-U = \operatorname{BlockDiag}(a_P, a_M)\]</p><p>where the predicted value of <span>$\mu_{\zeta_{Ms}}$</span> may depend on the random draw value of <span>$\zeta_P = \zeta_{r,P} + \mu_{\zeta_P}$</span>. By this construction HVI better supports the assumption that <span>$\zeta_{rM}$</span> is conditionally independent of <span>$\zeta_{rP}$</span>, which is required to macke the cholesky-factor, <span>$U$</span> of the covariance matrix block-diagonal.</p><p>The above procedure makes an additional subtle approximation. HVI allows the variance of <span>$\zeta_{M}$</span> to scale with its magnitude. In the computation of the correlation matrix, however, HVI uses the mean, <span>$\mu_{\zeta_{Mi}}$</span>, rather than the actual sampled value, <span>$\zeta_{Mi}$</span>. If it used the actual value, then the distribution of <span>$\zeta$</span> would need to be described as a general distribution, <span>$p(\zeta) = p(\zeta_{Ms}|\zeta_P) \, p(\zeta_P)$</span>, that would not be normal any more, and HVI could not compute the expectation by drawing centered normla random numbers.</p><h4 id="Implementation-of-the-cost-function"><a class="docs-heading-anchor" href="#Implementation-of-the-cost-function">Implementation of the cost function</a><a id="Implementation-of-the-cost-function-1"></a><a class="docs-heading-anchor-permalink" href="#Implementation-of-the-cost-function" title="Permalink"></a></h4><p>In practical terms the cost function </p><ul><li>generates normally distributed random values <span>$(\zeta_{rP}, \zeta_{rMs})$</span> based on the cholesky factor of the covariance matrix, which depends on optimized parameters <span>$(a_P, a_M)$</span></li><li>generates a sample of <span>$\zeta_P$</span> by adding optimized parameters <span>$\mu_{\zeta_P}$</span> to <span>$\zeta_{rP}$</span></li><li>computes expected value of <span>$\mu_{\zeta_{Ms}}$</span> using the machine learning model given covariates, <span>$X_M$</span>, given <span>$\zeta_P$</span>, and given optimized parameters <span>$\phi_g$</span>.</li><li>generates a sample of <span>$\zeta_{Ms}$</span> by adding the computed <span>$\mu_{\zeta_{Ms}}$</span> to <span>$\zeta_{rMs}$</span></li><li>transforms <span>$(\zeta_{P}, \zeta_{Ms})$</span> to the original scale to get a sample of model parameters <span>$(\theta_{rP}, \theta_{rMs})$</span></li><li>computes negative Log-density of observations for each sample using the physical model, <span>$f$</span>, and subtract the absolute determinant of the transformation, evaluated at the sample.</li><li>approximates the expected value of the former by taking the mean across the samples</li><li>subtract the entropy of the normal distribution approximator </li></ul><p>The automatic differentiation through this cost function including calls to <span>$g$</span>, T, and <span>$f$</span> allows to estimate parameters, <span>$\phi$</span>, by a stochastic gradient decent method.</p></article><nav class="docs-footer"><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="Tuesday 7 October 2025 13:41">Tuesday 7 October 2025</span>. Using Julia version 1.11.7.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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