdef\@journalfull{Journal of Low Power Electronics and Applications} \gdef\@doiabbr{jlpea} \gdef\@ISSN{2079-9268} } \DeclareOption{jmmp}{ \gdef\@journal{jmmp} \gdef\@journalshort{J. Manuf. Mater. Process.} \gdef\@journalfull{Journal of Manufacturing and Materials Processing} \gdef\@doiabbr{jmmp} \gdef\@ISSN{2504-4494} } \DeclareOption{jmse}{ \gdef\@journal{jmse} \gdef\@journalshort{J. Mar. Sci. Eng.} \gdef\@journalfull{Journal of Marine Science and Engineering} \gdef\@doiabbr{jmse} \gdef\@ISSN{2077-1312} } \DeclareOption{jnt}{ \gdef\@journal{jnt} \gdef\@journalshort{J. Nanotheranostics} \gdef\@journalfull{Journal of Nanotheranostics} \gdef\@doiabbr{jnt} \gdef\@ISSN{2624-845X} } \DeclareOption{jof}{ \gdef\@journal{jof} \gdef\@journalshort{J. Fungi} \gdef\@journalfull{Journal of Fungi} \gdef\@doiabbr{jof} \gdef\@ISSN{2309-608X} } \DeclareOption{joitmc}{ \gdef\@journal{joitmc} \gdef\@journalshort{J. Open Innov. Technol. Mark. Complex.} \gdef\@journalfull{Journal of Open Innovation: Technology, Market, and Complexity} \gdef\@doiabbr{joitmc} \gdef\@ISSN{2199-8531} } \DeclareOption{jpm}{ \gdef\@journal{jpm} \gdef\@journalshort{J. Pers. Med.} \gdef\@journalfull{Journal of Personalized Medicine} \gdef\@doiabbr{jpm} \gdef\@ISSN{2075-4426} } \DeclareOption{jrfm}{ \gdef\@journal{jrfm} \gdef\@journalshort{J. Risk Financial Manag.} \gdef\@journalfull{Journal of Risk and Financial Management} \gdef\@doiabbr{jrfm} \gdef\@ISSN{1911-8074} } \DeclareOption{jsan}{ \gdef\@journal{jsan} \gdef\@journalshort{J. Sens. Actuator Netw.} \gdef\@journalfull{Journal of Sensor and Actuator Networks} \gdef\@doiabbr{jsan} \gdef\@ISSN{2224-2708} } \DeclareOption{land}{ \gdef\@journal{land} \gdef\@journalshort{Land} \gdef\@journalfull{Land} \gdef\@doiabbr{land} \gdef\@ISSN{2073-445X} } \DeclareOption{languages}{ \gdef\@journal{languages} \gdef\@journalshort{Languages} \gdef\@journalfull{Languages} \gdef\@doiabbr{languages} \gdef\@ISSN{2226-471X} } \DeclareOption{laws}{ \gdef\@journal{laws} \gdef\@journalshort{Laws} \gdef\@journalfull{Laws} \gdef\@doiabbr{laws} \gdef\@ISSN{2075-471X} } \DeclareOption{life}{ \gdef\@journal{life} \gdef\@journalshort{Life} \gdef\@journalfull{Life} \gdef\@doiabbr{life} \gdef\@ISSN{2075-1729} } \DeclareOption{literature}{ \gdef\@journal{literature} \gdef\@journalshort{Literature} \gdef\@journalfull{Literature} \gdef\@doiabbr{} \gdef\@ISSN{2410-9789} } \DeclareOption{logistics}{ \gdef\@journal{logistics} \gdef\@journalshort{Logistics} \gdef\@journalfull{Logistics} \gdef\@doiabbr{logistics} \gdef\@ISSN{2305-6290} } \DeclareOption{lubricants}{ \gdef\@journal{lubricants} \gdef\@journalshort{Lubricants} \gdef\@journalfull{Lubricants} \gdef\@doiabbr{lubricants} \gdef\@ISSN{2075-4442} } \DeclareOption{machines}{ \gdef\@journal{machines} \gdef\@journalshort{Machines} \gdef\@journalfull{Machines} \gdef\@doiabbr{machines} \gdef\@ISSN{2075-1702} } \DeclareOption{magnetochemistry}{ \gdef\@journal{magnetochemistry} \gdef\@journalshort{Magnetochemistry} \gdef\@journalfull{Magnetochemistry} \gdef\@doiabbr{magnetochemistry} \gdef\@ISSN{2312-7481} } \DeclareOption{make}{ \gdef\@journal{make} \gdef\@journalshort{Mach. Learn. Knowl. Extr.} \gdef\@journalfull{Machine Learning and Knowledge Extraction} \gdef\@doiabbr{make} \gdef\@ISSN{2504-4990} } \DeclareOption{marinedrugs}{ \gdef\@journal{marinedrugs} \gdef\@journalshort{Mar. Drugs} \gdef\@journalfull{Marine Drugs} \gdef\@doiabbr{md} \gdef\@ISSN{1660-3397} } \DeclareOption{materials}{ \gdef\@journal{materials} \gdef\@journalshort{Materials} \gdef\@journalfull{Materials} \gdef\@doiabbr{ma} \gdef\@ISSN{1996-1944} } \DeclareOption{mathematics}{ \gdef\@journal{mathematics} \gdef\@journalshort{Mathematics} \gdef\@journalfull{Mathematics} \gdef\@doiabbr{math} \gdef\@ISSN{2227-7390} } \DeclareOption{mca}{ \gdef\@journal{mca} \gdef\@journalshort{Math. Comput. 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\DeclareOption{pharmacy}{ \gdef\@journal{pharmacy} \gdef\@journalshort{Pharmacy} \gdef\@journalfull{Pharmacy} \gdef\@doiabbr{pharmacy} \gdef\@ISSN{2226-4787} } \DeclareOption{philosophies}{ \gdef\@journal{philosophies} \gdef\@journalshort{Philosophies} \gdef\@journalfull{Philosophies} \gdef\@doiabbr{philosophies} \gdef\@ISSN{2409-9287} } \DeclareOption{photonics}{ \gdef\@journal{photonics} \gdef\@journalshort{Photonics} \gdef\@journalfull{Photonics} \gdef\@doiabbr{photonics} \gdef\@ISSN{2304-6732} } \DeclareOption{physics}{ \gdef\@journal{physics} \gdef\@journalshort{Physics} \gdef\@journalfull{Physics} \gdef\@doiabbr{physics} \gdef\@ISSN{2624-8174} } \DeclareOption{plants}{ \gdef\@journal{plants} \gdef\@journalshort{Plants} \gdef\@journalfull{Plants} \gdef\@doiabbr{plants} \gdef\@ISSN{2223-7747} } \DeclareOption{plasma}{ \gdef\@journal{plasma} \gdef\@journalshort{Plasma} \gdef\@journalfull{Plasma} \gdef\@doiabbr{plasma} \gdef\@ISSN{2571-6182} } \DeclareOption{polymers}{ 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\gdef\@journalshort{Psych} \gdef\@journalfull{Psych} \gdef\@doiabbr{psych} \gdef\@ISSN{2624-8611} } \DeclareOption{publications}{ \gdef\@journal{publications} \gdef\@journalshort{Publications} \gdef\@journalfull{Publications} \gdef\@doiabbr{publications} \gdef\@ISSN{2304-6775} } \DeclareOption{quantumrep}{ \gdef\@journal{quantumrep} \gdef\@journalshort{Quantum Rep.} \gdef\@journalfull{Quantum Reports} \gdef\@doiabbr{quantum} \gdef\@ISSN{2624-960X} } \DeclareOption{quaternary}{ \gdef\@journal{quaternary} \gdef\@journalshort{Quaternary} \gdef\@journalfull{Quaternary} \gdef\@doiabbr{quat} \gdef\@ISSN{2571-550X} } \DeclareOption{qubs}{ \gdef\@journal{qubs} \gdef\@journalshort{Quantum Beam Sci.} \gdef\@journalfull{Quantum Beam Science} \gdef\@doiabbr{qubs} \gdef\@ISSN{2412-382X} } \DeclareOption{reactions}{ \gdef\@journal{reactions} \gdef\@journalshort{Reactions} \gdef\@journalfull{Reactions} \gdef\@doiabbr{reactions} \gdef\@ISSN{2624-781X} } \DeclareOption{recycling}{ \gdef\@journal{recycling} \gdef\@journalshort{Recycling} \gdef\@journalfull{Recycling} \gdef\@doiabbr{recycling} \gdef\@ISSN{2313-4321} } \DeclareOption{religions}{ \gdef\@journal{religions} \gdef\@journalshort{Religions} \gdef\@journalfull{Religions} \gdef\@doiabbr{rel} \gdef\@ISSN{2077-1444} } \DeclareOption{remotesensing}{ \gdef\@journal{remotesensing} \gdef\@journalshort{Remote Sens.} \gdef\@journalfull{Remote Sensing} \gdef\@doiabbr{rs} \gdef\@ISSN{2072-4292} } \DeclareOption{reports}{ \gdef\@journal{reports} \gdef\@journalshort{Reports} \gdef\@journalfull{Reports} \gdef\@doiabbr{reports} \gdef\@ISSN{2571-841X} } \DeclareOption{resources}{ \gdef\@journal{resources} \gdef\@journalshort{Resources} \gdef\@journalfull{Resources} \gdef\@doiabbr{resources} \gdef\@ISSN{2079-9276} } \DeclareOption{risks}{ \gdef\@journal{risks} \gdef\@journalshort{Risks} \gdef\@journalfull{Risks} \gdef\@doiabbr{risks} \gdef\@ISSN{2227-9091} } \DeclareOption{robotics}{ \gdef\@journal{robotics} \gdef\@journalshort{Robotics} \gdef\@journalfull{Robotics} \gdef\@doiabbr{robotics} \gdef\@ISSN{2218-6581} } \DeclareOption{safety}{ \gdef\@journal{safety} \gdef\@journalshort{Safety} \gdef\@journalfull{Safety} \gdef\@doiabbr{safety} \gdef\@ISSN{2313-576X} } \DeclareOption{sci}{ \gdef\@journal{sci} \gdef\@journalshort{Sci} \gdef\@journalfull{Sci} \gdef\@doiabbr{sci} \gdef\@ISSN{2413-4155} } \DeclareOption{scipharm}{ \gdef\@journal{scipharm} \gdef\@journalshort{Sci. Pharm.} \gdef\@journalfull{Scientia Pharmaceutica} \gdef\@doiabbr{scipharm} \gdef\@ISSN{2218-0532} } \DeclareOption{sensors}{ \gdef\@journal{sensors} \gdef\@journalshort{Sensors} \gdef\@journalfull{Sensors} \gdef\@doiabbr{s} \gdef\@ISSN{1424-8220} } \DeclareOption{separations}{ \gdef\@journal{separations} \gdef\@journalshort{Separations} \gdef\@journalfull{Separations} \gdef\@doiabbr{separations} \gdef\@ISSN{2297-8739} } \DeclareOption{sexes}{ \gdef\@journal{sexes} \gdef\@journalshort{Sexes} \gdef\@journalfull{Sexes} \gdef\@doiabbr{} \gdef\@ISSN{2411-5118} } \DeclareOption{signals}{ \gdef\@journal{signals} \gdef\@journalshort{Signals} \gdef\@journalfull{Signals} \gdef\@doiabbr{signals} \gdef\@ISSN{2624-6120} } \DeclareOption{sinusitis}{ \gdef\@journal{sinusitis} \gdef\@journalshort{Sinusitis} \gdef\@journalfull{Sinusitis} \gdef\@doiabbr{sinusitis} \gdef\@ISSN{2309-107X} } \DeclareOption{smartcities}{ \gdef\@journal{smartcities} \gdef\@journalshort{Smart Cities} \gdef\@journalfull{Smart Cities} \gdef\@doiabbr{smartcities} \gdef\@ISSN{2624-6511} } \DeclareOption{sna}{ \gdef\@journal{sna} \gdef\@journalshort{Sinusitis Asthma} \gdef\@journalfull{Sinusitis and Asthma} \gdef\@doiabbr{sna} \gdef\@ISSN{2624-7003} } \DeclareOption{societies}{ \gdef\@journal{societies} \gdef\@journalshort{Societies} \gdef\@journalfull{Societies} \gdef\@doiabbr{soc} \gdef\@ISSN{2075-4698} } \DeclareOption{socsci}{ \gdef\@journal{socsci} \gdef\@journalshort{Soc. Sci.} \gdef\@journalfull{Social Sciences} \gdef\@doiabbr{socsci} \gdef\@ISSN{2076-0760} } \DeclareOption{soilsystems}{ \gdef\@journal{soilsystems} \gdef\@journalshort{Soil Syst.} \gdef\@journalfull{Soil Systems} \gdef\@doiabbr{soilsystems} \gdef\@ISSN{2571-8789} } \DeclareOption{sports}{ \gdef\@journal{sports} \gdef\@journalshort{Sports} \gdef\@journalfull{Sports} \gdef\@doiabbr{sports} \gdef\@ISSN{2075-4663} } \DeclareOption{standards}{ \gdef\@journal{standards} \gdef\@journalshort{Standards} \gdef\@journalfull{Standards} \gdef\@doiabbr{} \gdef\@ISSN{2305-6703} } \DeclareOption{stats}{ \gdef\@journal{stats} \gdef\@journalshort{Stats} \gdef\@journalfull{Stats} \gdef\@doiabbr{stats} \gdef\@ISSN{2571-905X} } \DeclareOption{surfaces}{ \gdef\@journal{surfaces} \gdef\@journalshort{Surfaces} \gdef\@journalfull{Surfaces} \gdef\@doiabbr{surfaces} \gdef\@ISSN{2571-9637} } \DeclareOption{surgeries}{ \gdef\@journal{surgeries} \gdef\@journalshort{Surgeries} \gdef\@journalfull{Surgeries} \gdef\@doiabbr{} \gdef\@ISSN{2017-2017} } \DeclareOption{sustainability}{ \gdef\@journal{sustainability} \gdef\@journalshort{Sustainability} \gdef\@journalfull{Sustainability} \gdef\@doiabbr{su} \gdef\@ISSN{2071-1050} } \DeclareOption{symmetry}{ \gdef\@journal{symmetry} \gdef\@journalshort{Symmetry} \gdef\@journalfull{Symmetry} \gdef\@doiabbr{sym} \gdef\@ISSN{2073-8994} } \DeclareOption{systems}{ \gdef\@journal{systems} \gdef\@journalshort{Systems} \gdef\@journalfull{Systems} \gdef\@doiabbr{systems} \gdef\@ISSN{2079-8954} } \DeclareOption{technologies}{ \gdef\@journal{technologies} \gdef\@journalshort{Technologies} \gdef\@journalfull{Technologies} \gdef\@doiabbr{technologies} \gdef\@ISSN{2227-7080} } \DeclareOption{test}{ \gdef\@journal{test} \gdef\@journalshort{Test} \gdef\@journalfull{Test} \gdef\@doiabbr{} \gdef\@ISSN{} } \DeclareOption{toxics}{ \gdef\@journal{toxics} \gdef\@journalshort{Toxics} \gdef\@journalfull{Toxics} \gdef\@doiabbr{toxics} \gdef\@ISSN{2305-6304} } \DeclareOption{toxins}{ \gdef\@journal{toxins} \gdef\@journalshort{Toxins} \gdef\@journalfull{Toxins} \gdef\@doiabbr{toxins} \gdef\@ISSN{2072-6651} } \DeclareOption{tropicalmed}{ \gdef\@journal{tropicalmed} \gdef\@journalshort{Trop. Med. Infect. Dis.} \gdef\@journalfull{Tropical Medicine and Infectious Disease} \gdef\@doiabbr{tropicalmed} \gdef\@ISSN{2414-6366} } \DeclareOption{universe}{ \gdef\@journal{universe} \gdef\@journalshort{Universe} \gdef\@journalfull{Universe} \gdef\@doiabbr{universe} \gdef\@ISSN{2218-1997} } \DeclareOption{urbansci}{ \gdef\@journal{urbansci} \gdef\@journalshort{Urban Sci.} \gdef\@journalfull{Urban Science} \gdef\@doiabbr{urbansci} \gdef\@ISSN{2413-8851} } \DeclareOption{vaccines}{ \gdef\@journal{vaccines} \gdef\@journalshort{Vaccines} \gdef\@journalfull{Vaccines} \gdef\@doiabbr{vaccines} \gdef\@ISSN{2076-393X} } \DeclareOption{vehicles}{ \gdef\@journal{vehicles} \gdef\@journalshort{Vehicles} \gdef\@journalfull{Vehicles} \gdef\@doiabbr{vehicles} \gdef\@ISSN{2624-8921} } \DeclareOption{vetsci}{ \gdef\@journal{vetsci} \gdef\@journalshort{Vet. Sci.} \gdef\@journalfull{Veterinary Sciences} \gdef\@doiabbr{vetsci} \gdef\@ISSN{2306-7381} } \DeclareOption{vibration}{ \gdef\@journal{vibration} \gdef\@journalshort{Vibration} \gdef\@journalfull{Vibration} \gdef\@doiabbr{vibration} \gdef\@ISSN{2571-631X} } \DeclareOption{viruses}{ \gdef\@journal{viruses} \gdef\@journalshort{Viruses} \gdef\@journalfull{Viruses} \gdef\@doiabbr{v} \gdef\@ISSN{1999-4915} } \DeclareOption{vision}{ \gdef\@journal{vision} \gdef\@journalshort{Vision} \gdef\@journalfull{Vision} \gdef\@doiabbr{vision} \gdef\@ISSN{2411-5150} } \DeclareOption{water}{ \gdef\@journal{water} \gdef\@journalshort{Water} \gdef\@journalfull{Water} \gdef\@doiabbr{w} \gdef\@ISSN{2073-4441} } \DeclareOption{wem}{ \gdef\@journal{wem} \gdef\@journalshort{Wildl. Ecol. Manag.} \gdef\@journalfull{Wildlife Ecology and Management} \gdef\@doiabbr{} \gdef\@ISSN{1234-4321} } \DeclareOption{wevj}{ \gdef\@journal{wevj} \gdef\@journalshort{World Electric Vehicle Journal} \gdef\@journalfull{World Electric Vehicle Journal} \gdef\@doiabbr{wevj} \gdef\@ISSN{2032-6653} }%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} \label{sec:Introduction} Bayesian inference is a cornerstone of modern cosmology and astrophysics. It is frequently employed to derive parameter constraints on key signal parameters from data sets such as the Dark Energy Survey~\citep[DES, ][]{DES_Year1_2018, DES_year3_2021}, Planck \cite{Planck_cosmo_2020}, REACH \cite{REACH} and SARAS2 \cite{SARAS2} among others. Often experiments are sensitive to different aspects of the same physics, and by combining constraints across probes we can improve our understanding of the Universe or reveal tensions between different experiments. However, this can become a computationally expensive task as many experiments feature systematics, instrumental effects and contamination from other physical signals that need to be model alongside the signal or parameters of interest. For individual experiments, this can lead to high dimensional problems with $\gtrsim 20$ parameters of which the majority can be considered `nuisance' parameters. The problem is compounded when combining different data sets with different models for common nuisance components and different systematics or instrumental effects that have to be modelled. In this work we demonstrate that we can use density estimators, such as Kernel Density Estimators \cite{parzen_KDE_1962, rosenblatt_KDE_1956} and Masked Autoregressive Flows \cite{Papamarkarios_MAF_2017}, to rapidly calculate reliable and reusable representations of marginal probability densities and marginal Bayesian summary statistics for key signal or cosmological parameters. This gives us access to the nuisance-free likelihood functions and allows us to combine parameter constraints from different data sets in a computationally efficient manner given marginal posterior samples from the different experiments. We use the publicly available code\footnote{https://github.com/htjb/margarine} \textsc{margarine} \cite{margarine_neurips} to generate density estimators. In \cref{sec:theory} we mathematically demonstrate that the application of \textsc{margarine} to the problem of combining the marginal posteriors from two data sets is equivalent to running a full nested sampling run including all `nuisance' parameters. We define in this section the nuisance-free likelihood. \Cref{sec:methods} briefly discusses the methodology behind \textsc{margarine} with reference to a previously published work \cite{margarine_neurips}. Finally, we show the results of combining samples from DES and Planck in \cref{sec:results} and conclude in \cref{sec:conclusion}. \section{Theory} \label{sec:theory} \subsection*{Notation} Given a likelihood $\mathcal{L}(\Theta)\equiv P(D|\Theta,\mathcal{M})$ representing the probability of data $D$ given some model $\mathcal{M}$ with parameters $\Theta$, Bayesian inference proceeds by defining a prior $\pi(\Theta)\equiv P(\Theta|\mathcal{M})$, and then through Bayes theorem computing a posterior distribution $\mathcal{P}(\Theta)\equiv P(\Theta|D,\mathcal{M})$ for the purposes of parameter estimation and an evidence $\mathcal{Z}\equiv P(D|\mathcal{M})$ in order to perform model comparison. In our notation we suppress model dependence, but where we wish to refer to the likelihoods derived from different datasets, we denote this with a subscript so for example $\mathcal{L}_A(\Theta)\equiv P(D_A|\Theta,\mathcal{M})$, and $\mathcal{L}_B(\Theta)\equiv P(D_B|\Theta,\mathcal{M})$. In our setting, the parameter vector is split into two sub-vectors $\Theta=(\theta,\alpha)$, where $\theta$ are parameters of scientific interest, and $\alpha$ are \emph{nuisance} parameters, included for the purposes of data analysis. Such situations are common in astrophysics, where for example $\theta$ might be parameters governing the Universe's evolution, whilst $\alpha$ might be associated with instrument calibration and foreground removal \cite{Planck_cosmo_2020, REACH}. The $\alpha$ parameters are generally ``marginalised out'' and not considered further in final or future analyses. \subsection*{Definitions} With this notation established, the version of Bayes theorem including nuisance parameters takes the form \begin{equation} \mathcal{L}(\theta,\alpha)\times \pi(\theta,\alpha) =\mathcal{P}(\theta,\alpha)\times \mathcal{Z}, \label{eqn:bayes} \end{equation} where we have placed the inputs of inference (likelihood and prior) on the left-hand side, and the outputs (posterior and evidence) on the right. The evidence is as usual equivalent to the (fully-)marginalised likelihood $\mathcal{Z} = \int \mathcal{L}(\theta,\alpha) \pi(\theta,\alpha)d\theta d\alpha$. We may marginalise any probability distribution so can straightforwardly define the nuisance marginalised posterior and prior by integrating over $\alpha$ \begin{equation} \mathcal{P}(\theta) = \int \mathcal{P}(\theta,\alpha)d\alpha, \qquad \pi(\theta) = \int \pi(\theta,\alpha)d\alpha. \label{eqn:marginal} \end{equation} The nuisance marginalised version of Bayes theorem~\cref{eqn:bayes} takes the form \begin{equation} \mathcal{L}(\theta)\times\pi(\theta) = \mathcal{P}(\theta)\times\mathcal{Z}. \label{eqn:marginal_bayes} \end{equation} Here $\mathcal{Z}$ is the original evidence, whilst $\mathcal{L}(\theta)$ is non-trivially the \emph{nuisance-free likelihood} \begin{equation} \mathcal{L}(\theta) \equiv \frac{\int\mathcal{L}(\theta,\alpha)\pi(\theta,\alpha)d\alpha}{\int \pi(\theta,\alpha)d\alpha} = \frac{\mathcal{P}(\theta)\mathcal{Z}}{\pi(\theta)}, \label{eqn:partial} \end{equation} where the above is motivated by marginalising over $\alpha$ of the full Bayes theorem~\cref{eqn:bayes} and substituting the definitions in \cref{eqn:partial,eqn:marginal} recovers the marginalised Bayes theorem~\cref{eqn:marginal_bayes}. The nuisance-free likelihood \cref{eqn:partial} is straightforward to compute in our framework since we (uniquely) have NS-computed evidences $\mathcal{Z}$ combined with $\textsc{margarine}$ trained distributions $\mathcal{P}(\theta)$ and $\pi(\theta)$ \cite{margarine_neurips} (see \cref{sec:methods}). We now explain why \cref{eqn:partial} is a useful definition \begin{Theorem} Let $\mathcal{L}_A(\theta,\alpha_A)$ and $\mathcal{L}_B(\theta,\alpha_B)$ be two likelihoods with distinct datasets, each with their own nuisance parameters. The nuisance-free likelihoods $\mathcal{L}_A(\theta)$, $\mathcal{L}_B(\theta)$ form a lossless compression in $\theta$. This means that we can recover the same (marginal) inference in combination that we would have made when performing a combined analysis with all nuisance parameters: \begin{align} \mathcal{L}_A(\theta,\alpha_A)\mathcal{L}_B(\theta,\alpha_B)\pi_{AB}(\theta,\alpha_A,\alpha_B) &= \mathcal{P}_{AB}(\theta,\alpha_A,\alpha_B)\mathcal{Z}_{AB}, \label{eqn:combined_bayes}\\ \Rightarrow \mathcal{L}_A(\theta)\mathcal{L}_B(\theta)\pi(\theta) &= \mathcal{P}_{AB}(\theta)\mathcal{Z}_{AB}, \label{eqn:marginal_combined_bayes} \end{align} if their respective priors $\pi_A(\theta,\alpha_A)$ and $\pi_B(\theta,\alpha_B)$ satisfy the marginal consistency relations: \begin{gather} \pi(\theta) = \int \pi_A(\theta,\alpha_A)d\alpha_A = \int \pi_B(\theta,\alpha_B)d\alpha_B, \label{eqn:marginal_A}\\ \int \pi_{AB}(\theta,\alpha_A,\alpha_B)d\alpha_A = \pi_B(\theta,\alpha_B), \qquad \int \pi_{AB}(\theta,\alpha_A,\alpha_B)d\alpha_B = \pi_A(\theta,\alpha_A). \label{eqn:marginal_B} \end{gather} This process is represented graphically in \cref{fig:pipeline} and \cref{fig:pipelineB}. \end{Theorem} \begin{proof} Integrating the combined Bayes theorem \cref{eqn:combined_bayes} with respect to $\alpha_B$, applying the definition of the marginal posterior \cref{eqn:marginal} on the right-hand side, and drawing out terms independent of $\alpha_B$ on the left, yields \begin{equation} \mathcal{L}_A(\theta,\alpha_A)\int\mathcal{L}_B(\theta,\alpha_B)\pi_{AB}(\theta,\alpha_A,\alpha_B)d\alpha_B = \mathcal{P}_{AB}(\theta,\alpha_A)\mathcal{Z}_{AB}. \label{eqn:temp1} \end{equation} From the definition of a nuisance-free likelihood~\cref{eqn:partial}, and the marginal consistency~\cref{eqn:marginal_B}, we can say that the integral on the left-hand side becomes: \begin{align} &\int \mathcal{L}_B(\theta,\alpha_B) \pi_{AB}(\theta,\alpha_A,\alpha_B)d\alpha_B \nonumber\\ &= \int \mathcal{L}_B(\theta,\alpha_A,\alpha_B) \pi_{AB}(\theta,\alpha_A,\alpha_B)d\alpha_B &\left[\text{$\mathcal{L}_B(\theta,\alpha_B)\equiv \mathcal{L}_B(\theta,\alpha_A,\alpha_B)$ since $\mathcal{L}_B$ indep of $\alpha_A$}\right] \nonumber\\ &= \mathcal{L}_B(\theta,\alpha_A) {\int \pi_{AB}(\theta,\alpha_A,\alpha_B) d\alpha_B} &\left[\text{Using \cref{eqn:partial} for $\mathcal{L}_B$}\right] \nonumber\\ &= \mathcal{L}_B(\theta) {\int \pi_{AB}(\theta,\alpha_A,\alpha_B) d\alpha_B} &\left[\text{$\mathcal{L}_B(\theta,\alpha_A)\equiv \mathcal{L}_B(\theta)$ since $\mathcal{L}_B$ indep of $\alpha_A$}\right] \nonumber\\ &= \mathcal{L}_B(\theta) \pi_A(\theta,\alpha_A). &\left[\text{Using marginal consistency \cref{eqn:marginal_B}}\right] \label{eqn:temp2} \end{align} Substituting \cref{eqn:temp2} back into \cref{eqn:temp1} we find \begin{equation} \mathcal{L}_A(\theta,\alpha_A)\mathcal{L}_B(\theta)\pi_A(\theta,\alpha_A) = \mathcal{P}_{AB}(\theta,\alpha_A)\mathcal{Z}_{AB}. \end{equation} Proceeding with a similar manipulation to \cref{eqn:temp2}, marginalising with respect to $\alpha_A$, and applying the definition of the nuisance-free likelihood $\mathcal{L}_A(\theta)$ \cref{eqn:partial} and the marginal prior consistency~\cref{eqn:marginal_A} we recover \cref{eqn:marginal_combined_bayes} \begin{equation} \mathcal{L}_A(\theta)\mathcal{L}_B(\theta)\pi(\theta) = \mathcal{P}_{AB}(\theta)\mathcal{Z}_{AB}. \nonumber \end{equation} \end{proof} \subsection*{Discussion} \Cref{eqn:combined_bayes} represents Bayes theorem for the combined likelihood of both datasets $\mathcal{L}_{AB}(\theta,\alpha_A,\alpha_B) = \mathcal{L}_A(\theta,\alpha_A)\mathcal{L}_B(\theta,\alpha_B)$, using the combined prior $\pi_{AB}(\theta,\alpha_A,\alpha_B)$. We assume the combined prior is marginally consistent, \cref{eqn:marginal_A,eqn:marginal_B}, which is reasonable, merely demanding that the priors are identical in the parameter spaces where they overlap. In practice, this would usually be achieved by assuming separability between signal and nuisance parameter spaces $\pi(\theta,\alpha) = \pi(\theta)\pi(\alpha)$, but \cref{eqn:marginal_A,eqn:marginal_B} are a slightly less restrictive requirement and therefore more general. The upshot of this is that if you have performed inference for two datasets separately, such that you are able to compute the nuisance-free likelihoods with \textsc{margarine}, you may discard the nuisance parameters for the next set of analyses when you combine the datasets. \section{Methods} \label{sec:methods} \textsc{margarine} was first introduced in \cite{margarine_neurips} and uses density estimation to approximate probability distributions such as $\mathcal{P}(\theta)$ and $\pi(\theta)$ given sets of representative samples. The code was developed initially to calculate marginal Kullback-Leibler~(KL) divergences \cite{kullback_information_1951} and Bayesian Model Dimensionalities~(BMD) \cite{Handley_dimensionality_2019} however as discussed in \cref{sec:theory} it can be used to calculate the nuisance-free likelihoods. This in turn means that we can use \textsc{margarine} alongside an implementation of the nested sampling algorithm to sample the product $\mathcal{L}_A(\theta)\mathcal{L}_B(\theta)$. In this manner, \textsc{margarine} allows us to combine constraints on common parameters across different data sets. We refer the reader to \cite{margarine_neurips} for a complete discussion of how \textsc{margarine} works, however, we discuss briefly the density estimation here. \textsc{margarine} uses two different types of density estimator to model posterior and prior samples, namely Masked Autoregressive Flows~(MAFs, \cite{Papamarkarios_MAF_2017}) and Kernel Density Estimators~(KDEs, \cite{parzen_KDE_1962,rosenblatt_KDE_1956}). \begin{figure} \centering \begin{tikzpicture}[squarednodeA/.style={rectangle, draw=red!60, fill=red!5, very thick, minimum size=5mm}, squarednodeB/.style={rectangle, draw=blue!60, fill=blue!5, very thick, minimum size=5mm}, squarednodeC/.style={rectangle, draw=green!60, fill=green!5, very thick, minimum size=5mm}] \node[squarednodeA, text width=3cm, align=center](inference3) at (17, -1.5) {Nested Sampling with $\theta$, $\alpha_A$ and $\alpha_B$}; \node[squarednodeB](fulllikelihood1) at (15, 1.5){$ \mathcal{L}_A(\theta,\alpha_A)$}; \node[squarednodeB](fulllikelihood2) at (16.85, 1.5){$ \mathcal{L}_B(\theta,\alpha_B)$}; \node[squarednodeB](fulljointlikelihood) at (16, 0){$ \mathcal{L}_A(\theta,\alpha_A) \mathcal{L}_B(\theta,\alpha_B)$}; \node[squarednodeB](fullprior) at (19, 1.5){$ \pi_{AB}(\theta,\alpha_A, \alpha_B)$}; \draw[->](fulllikelihood1.south) -- (15.5, 0.3); \draw[->](fulllikelihood2.south) -- (16.5, 0.3); \draw[->](fullprior.south) -- (inference3.north); \draw[->](fulljointlikelihood.south) -- (inference3.north); \node[squarednodeB](jointEvidence2) at (15, -3){$ \mathcal{Z}_{AB}$}; \node[squarednodeB](jointPosterior2) at (17, -3){$ \{\theta\}_{\mathcal{P}_{AB}}$}; \draw[<-](jointEvidence2.north) -- (16, -2); \draw[<-](jointPosterior2.north) -- (inference3.south); \node[squarednodeB](jointPosteriorNuisance) at (19, -3){$ \{\alpha_A, \alpha_B\}_{\mathcal{P}_{AB}}$}; \draw[->](18, -2) -- (jointPosteriorNuisance.north); \draw[blue,thick](16.2, -3.5) -- (20.1, -3.5) -- (20.1, -2.5) -- (16.2, -2.5) -- (16.2, -3.5); \end{tikzpicture} \caption{A graphical representation of combining constraints from different data sets via a full nested sampling run over both cosmological and nuisance parameters (\cref{eqn:combined_bayes}).} \label{fig:pipeline} \end{figure} \begin{figure} \centering \begin{tikzpicture}[squarednodeA/.style={rectangle, draw=red!60, fill=red!5, very thick, minimum size=5mm}, squarednodeB/.style={rectangle, draw=blue!60, fill=blue!5, very thick, minimum size=5mm}, squarednodeC/.style={rectangle, draw=green!60, fill=green!5, very thick, minimum size=5mm}] \node[squarednodeA](inference) at (0, 0) {Nested Sampling}; \node[squarednodeB](likelihood) at (-1, 1.5){$ \mathcal{L}(\theta,\alpha)$}; \node[squarednodeB](prior) at (1, 1.5){$ \pi(\theta,\alpha)$}; \node[squarednodeB](evidence) at (-1, -1.5){$ \mathcal{Z}$}; \node[squarednodeB](posterior) at (1, -1.5){$ \{\theta,\alpha\}_\mathcal{P}$}; \node[squarednodeB](priorSamples) at (2.8, -1.5){$ \{\theta,\alpha\}_\pi$}; \node[squarednodeA](margarine) at (2, -3) {\textsc{margarine}}; \node[squarednodeB](marginalPrior) at (2.8, -5.5){$ \pi(\theta)$}; \node[squarednodeB](marginalPosterior) at (1.5, -4.5){$ \mathcal{P}(\theta)$}; \node[squarednodeB](marginalLikelihood) at (0, -6){$ \mathcal{L}(\theta)$}; \draw[green, ultra thick] (-1.8, 0.5) -- (-1.8, -5) -- (3.8, -5)-- (3.8, 0.5) -- (-1.8, 0.5); \draw[->](likelihood.south) -- (-1, 0.3); \draw[->](prior.south) -- (1, 0.3); \draw[->](-1, -0.3) -- (evidence.north); \draw[->](1, -0.3) -- (posterior.north); \draw[->](posterior.south) -- (1.5, -2.7); \draw[dashed,->](prior.east) to[out=-20, in=90] (2.8, -1.2); \draw[->](priorSamples.south) -- (2.8, -2.7); \draw[->](2.8, -3.3) -- (marginalPrior.north); \draw[->](1.5, -3.3) -- (1.5, -4.2); \draw[->](evidence.south) -- (marginalLikelihood.north); \draw[->](marginalPosterior.south) -- (marginalLikelihood.north); \draw[->](marginalPrior.west) -- (marginalLikelihood.east); \draw[dashed,->](inference.east) -- (priorSamples.north); \draw[black, ultra thick, dashed] (4.5, 2) -- (4.5, -7); \node[squarednodeB](likelihood1) at (6, 1.5){$ \mathcal{L}_A(\theta,\alpha_A)$}; \node[squarednodeB](prior1) at (8, 1.5){$ \pi_A(\theta,\alpha_A)$}; \node[squarednodeC, text width=3cm, align=center](NestedMarg1) at (7, 0){Nested Sampling + \textsc{Margarine}}; \draw[->](likelihood1.south) -- (6, 0.5); \draw[->](prior1.south) -- (8, 0.5); \node[squarednodeB](marglike1) at (6, -1.5){$ \mathcal{L}_A(\theta)$}; \node[squarednodeB](margprior1) at (9, -1.5){$ \pi(\theta)$}; \draw[->](6, -0.5) -- (6, -1.2); \draw[->](8, -0.5) -- (9, -1.2); \node[squarednodeB](likelihood2) at (10, 1.5){$ \mathcal{L}_B(\theta,\alpha_B)$}; \node[squarednodeB](prior2) at (12, 1.5){$ \pi_B(\theta,\alpha_B)$}; \draw[->](likelihood2.south) -- (10, 0.5); \draw[->](prior2.south) -- (12, 0.5); \node[squarednodeC, text width=3cm, align=center](NestedMarg1) at (11, 0){Nested Sampling + \textsc{Margarine}}; \node[squarednodeB](marglike2) at (12, -1.5){$ \mathcal{L}_B(\theta)$}; \draw[->](12, -0.5) -- (12, -1.2); \draw[->](10, -0.5) -- (9, -1.2); \node[squarednodeB](combinedlike) at (7, -3){$ \mathcal{L}_A(\theta) \mathcal{L}_B(\theta)$}; \draw[->](marglike1.south) -- (combinedlike.north); \draw[->](marglike2.south) -- (combinedlike.north); \node[squarednodeA, text width=3cm, align=center](inference2) at (9, -4.5) {Nested Sampling with $\theta$}; \draw[->](combinedlike.south) -- (8, -4); \draw[->](margprior1.south) -- (10, -4); \node[squarednodeB](jointEvidence) at (8, -6){$ \mathcal{Z}_{AB}$}; \node[squarednodeB](jointPosterior) at (10, -6){$ \{\theta\}_{\mathcal{P}_{AB}}$}; \draw[->](8, -5) -- (jointEvidence.north); \draw[->](10, -5) -- (jointPosterior.north); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{tikzpicture} \caption{A graphical representation of combining constraints from two data sets via \textsc{margarine} (\cref{eqn:marginal_combined_bayes}). Left of the dashed line illustrates the derivation of a nuisance-free likelihood function for one experimental data set.} \label{fig:pipelineB} \end{figure} MAFs transform a multivariate base distribution, the standard normal, into a target distribution via a series of shifts and scaling which are estimated by autoregressive neural networks. To improve the performance of the MAF the samples representing the target distribution, in our case $\mathcal{P}(\theta)$ and $\pi(\theta)$, are transformed into a gaussianized space. We implement the MAFs using \textsc{tensorflow} and the \textsc{keras} backend \cite{tensorflow2015-whitepaper}. KDEs use a kernel to approximate the multivariate probability density of a series of samples. In our case, the kernel is Gaussian and the probability density is a sum of Gaussians centred on the sample points with a given bandwidth. Again we transform the target samples into a gaussianized parameter space allowing the KDE to better capture the distribution. The KDEs are implemented with \textsc{SciPy} in \textsc{margarine} \cite{rosenblatt_KDE_1956, parzen_KDE_1962, 2020SciPy-NMeth}. Since both types of density estimator build approximations to the target distribution using known distributions, the approximate log-probabilities of the target distribution can be easily calculated. %Since we are working in a gaussianized parameter space the log-probabilities have to be corrected for the change of variables if we want to evaluate them in the original parameter space but this is trivial and fully implemented in \textsc{margarine}. The evaluation of normalized log-probabilities for the marginal posterior and marginal prior allows us to calculate the nuisance-free likelihoods, as discussed, along with marginal Kullback-Leibler divergences \begin{equation} \mathcal{D}(\mathcal{P}||\pi) = \int \mathcal{P}(\theta) \log\frac{\mathcal{P}(\theta)}{\pi(\theta)} d\theta, \label{eq:kl_divergence} \end{equation} which quantifies the amount of information gained when moving from the marginal prior to posterior. \section{Cosmological Example} \label{sec:results} It has previously been demonstrated that \textsc{margarine} is capable of replicating complex probability distributions and approximating marginal Bayesian statistics such as the KL divergence and the BMD \cite{margarine_neurips}. Here we demonstrate the theory discussed in \cref{sec:theory} by combining samples from the Dark Energy Survey~(DES) Year 1 posterior \cite{DES_Year1_2018} and Planck posterior \cite{Planck_cosmo_2020} using \textsc{margarine} to estimate nuisance-free likelihoods. DES surveys supernovae, galaxies and large scale cosmic structures in the universe in an effort to measure dark matter and dark energy densities and model the dark energy equation of state. In contrast, Planck mapped the anisotropies in the Cosmic Microwave Background~(CMB) and correspondingly provided constraints on key cosmological parameters. The constraints from DES and Planck have previously been combined using a full nested sampling run over all parameters including a multitude of `nuisance' parameters in a computationally expensive exercise \cite{Handley_tensions_2019}. This corresponds to the flow chart in \cref{fig:pipeline} and the previous analysis gives us access to the combined DES and Planck evidence, which is found to have a value of $\log(\mathcal{Z}) = -5965.7 \pm 0.3$. In \cref{fig:joint} we show the DES, Planck and joint posteriors for the six cosmological parameters derived in this work using \textsc{margarine} and the flow chart in \cref{fig:pipelineB}. The constrained parameters are the baryon and dark matter density parameters, $\Omega_b h^2$ and $\Omega_c h^2$, the angular size of the sound horizon at recombination, $\theta_{MC}$, the CMB optical depth, $\tau$, the amplitude of the power spectrum, $A_s$, and the corresponding spectral index, $n_s$. These make up the set $\theta = (\Omega_b h^2, \Omega_c h^2, \theta_{MC}, \tau, A_s, n_s)$. We use the nested sampling algorithm \textsc{polychord} in our analysis \cite{Handley2015a, Handley2015b}. We use a uniform prior that is defined to be three sigma around the Planck posterior mean. This is done to improve the efficiency of our nested sampling run. However, we subsequently have to re-weight the samples and correct the evidence for the difference between the priors used here and in the previous full nested sampling run \cite{Handley_tensions_2019} for comparison. If we define \begin{equation} \mathcal{Z}_A = \int \mathcal{L}(\theta) \pi_A(\theta) d\theta, \qquad \mathcal{Z}_B = \int \mathcal{L}(\theta) \pi_B(\theta) d\theta, \end{equation} where $A$ is our uniform prior space and $B$ is our target prior space from the previous work \cite{Handley_tensions_2019}, then \begin{equation} \mathcal{Z}_B = \int \mathcal{L}(\theta) \pi_B(\theta) d\theta = \int \mathcal{L}(\theta) {\pi_A(\theta)}\frac{\pi_B(\theta)}{\pi_A(\theta)} d\theta = \mathcal{Z}_A\left\langle \frac{\pi_B(\theta)}{\pi_A(\theta)}\right\rangle_{\mathcal{P}_A} \end{equation} giving \begin{equation} \mathcal{Z}_B = \mathcal{Z}_A\left\langle \frac{\pi_B(\theta)}{\pi_A(\theta)}\right\rangle_{\mathcal{P}_A}. \end{equation} Then following similar arguments we can transform our posteriors by re-weighting the distributions with the following \begin{equation} w^{(i)}_B = w^{(i)}_A \frac{\pi_B(\theta^{(i)})}{\pi_A(\theta^{(i)})}. \end{equation} We see from the figure and corresponding table that with our joint analysis we are able to derive a log-evidence that is approximately consistent with that found in \cite{Handley_tensions_2019} validating the theory discussed and its implementation with \textsc{margarine}. We note that the re-weighting described above relies on calculation of the two prior log-probabilities for which we use \textsc{margarine} and currently do not have an estimate of the error for. As a result, the error in the combined evidence, $Z_B$, is given by the error in $Z_A$ from the nested samples and is likely underestimated. Using \textsc{margarine} \cite{margarine_neurips} we can also derive the combined KL divergence, also reported in \cref{fig:joint}, which we find is consistent with the result in the literature of $\mathcal{D} = 6.17 \pm 0.36$ \cite{Handley_dimensionality_2019}. Similarly, we derive marginal KL divergences for the DES and Planck cosmological parameters using \textsc{margarine}. A full discussion of the implications of combining the two data sets for our understanding of cosmology can be found in the literature \cite[e.g][]{Handley_tensions_2019, Handley_dimensionality_2019}. By reducing the number of parameters that need to be sampled, we significantly reduce the nested sampling runtime. For \textsc{polychord} the runtime scales as the cube of the number of dimensions \cite{supernest}. This can be seen by assessing the time complexity of the algorithm where, $T \propto n_\mathrm{live} \times \langle T\{\mathcal{L}(\theta)\}\rangle \times \langle T\{Impl.\}\rangle \times \mathcal{D}(\mathcal{P}||\pi)$. Here $n_\mathrm{live}$ scales with the number of dimensions, $d$, as does the Kullback-Leibler divergence. For \textsc{polychord}, the specific implementation time complexity factor, $\langle T\{Impl.\}\rangle$, representing the impact of replacing dead points with higher likelihood live points on the runtime, scales linearly with $d$. Together this gives $T \propto d^3 \times \langle T\{\mathcal{L}(\theta)\}\rangle$. Therefore, by using nuisance-free likelihoods and sampling over 6 parameters rather than 41 parameters (cosmological plus 20 nuisance parameters for DES and 15 different nuisance parameters for Planck) we reduce the runtime by a factor of $(41/6)^3 \approx 319$ with further improvements in $\langle T\{\mathcal{L}(\theta)\}\rangle$. Using \textsc{margarine}, $\langle T\{\mathcal{L}(\theta)\}\rangle$ is typically reduced since analytic likelihoods are computationally more expensive than emulated likelihoods. \begin{figure} \centering \includegraphics{Figs/paper_plot.pdf} \caption{The combined posterior (in grey) found when combining the constraints on the cosmological parameters from DES and Planck using \textsc{margarine}. For DES and Planck, we calculate the marginal KL divergences using \textsc{margarine}, whereas the Bayesian evidences are calculated using \textsc{anesthetic}. The joint evidence and joint KL divergence are calculated with a combination of the two codes and are found to be approximately consistent with those found in the literature \cite{Handley_tensions_2019, Handley_dimensionality_2019}. Note that the error on the joint evidence is likely underestimated as it relies on evaluations of log-probabilities for various distributions, for which \textsc{margarine} does not currently provide errors. The figure produced with \textsc{anesthetic} \cite{anesthetic}.} \label{fig:joint} \end{figure} %As a practical note, when combining data sets we need to translate both sets of posteriors and their corresponding evidences onto a common prior to satisfy \cref{eqn:marginal_A}. \section{Conclusion} \label{sec:conclusion} In the paper, we have demonstrated the consistency between combining constraints from different experiments in a marginal framework using density estimators and the code \textsc{margarine} with a full nested sampling run over all parameters, including those describing `nuisance' components of the data. We have shown this consistency mathematically and with a cosmological example. For the combination of Planck and DES, we find a Bayesian evidence and KL divergence that is consistent with previous results \cite{Handley_tensions_2019, Handley_dimensionality_2019}. The analysis in this paper is only possible because (a) we are able to estimate densities in the (much smaller) cosmological parameter space $\theta$ using \textsc{margarine}, and (b) because we have evidences, $\mathcal{Z}$, from our original nested sampling runs. It is this unique combination which allows us to compress away or discard nuisance parameters once they have been used. We note also that working in the marginal space results in a compression that is lossless in information on $\theta$ as it recovers an identical marginal posterior and total evidence as is found during a full Bayesian inference. Finally, through the nuisance-free likelihood we can significantly reduce the dimensionality of our problems and since it is faster to emulate a likelihood rather than analytically evaluate, \textsc{margarine} offers a much more computationally efficient path to combined Bayesian analysis. In principle, our work paves the way for the development of a publicly available library of cosmological density estimators modelled with \textsc{margarine} that can be combined with new data sets using the proposed method in a more efficient manner than currently implemented techniques. However, the work has implications outside of cosmology in any field where multiple experiments probe different aspects of the same physics. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% optional %\supplementary{The following are available online at \linksupplementary{s1}, Figure S1: title, Table S1: title, Video S1: title.} % Only for the journal Methods and Protocols: % If you wish to submit a video article, please do so with any other supplementary material. % \supplementary{The following are available at \linksupplementary{s1}, Figure S1: title, Table S1: title, Video S1: title. A supporting video article is available at doi: link.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\authorcontributions{For research articles with several authors, a short paragraph specifying their individual contributions must be provided. The following statements should be used ``conceptualization, X.X. and Y.Y.; methodology, X.X.; software, X.X.; validation, X.X., Y.Y. and Z.Z.; formal analysis, X.X.; investigation, X.X.; resources, X.X.; data curation, X.X.; writing--original draft preparation, X.X.; writing--review and editing, X.X.; visualization, X.X.; supervision, X.X.; project administration, X.X.; funding acquisition, Y.Y.'', please turn to the \href{http://img.mdpi.org/data/contributor-role-instruction.pdf}{CRediT taxonomy} for the term explanation. Authorship must be limited to those who have contributed substantially to the work reported.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \authorcontributions{Conceptualization, W.H. and H.B.; methodology, W.H. and H.B.; formal analysis, H.B.; software, H.B.; writing---original draft preparation, H.B.; writing---review and editing, H.B., W.H., P.L. and P.S.; supervision, W.H., E.d.L.A. and A.F.. All authors have read and agreed to the published version of the manuscript.} \funding{H.B. acknowledges the support of the Science and Technology Facilities Council (STFC) through grant number ST/T505997/1 and Fitzwilliam College, Cambridge. W.H. and A.F. were supported by Royal Society University Research Fellowships. PHS acknowledges support from a McGill Space Institute Fellowship and the Canada 150 Research Chairs Program. E.d.L.A. was supported by the STFC through the Ernest Rutherford Fellowship.} %\dataavailability{The Planck and DES posterior samples used in this paper are available at \url{https://zenodo.org/record/4116393#.Y1k18y8w1hA}.} \conflictsofinterest{The authors declare no conflict of interest.} % LaTeX support: latex@mdpi.com % In case you need support, please attach all files that are necessary for compiling as well as the log file, and specify the details of your LaTeX setup (which operating system and LaTeX version / tools you are using). %================================================================= \documentclass[preprints,article,accept,moreauthors,pdftex]{Definitions/mdpi} % If you would like to post an early version of this manuscript as a preprint, you may use preprint as the journal and change 'submit' to 'accept'. The document class line would be, e.g., \documentclass[preprints,article,accept,moreauthors,pdftex]{mdpi}. This is especially recommended for submission to arXiv, where line numbers should be removed before posting. 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This will only make changes to the frontpage (e.g., the logo of the journal will get visible), the headings, and the copyright information. Also, line numbering will be removed. Journal info and pagination for accepted papers will also be assigned by the Editorial Office. %------------------ % moreauthors %------------------ % If there is only one author the class option oneauthor should be used. Otherwise use the class option moreauthors. %--------- % pdftex %--------- % The option pdftex is for use with pdfLaTeX. If eps figures are used, remove the option pdftex and use LaTeX and dvi2pdf. %================================================================= \firstpage{1} \makeatletter \setcounter{page}{\@firstpage} \makeatother \pubvolume{xx} \issuenum{x} \articlenumber{x} \pubyear{2022} \copyrightyear{2022} %\externaleditor{Academic Editor: name} \history{Received: date; Accepted: date; Published: date} %\updates{yes} % If there is an update available, un-comment this line %% MDPI internal command: uncomment if new journal that already uses continuous page numbers %\continuouspages{yes} %------------------------------------------------------------------ % The following line should be uncommented if the LaTeX file is uploaded to arXiv.org %\pdfoutput=1 %================================================================= % Add packages and commands here. The following packages are loaded in our class file: fontenc, calc, indentfirst, fancyhdr, graphicx, lastpage, ifthen, lineno, float, amsmath, setspace, enumitem, mathpazo, booktabs, titlesec, etoolbox, amsthm, hyphenat, natbib, hyperref, footmisc, geometry, caption, url, mdframed, tabto, soul, multirow, microtype, tikz % macros provided by Ali Mohammad-Djafari % Use only if needed \usepackage{color,xspace} \usepackage{Definitions/macros_gpi} \usepackage{amssymb} %\usepackage{algorithm2e} %\usepackage{cases} %\usepackage{caption} %\usepackage{subcaption} %\usepackage{array} %\usepackage{xcolor} %\usepackage{multirow} %\usepackage{cite} \usepackage{hyperref} \usepackage{cleveref} \usepackage{tikz} \usepackage{pdflscape} \newcommand{\TikZ}{Ti\textit{k}Z\xspace} \newcommand{\mean}[2][]{\left\langle#2\right\rangle_{#1}} % Use in case you have great number of graphics file. Put all of them in Figs sub-directory \graphicspath{{Figs/}} \setitemize{parsep=6pt,itemsep=0pt,leftmargin=*,labelsep=5.5mm} \setenumerate{parsep=6pt,itemsep=0pt,leftmargin=*,labelsep=5.5mm} \setlist[description]{itemsep=0mm} \usepackage{environ} \NewEnviron{myequation}{% \begin{equation} \scalebox{0.95}{$\BODY$} \end{equation} } %================================================================= %% Please use the following mathematics environments: Theorem, Lemma, Corollary, Proposition, Characterization, Property, Problem, Example, ExamplesandDefinitions, Hypothesis, Remark, Definition, Notation, Assumption %% For proofs, please use the proof environment (the amsthm package is loaded by the MDPI class). %================================================================= % Full title of the paper (Capitalized) % REPLACE WITH YOUR TITLE? AUTHOR? AFFILIATION, .. \Title{ Marginal Bayesian Statistics Using Masked Autoregressive Flows and Kernel Density Estimators with Examples in Cosmology$^\dagger$} % Author Orchid ID: enter ID or remove command %\newcommand{\orcidauthorA}{0000-0002-4367-3550} % Add \orcidA{} behind the author's name %\newcommand{\orcidauthorB}{0000-0000-000-000X} % Add \orcidB{} behind the author's name % Authors, for the paper (add full first names) \Author{Harry Bevins $^{1, 2}$, Will Handley $^{1, 2}$, %\ldots Pablo Lemos $^{3, 4}$, Peter Sims $^{5}$ , Eloy de Lera Acedo $^{1, 2}$, Anastasia Fialkov $^{2, 3}$} %Please carefully check the accuracy of names and affiliations. Changes will not be possible after proofreading. % Authors, for metadata in PDF \AuthorNames{Harry T. J. Bevins, William J. Handley, Pablo Lemos, Peter Sims, Eloy de Lera Acedo, Anastasia Fialkov} % Affiliations / Addresses (Add [1] after \address if there is only one affiliation.) \address{% $^1$ \quad Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK \\ $^{2}$ \quad Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, UK \\ $^{3}$ \quad Department of Physics \& Astronomy, University College London, Gower Street, London, WC1E 6BT, UK \\ $^{4}$ \quad Department of Physics and Astronomy, Pevensey Building, University of Sussex, Brighton, BN1 9QH, UK \\ $^{5}$ \quad Department of Physics and McGill Space Institute, McGill University, 3600 University Street, Montreal, QC H3A 2T8, Canada \\ $^{6}$ \quad Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK \\ } %\corres{Harry T. J. Bevins} % Current address and/or shared authorship \firstnote{Submitted to International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, IHP, Paris, July 18-22, 2022.} %\secondnote{These authors contributed equally to this work.} % The commands \thirdnote{} till \eighthnote{} are available for further notes %\simplesumm{} % Simple summary %\conference{} % An extended version of a conference paper % PUT YOUR ABSTRACT HERE % Abstract (Do not insert blank lines, i.e. \\) \abstract{ Cosmological experiments often employ Bayesian workflows to derive constraints on cosmological and astrophysical parameters from their data. It has been shown that these constraints can be combined across different probes such as Planck and the Dark Energy Survey and that this can be a valuable exercise to improve our understanding of the universe and quantify tension between multiple experiments. However, these experiments are typically plagued by differing systematics, instrumental effects and contaminating signals, which we collectively refer to as `nuisance' components, that have to be modelled alongside target signals of interest. This leads to high dimensional parameter spaces, especially when combining data sets, with $ \gtrsim 20$ dimensions of which only $\sim 5$ correspond to key physical quantities. We present a means by which to combine constraints from different data sets in a computationally efficient manner by generating rapid, reusable and reliable marginal probability density estimators, giving us access to nuisance-free likelihoods. This is possible through the unique combination of nested sampling, which gives us access to Bayesian evidences, and the marginal Bayesian statistics code \textsc{margarine}. Our method is lossless in the signal parameters, resulting in the same posterior distributions as would be found from a full nested sampling run over all nuisance parameters, and typically quicker than evaluating full likelihoods. We demonstrate our approach by applying it to the combination of posteriors from the Dark Energy Survey and Planck. } % PUT YOUR KEYWORDS HERE % Keywords \keyword{\textls[-15]{Bayesian Analysis, Normalizing Flows, Kullback-Leibler, Cosmology}} %\setcounter{secnumdepth}{4} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % For abstract only, no need % When your abstract is accepted, you can put % your main text here. The same can be used for the final selected paper submission. \input{main_text.tex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \reftitle{References} \bibliography{refs, journals} \end{document} ``` 4. **Bibliographic Information:** ```bbl \begin{thebibliography}{-------} \providecommand{\natexlab}[1]{#1} \bibitem[{DES Collaboration}(2018)]{DES_Year1_2018} {DES Collaboration}. \newblock {Dark Energy Survey year 1 results: Cosmological constraints from galaxy clustering and weak lensing}. \newblock {\em Phys.\ Rev.~D} {\bf 2018}, {\em 98},~043526, \href{http://xxx.lanl.gov/abs/1708.01530}{{\normalfont [arXiv:astro-ph.CO/1708.01530]}}. \newblock doi:{\changeurlcolor{black}\href{https://doi.org/10.1103/PhysRevD.98.043526}{\detokenize{10.1103/PhysRevD.98.043526}}}. \bibitem[{DES Collaboration}(2021)]{DES_year3_2021} {DES Collaboration}. \newblock Dark Energy Survey year 3 results: covariance modelling and its impact on parameter estimation and quality of fit. \newblock {\em MNRAS} {\bf 2021}, {\em 508},~3125--3165, \href{http://xxx.lanl.gov/abs/2012.08568}{{\normalfont [arXiv:astro-ph.CO/2012.08568]}}. \newblock doi:{\changeurlcolor{black}\href{https://doi.org/10.1093/mnras/stab2384}{\detokenize{10.1093/mnras/stab2384}}}. \bibitem[{Planck Collaboration}(2020)]{Planck_cosmo_2020} {Planck Collaboration}. \newblock {Planck 2018 results. VI. Cosmological parameters}. \newblock {\em A\&A} {\bf 2020}, {\em 641}, \href{http://xxx.lanl.gov/abs/1807.06209}{{\normalfont [arXiv:astro-ph.CO/1807.06209]}}. \newblock doi:{\changeurlcolor{black}\href{https://doi.org/10.1051/0004-6361/201833910}{\detokenize{10.1051/0004-6361/201833910}}}. \bibitem[Anstey \em{et~al.}(2020)Anstey, Acedo, and Handley]{REACH} Anstey, D.; Acedo, E.d.L.; Handley, W. \newblock A {General} {Bayesian} {Framework} for {Foreground} {Modelling} and {Chromaticity} {Correction} for {Global} 21cm {Experiments}. \newblock {\em arXiv:2010.09644 [astro-ph]} {\bf 2020}. \newblock arXiv: 2010.09644. \bibitem[{Bevins} \em{et~al.}(2022){Bevins}, {de Lera Acedo}, {Fialkov}, {Handley}, {Singh}, {Subrahmanyan}, and {Barkana}]{SARAS2} {Bevins}, H.T.J.; {de Lera Acedo}, E.; {Fialkov}, A.; {Handley}, W.J.; {Singh}, S.; {Subrahmanyan}, R.; {Barkana}, R. \newblock {A Comprehensive Bayesian re-analysis of the SARAS2 data from the Epoch of Reionization}. \newblock {\em arXiv e-prints} {\bf 2022}, p. arXiv:2201.11531, \href{http://xxx.lanl.gov/abs/2201.11531}{{\normalfont [arXiv:astro-ph.CO/2201.11531]}}. \bibitem[Parzen(1962)]{parzen_KDE_1962} Parzen, E. \newblock On {Estimation} of a {Probability} {Density} {Function} and {Mode}. \newblock {\em The Annals of Mathematical Statistics} {\bf 1962}, {\em 33},~1065--1076. \newblock Publisher: Institute of Mathematical Statistics, doi:{\changeurlcolor{black}\href{https://doi.org/10.1214/aoms/1177704472}{\detokenize{10.1214/aoms/1177704472}}}. \bibitem[Rosenblatt(1956)]{rosenblatt_KDE_1956} Rosenblatt, M. \newblock Remarks on {Some} {Nonparametric} {Estimates} of a {Density} {Function}. \newblock {\em The Annals of Mathematical Statistics} {\bf 1956}, {\em 27},~832--837. \newblock Publisher: Institute of Mathematical Statistics, doi:{\changeurlcolor{black}\href{https://doi.org/10.1214/aoms/1177728190}{\detokenize{10.1214/aoms/1177728190}}}. \bibitem[{Papamakarios} \em{et~al.}(2017){Papamakarios}, {Pavlakou}, and {Murray}]{Papamarkarios_MAF_2017} {Papamakarios}, G.; {Pavlakou}, T.; {Murray}, I. \newblock {Masked Autoregressive Flow for Density Estimation}. \newblock {\em arXiv e-prints} {\bf 2017}, p. arXiv:1705.07057, \href{http://xxx.lanl.gov/abs/1705.07057}{{\normalfont [arXiv:stat.ML/1705.07057]}}. \bibitem[{Bevins} \em{et~al.}(2022){Bevins}, {Handley}, {Lemos}, {Sims}, {de Lera Acedo}, {Fialkov}, and {Alsing}]{margarine_neurips} {Bevins}, H.T.J.; {Handley}, W.J.; {Lemos}, P.; {Sims}, P.H.; {de Lera Acedo}, E.; {Fialkov}, A.; {Alsing}, J. \newblock {Removing the fat from your posterior samples with margarine}. \newblock {\em arXiv e-prints} {\bf 2022}, p. arXiv:2205.12841, \href{http://xxx.lanl.gov/abs/2205.12841}{{\normalfont [arXiv:astro-ph.IM/2205.12841]}}. \bibitem[Kullback and Leibler(1951)]{kullback_information_1951} Kullback, S.; Leibler, R.A. \newblock On {Information} and {Sufficiency}. \newblock {\em The Annals of Mathematical Statistics} {\bf 1951}, {\em 22},~79--86. \newblock Publisher: Institute of Mathematical Statistics, doi:{\changeurlcolor{black}\href{https://doi.org/10.1214/aoms/1177729694}{\detokenize{10.1214/aoms/1177729694}}}. \bibitem[{Handley} and {Lemos}(2019)]{Handley_dimensionality_2019} {Handley}, W.; {Lemos}, P. \newblock {Quantifying dimensionality: Bayesian cosmological model complexities}. \newblock {\em Phys.\ Rev.~D} {\bf 2019}, {\em 100},~023512, \href{http://xxx.lanl.gov/abs/1903.06682}{{\normalfont [arXiv:astro-ph.CO/1903.06682]}}. \newblock doi:{\changeurlcolor{black}\href{https://doi.org/10.1103/PhysRevD.100.023512}{\detokenize{10.1103/PhysRevD.100.023512}}}. \bibitem[Abadi \em{et~al.}(2015)Abadi, Agarwal, Barham, Brevdo, Chen, Citro, Corrado, Davis, Dean, Devin, Ghemawat, Goodfellow, Harp, Irving, Isard, Jia, Jozefowicz, Kaiser, Kudlur, Levenberg, Man\'{e}, Monga, Moore, Murray, Olah, Schuster, Shlens, Steiner, Sutskever, Talwar, Tucker, Vanhoucke, Vasudevan, Vi\'{e}gas, Vinyals, Warden, Wattenberg, Wicke, Yu, and Zheng]{tensorflow2015-whitepaper} Abadi, M.; Agarwal, A.; Barham, P.; Brevdo, E.; Chen, Z.; Citro, C.; Corrado, G.S.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Goodfellow, I.; Harp, A.; Irving, G.; Isard, M.; Jia, Y.; Jozefowicz, R.; Kaiser, L.; Kudlur, M.; Levenberg, J.; Man\'{e}, D.; Monga, R.; Moore, S.; Murray, D.; Olah, C.; Schuster, M.; Shlens, J.; Steiner, B.; Sutskever, I.; Talwar, K.; Tucker, P.; Vanhoucke, V.; Vasudevan, V.; Vi\'{e}gas, F.; Vinyals, O.; Warden, P.; Wattenberg, M.; Wicke, M.; Yu, Y.; Zheng, X. \newblock {TensorFlow}: Large-Scale Machine Learning on Heterogeneous Systems, 2015. \newblock Software available from tensorflow.org. \bibitem[Virtanen \em{et~al.}(2020)Virtanen, Gommers, Oliphant, Haberland, Reddy, Cournapeau, Burovski, Peterson, Weckesser, Bright, {van der Walt}, Brett, Wilson, Millman, Mayorov, Nelson, Jones, Kern, Larson, Carey, Polat, Feng, Moore, {VanderPlas}, Laxalde, Perktold, Cimrman, Henriksen, Quintero, Harris, Archibald, Ribeiro, Pedregosa, {van Mulbregt}, and {SciPy 1.0 Contributors}]{2020SciPy-NMeth} Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; {van der Walt}, S.J.; Brett, M.; Wilson, J.; Millman, K.J.; Mayorov, N.; Nelson, A.R.J.; Jones, E.; Kern, R.; Larson, E.; Carey, C.J.; Polat, {\.I}.; Feng, Y.; Moore, E.W.; {VanderPlas}, J.; Laxalde, D.; Perktold, J.; Cimrman, R.; Henriksen, I.; Quintero, E.A.; Harris, C.R.; Archibald, A.M.; Ribeiro, A.H.; Pedregosa, F.; {van Mulbregt}, P.; {SciPy 1.0 Contributors}. \newblock {{SciPy} 1.0: Fundamental Algorithms for Scientific Computing in Python}. \newblock {\em Nature Methods} {\bf 2020}, {\em 17},~261--272. \newblock doi:{\changeurlcolor{black}\href{https://doi.org/10.1038/s41592-019-0686-2}{\detokenize{10.1038/s41592-019-0686-2}}}. \bibitem[{Handley} and {Lemos}(2019)]{Handley_tensions_2019} {Handley}, W.; {Lemos}, P. \newblock {Quantifying tensions in cosmological parameters: Interpreting the DES evidence ratio}. \newblock {\em Phys.\ Rev.~D} {\bf 2019}, {\em 100},~043504, \href{http://xxx.lanl.gov/abs/1902.04029}{{\normalfont [arXiv:astro-ph.CO/1902.04029]}}. \newblock doi:{\changeurlcolor{black}\href{https://doi.org/10.1103/PhysRevD.100.043504}{\detokenize{10.1103/PhysRevD.100.043504}}}. \bibitem[Handley \em{et~al.}(2015{\natexlab{a}})Handley, Hobson, and Lasenby]{Handley2015a} Handley, W.J.; Hobson, M.P.; Lasenby, A.N. \newblock {PolyChord}: nested sampling for cosmology. \newblock {\em Mon. Not. of the R. Astron. Soc.} {\bf 2015}, {\em 450},~L61--L65. \newblock arXiv: 1502.01856, doi:{\changeurlcolor{black}\href{https://doi.org/10.1093/mnrasl/slv047}{\detokenize{10.1093/mnrasl/slv047}}}. \bibitem[Handley \em{et~al.}(2015{\natexlab{b}})Handley, Hobson, and Lasenby]{Handley2015b} Handley, W.J.; Hobson, M.P.; Lasenby, A.N. \newblock {PolyChord}: next-generation nested sampling. \newblock {\em Mon. Not. of the R. Astron. Soc.} {\bf 2015}, {\em 453},~4385--4399. \newblock arXiv: 1506.00171, doi:{\changeurlcolor{black}\href{https://doi.org/10.1093/mnras/stv1911}{\detokenize{10.1093/mnras/stv1911}}}. \bibitem[Petrosyan and Handley(2022)]{supernest} Petrosyan, A.; Handley, W.J. \newblock supernest: accelerated nested sampling applied to astrophysics and cosmology. \newblock Accepted as proceedings to International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, 2022. \bibitem[Handley(2019)]{anesthetic} Handley, W. \newblock anesthetic: nested sampling visualisation. \newblock {\em The Journal of Open Source Software} {\bf 2019}, {\em 4},~1414. \newblock doi:{\changeurlcolor{black}\href{https://doi.org/10.21105/joss.01414}{\detokenize{10.21105/joss.01414}}}. \end{thebibliography} ``` 5. **Author Information:** - Lead Author: {'name': 'Harry Bevins'} - Full Authors List: ```yaml Harry Bevins: coi: start: 2023-10-01 thesis: null phd: start: 2019-10-01 end: 2023-03-31 supervisors: - Will Handley - Eloy de Lera Acedo - Anastasia Fialkov thesis: A Machine Learning-enhanced Toolbox for Bayesian 21-cm Data Analysis and Constraints on the Astrophysics of the Early Universe original_image: images/originals/harry_bevins.jpeg image: /assets/group/images/harry_bevins.jpg links: Webpage: https://htjb.github.io/ GitHub: https://github.com/htjb ADS: https://ui.adsabs.harvard.edu/search/q=author%3A%22Bevins%2C%20H.%20T.%20J.%22&sort=date%20desc%2C%20bibcode%20desc&p_=0 Publons: https://publons.com/researcher/5239833/harry-bevins/ destination: 2023-04-01: Postdoc in Cambridge (Eloy) 2023-10-01: Cambridge Kavli Fellowship 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 Pablo Lemos: {} Peter Sims: {} 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 Anastasia Fialkov: coi: start: 2019-10-01 thesis: null image: https://www.ast.cam.ac.uk/sites/default/files/styles/inline/public/anastasia-fialkov-20180213-sq2.jpg?itok=am4DF9YQ links: Department webpage: https://www.ast.cam.ac.uk/people/Anastasia.Fialkov ``` 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 [2207.11457](https://arxiv.org/abs/2207.11457) is featured in the first sentence. Generate only the final Markdown output that meets all these requirements. {% endraw %}