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In engineering applications, the objective function is typically calculated with a numerically costly black-box simulation." /> <title>(PDF) A comparison of mixed-variables Bayesian optimization approaches | Jhouben Cuesta Ramirez - Academia.edu</title> <link rel="canonical" href="https://www.academia.edu/81962540/A_comparison_of_mixed_variables_Bayesian_optimization_approaches" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '2f444c8e4ed6011e01999d24d77abd5ec19178fb'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1732740429000); window.Aedu.timeDifference = new Date().getTime() - 1732740429000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"Most real optimization problems are defined over a mixed search space where the variables are both discrete and continuous. In engineering applications, the objective function is typically calculated with a numerically costly black-box simulation. General mixed and costly optimization problems are therefore of a great practical interest, yet their resolution remains in a large part an open scientific question. In this article, costly mixed problems are approached through Gaussian processes where the discrete variables are relaxed into continuous latent variables. The continuous space is more easily harvested by classical Bayesian optimization techniques than a mixed space would. Discrete variables are recovered either subsequently to the continuous optimization, or simultaneously with an additional continuous-discrete compatibility constraint that is handled with augmented Lagrangians. Several possible implementations of such Bayesian mixed optimizers are compared. In particular, th...","author":[{"@context":"https://schema.org","@type":"Person","name":"Jhouben Cuesta Ramirez"}],"contributor":[],"dateCreated":"2022-06-21","dateModified":null,"datePublished":"2021-01-01","headline":"A comparison of mixed-variables Bayesian optimization approaches","inLanguage":"en","keywords":["Mathematics","Computer Science","Optimization Problem","Latent variable","Mathematical Optimization","Discrete Optimization","Bayesian Optimization","Continuous Optimization","Constrained Optimization","arXiv"],"locationCreated":null,"publication":null,"publisher":{"@context":"https://schema.org","@type":"Organization","name":"Research Square Platform LLC"},"image":null,"thumbnailUrl":null,"url":"https://www.academia.edu/81962540/A_comparison_of_mixed_variables_Bayesian_optimization_approaches","sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"utp-co"}]}</script><link rel="stylesheet" media="all" 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In engineering applications, the objective function is typically calculated with a numerically costly black-box simulation. General mixed and costly optimization problems are therefore of a great practical interest, yet their resolution remains in a large part an open scientific question. In this article, costly mixed problems are approached through Gaussian processes where the discrete variables are relaxed into continuous latent variables. The continuous space is more easily harvested by classical Bayesian optimization techniques than a mixed space would. Discrete variables are recovered either subsequently to the continuous optimization, or simultaneously with an additional continuous-discrete compatibility constraint that is handled with augmented Lagrangians. Several possible implementations of such Bayesian mixed optimizers are compared. In particular, th...","publisher":"Research Square Platform LLC","publication_date":"2021,,"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"A comparison of mixed-variables Bayesian optimization approaches","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [23422185]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loswp.appleClientId = 'edu.academia.applesignon';</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":87823281,"attachmentType":"pdf"}"><img alt="First page of “A comparison of mixed-variables Bayesian optimization approaches”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/87823281/mini_magick20220621-22135-1i4upuq.png?1655801425" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/assets/single_work_splash/adobe.icon-574afd46eb6b03a77a153a647fb47e30546f9215c0ee6a25df597a779717f9ef.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">A comparison of mixed-variables Bayesian optimization approaches</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="23422185" href="https://utp-co.academia.edu/JhoubenCuestaRamirez"><img alt="Profile image of Jhouben Cuesta Ramirez" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/23422185/6917557/72780232/s65_jhouben.cuesta_ramirez.png" />Jhouben Cuesta Ramirez</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2021</p></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Most real optimization problems are defined over a mixed search space where the variables are both discrete and continuous. In engineering applications, the objective function is typically calculated with a numerically costly black-box simulation. General mixed and costly optimization problems are therefore of a great practical interest, yet their resolution remains in a large part an open scientific question. In this article, costly mixed problems are approached through Gaussian processes where the discrete variables are relaxed into continuous latent variables. The continuous space is more easily harvested by classical Bayesian optimization techniques than a mixed space would. Discrete variables are recovered either subsequently to the continuous optimization, or simultaneously with an additional continuous-discrete compatibility constraint that is handled with augmented Lagrangians. Several possible implementations of such Bayesian mixed optimizers are compared. In particular, th...</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":87823281,"attachmentType":"pdf","workUrl":"https://www.academia.edu/81962540/A_comparison_of_mixed_variables_Bayesian_optimization_approaches"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":87823281,"attachmentType":"pdf","workUrl":"https://www.academia.edu/81962540/A_comparison_of_mixed_variables_Bayesian_optimization_approaches"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="87823281" data-landing_url="https://www.academia.edu/81962540/A_comparison_of_mixed_variables_Bayesian_optimization_approaches" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="120517861" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/120517861/Bayesian_optimization_for_mixed_variable_multi_objective_problems">Bayesian optimization for mixed-variable, multi-objective problems</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48815719" href="https://independent.academia.edu/HarisSheikh5">Haris Sheikh</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Structural and Multidisciplinary Optimization</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Bayesian optimization for mixed-variable, multi-objective problems","attachmentId":115641298,"attachmentType":"pdf","work_url":"https://www.academia.edu/120517861/Bayesian_optimization_for_mixed_variable_multi_objective_problems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/120517861/Bayesian_optimization_for_mixed_variable_multi_objective_problems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="83707493" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/83707493/Bayesian_Optimization_For_Multi_Objective_Mixed_Variable_Problems">Bayesian Optimization For Multi-Objective Mixed-Variable Problems</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="48815719" href="https://independent.academia.edu/HarisSheikh5">Haris Sheikh</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2022</p><p class="ds-related-work--abstract ds2-5-body-sm">Optimizing multiple, non-preferential objectives for mixed-variable, expensive black-box problems is important in many areas of engineering and science. The expensive, noisy black-box nature of these problems makes them ideal candidates for Bayesian optimization (BO). Mixed-variable and multi-objective problems, however, are a challenge due to the BO&#39;s underlying smooth Gaussian process surrogate model. Current multi-objective BO algorithms cannot deal with mixed-variable problems. We present MixMOBO, the first mixed variable multi-objective Bayesian optimization framework for such problems. Using a genetic algorithm to sample the surrogate surface, optimal Pareto-fronts for multi-objective, mixed-variable design spaces can be found efficiently while ensuring diverse solutions. The method is sufficiently flexible to incorporate many different kernels and acquisition functions, including those that were developed for mixed-variable or multi-objective problems by other authors. We...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Bayesian Optimization For Multi-Objective Mixed-Variable Problems","attachmentId":88964179,"attachmentType":"pdf","work_url":"https://www.academia.edu/83707493/Bayesian_Optimization_For_Multi_Objective_Mixed_Variable_Problems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/83707493/Bayesian_Optimization_For_Multi_Objective_Mixed_Variable_Problems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="64281713" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/64281713/Discrete_Bayesian_Optimization_Algorithms_and_Applications">Discrete Bayesian Optimization Algorithms and Applications</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="5817647" href="https://independent.academia.edu/raffaeledamiano">raffaele damiano</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2020</p><p class="ds-related-work--abstract ds2-5-body-sm">Dealing with expensive-to-evaluate objective functions is a hard problem in optimization. Bayesian Optimization (BO) is a methodology allowing one to efficiently approximate the objective function and perform the optimization with as few evaluations as possible. This is achieved by introducing a surrogate model, i.e., a statistical model for the objective function, and an acquisition function that let us move through the feature space. The most common surrogate models are Gaussian Processes. While BO algorithms based on Gaussian Processes typically perform well over continuous domains, these techniques prove not so efficient when dealing with discrete or categorical variables and different approaches and settings are required. In this thesis, the Separable Bayesian Optimization algorithm (SBO) is proposed to overcome the limitations of classical BO. It moves from the idea of considering the discrete variables as nodes of a graph, over which a statistical model is built. This model i...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Discrete Bayesian Optimization Algorithms and Applications","attachmentId":76387901,"attachmentType":"pdf","work_url":"https://www.academia.edu/64281713/Discrete_Bayesian_Optimization_Algorithms_and_Applications","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/64281713/Discrete_Bayesian_Optimization_Algorithms_and_Applications"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="117709161" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/117709161/Constrained_Bayesian_Optimization_Over_Mixed_Categorical_Variables_with_Application_to_Aircraft_Design">Constrained Bayesian Optimization Over Mixed Categorical Variables, with Application to Aircraft Design</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="36416992" href="https://independent.academia.edu/NathalieBartoli">Nathalie Bartoli</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Zenodo (CERN European Organization for Nuclear Research), 2021</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Constrained Bayesian Optimization Over Mixed Categorical Variables, with Application to Aircraft Design","attachmentId":113497872,"attachmentType":"pdf","work_url":"https://www.academia.edu/117709161/Constrained_Bayesian_Optimization_Over_Mixed_Categorical_Variables_with_Application_to_Aircraft_Design","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/117709161/Constrained_Bayesian_Optimization_Over_Mixed_Categorical_Variables_with_Application_to_Aircraft_Design"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="64382777" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/64382777/Bayesian_Optimization_Under_Uncertainty">Bayesian Optimization Under Uncertainty</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39253808" href="https://utoronto.academia.edu/JustinBeland">Justin Beland</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2017</p><p class="ds-related-work--abstract ds2-5-body-sm">We consider the problem of robust optimization, where it is sought to design a system such that it sustains a specified measure of performance under uncertainty. 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Numerical studies on a test problem are presented to demonstrate the efficacy of the proposed approach.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Bayesian Optimization Under Uncertainty","attachmentId":76444724,"attachmentType":"pdf","work_url":"https://www.academia.edu/64382777/Bayesian_Optimization_Under_Uncertainty","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/64382777/Bayesian_Optimization_Under_Uncertainty"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="50220187" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/50220187/Improved_Gaussian_Process_Acquisition_for_Targeted_Bayesian_Optimization">Improved Gaussian Process Acquisition for Targeted Bayesian Optimization</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="24980269" href="https://independent.academia.edu/PeterMitic">Peter Mitic</a></div><p class="ds-related-work--metadata ds2-5-body-xs">International Journal of Modeling and Optimization</p><p class="ds-related-work--abstract ds2-5-body-sm">A black-box optimization problem is considered, in which the function to be optimized can only be expressed in terms of a complicated stochastic algorithm that takes a long time to evaluate. The value returned is required to be sufficiently near to a target value, and uses data that has a significant noise component. Bayesian Optimization with an underlying Gaussian Process is used as an optimization solution, and its effectiveness is measured in terms of the number of function evaluations required to attain the target. To improve results, a simple modification of the Gaussian Process ‘Lower Confidence Bound’ (LCB) acquisition function is proposed. The expression used for the confidence bound is squared in order to better comply with the target requirement. With this modification, much improved results compared to random selection methods and to other commonly used acquisition functions are obtained.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Improved Gaussian Process Acquisition for Targeted Bayesian Optimization","attachmentId":68287761,"attachmentType":"pdf","work_url":"https://www.academia.edu/50220187/Improved_Gaussian_Process_Acquisition_for_Targeted_Bayesian_Optimization","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/50220187/Improved_Gaussian_Process_Acquisition_for_Targeted_Bayesian_Optimization"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="113178829" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/113178829/Bayesian_Optimization_Under_Uncertainty_for_Chance_Constrained_Problems">Bayesian Optimization Under Uncertainty for Chance Constrained Problems</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="65884350" href="https://independent.academia.edu/mohamedelamri5">mohamed el amri</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2019</p><p class="ds-related-work--abstract ds2-5-body-sm">Chance constraint is an important tool for modeling the reliability on decision making in the presence of uncertainties. Indeed, the chance constraint enforces that the constraint is satisfied with probability 1 − α ( 0 &lt; α &lt; 1 ) at least. In addition, we consider that the objective func- tion is affected by uncertainties. This problem is challenging since modeling a complex system under uncertainty can be expensive and for most real-world stochastic optimization will not be computationally viable. In this talk, we propose a Bayesian methodology to efficiently solve such class of problems. The central idea is to use Gaussian Process (GP) models [1] together with appropriate acquisi- tion functions to guide the search for an optimal solution. We first show that by specifying a GP prior to the objective function, the loss function becomes tractable [2]. Similarly, using GP models for the constraints, the probability satisfaction can be efficiently approximated. Sub- sequently, w...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Bayesian Optimization Under Uncertainty for Chance Constrained Problems","attachmentId":110205839,"attachmentType":"pdf","work_url":"https://www.academia.edu/113178829/Bayesian_Optimization_Under_Uncertainty_for_Chance_Constrained_Problems","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/113178829/Bayesian_Optimization_Under_Uncertainty_for_Chance_Constrained_Problems"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="93547845" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/93547845/Bayesian_optimization_of_generalized_data">Bayesian optimization of generalized data</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="250891876" href="https://independent.academia.edu/VladimirSobes">Vladimir Sobes</a></div><p class="ds-related-work--metadata ds2-5-body-xs">EPJ Nuclear Sciences &amp; Technologies, 2018</p><p class="ds-related-work--abstract ds2-5-body-sm">Direct application of Bayes&#39; theorem to generalized data yields a posterior probability distribution function (PDF) that is a product of a prior PDF of generalized data and a likelihood function, where generalized data consists of model parameters, measured data, and model defect data. The prior PDF of generalized data is defined by prior expectation values and a prior covariance matrix of generalized data that naturally includes covariance between any two components of generalized data. A set of constraints imposed on the posterior expectation values and covariances of generalized data via a given model is formally solved by the method of Lagrange multipliers. Posterior expectation values of the constraints and their covariance matrix are conventionally set to zero, leading to a likelihood function that is a Dirac delta function of the constraining equation. It is shown that setting constraints to values other than zero is analogous to introducing a model defect. 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