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class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/EarthyScience/HybridVariationalInference.jl/blob/main/docs/src/problem.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Problem"><a class="docs-heading-anchor" href="#Problem">Problem</a><a id="Problem-1"></a><a class="docs-heading-anchor-permalink" href="#Problem" title="Permalink"></a></h1><p>Consider the case of Parameter learning, a special case of hybrid models, where a machine learning model, <span>$g_{\phi_g}$</span>, uses known covariates <span>$x_{Mi}$</span> at site <span>$i$</span>, to predict a subset of the parameters, <span>$\theta$</span> of the process based model, <span>$f$</span>.</p><p>We are interested in both,</p><ul><li>the uncertainty of hybrid model predictions, <span>$ŷ$</span> (predictive posterior), and</li><li>the uncertainty of process-model parameters <span>$\theta$</span>, including their correlations (posterior)</li></ul><p>For example we have soil organic matter process-model that predicts carbon stocks for different sites. We need to parameterize the unknown carbon use efficiency (CUE) of the soil microbial community that differs by site, but is hypothesized to correlate with climate variables and pedogenic factors, such as clay content. We apply a machine learning model to estimate CUE and fit it end-to-end with other parameters of the process-model to observed carbon stocks. In addition to the predicted CUE, we are interested in the uncertainty of CUE and its correlation with other parameters, such a the capacity of the soil minerals to bind carbon. I.e. we are interetes in the entire posterior probability distribution of the model parameters.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../tutorials/basic_cpu/">Basic workflow »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.14.1 on <span class="colophon-date" title="Wednesday 20 August 2025 08:39">Wednesday 20 August 2025</span>. 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