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{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-18T00:21:43","documenter_version":"1.8.0"}}
{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2025-01-19T00:23:10","documenter_version":"1.8.0"}}
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4 changes: 2 additions & 2 deletions dev/examples/initializing-hmc/index.html

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6 changes: 3 additions & 3 deletions dev/examples/quickstart/index.html
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prob_banana = BananaProblem()</code></pre><p>and then visualise it:</p><pre><code class="language-julia hljs">xrange = -3.5:0.05:3.5
yrange = -3:0.05:7
logp_banana(x) = LogDensityProblems.logdensity(prob_banana, x)
contour(xrange, yrange, exp ∘ logp_banana ∘ Base.vect; xlabel=&quot;x₁&quot;, ylabel=&quot;x₂&quot;)</code></pre><img src="4bcc5d0d.svg" alt="Example block output"/><p>Now we run <a href="../../lib/public/#Pathfinder.pathfinder"><code>pathfinder</code></a>.</p><pre><code class="language-julia hljs">result = pathfinder(prob_banana; init_scale=10)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Single-path Pathfinder result
contour(xrange, yrange, exp ∘ logp_banana ∘ Base.vect; xlabel=&quot;x₁&quot;, ylabel=&quot;x₂&quot;)</code></pre><img src="b522a81d.svg" alt="Example block output"/><p>Now we run <a href="../../lib/public/#Pathfinder.pathfinder"><code>pathfinder</code></a>.</p><pre><code class="language-julia hljs">result = pathfinder(prob_banana; init_scale=10)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">Single-path Pathfinder result
tries: 1
draws: 5
fit iteration: 10 (total: 17)
@@ -127,7 +127,7 @@

contour(xrange, yrange, exp ∘ logp_banana ∘ Base.vect)
scatter!(x₁_approx, x₂_approx; msw=0, ms=2, alpha=0.5, color=1)
plot!(xlims=extrema(xrange), ylims=extrema(yrange), xlabel=&quot;x₁&quot;, ylabel=&quot;x₂&quot;, legend=false)</code></pre><img src="afa2cd6a.svg" alt="Example block output"/><p>While the draws do a poor job of covering the tails of the distribution, they are still useful for identifying the nonlinear correlation between these two parameters.</p><h2 id="A-100-dimensional-funnel"><a class="docs-heading-anchor" href="#A-100-dimensional-funnel">A 100-dimensional funnel</a><a id="A-100-dimensional-funnel-1"></a><a class="docs-heading-anchor-permalink" href="#A-100-dimensional-funnel" title="Permalink"></a></h2><p>As we have seen above, running multi-path Pathfinder is much more useful for target distributions that are far from normal. One particularly difficult distribution to sample is Neal&#39;s funnel:</p><p class="math-container">\[\begin{aligned}
plot!(xlims=extrema(xrange), ylims=extrema(yrange), xlabel=&quot;x₁&quot;, ylabel=&quot;x₂&quot;, legend=false)</code></pre><img src="558a0a05.svg" alt="Example block output"/><p>While the draws do a poor job of covering the tails of the distribution, they are still useful for identifying the nonlinear correlation between these two parameters.</p><h2 id="A-100-dimensional-funnel"><a class="docs-heading-anchor" href="#A-100-dimensional-funnel">A 100-dimensional funnel</a><a id="A-100-dimensional-funnel-1"></a><a class="docs-heading-anchor-permalink" href="#A-100-dimensional-funnel" title="Permalink"></a></h2><p>As we have seen above, running multi-path Pathfinder is much more useful for target distributions that are far from normal. One particularly difficult distribution to sample is Neal&#39;s funnel:</p><p class="math-container">\[\begin{aligned}
\tau &amp;\sim \mathrm{Normal}(\mu=0, \sigma=3)\\
\beta_i &amp;\sim \mathrm{Normal}(\mu=0, \sigma=e^{\tau/2})
\end{aligned}\]</p><p>Such funnel geometries appear in other models (e.g. many hierarchical models) and typically frustrate MCMC sampling. Multi-path Pathfinder can&#39;t sample the funnel well, but it can quickly give us draws that can help us diagnose that we have a funnel.</p><p>In this example, we draw from a 100-dimensional funnel and visualize 2 dimensions.</p><pre><code class="language-julia hljs">using ReverseDiff, ADTypes
@@ -165,4 +165,4 @@

contour(τ_range, β₁_range, exp ∘ logp_funnel ∘ Base.vect)
scatter!(τ_approx, β₁_approx; msw=0, ms=2, alpha=0.5, color=1)
plot!(; xlims=extrema(τ_range), ylims=extrema(β₁_range), xlabel=&quot;τ&quot;, ylabel=&quot;β₁&quot;, legend=false)</code></pre><img src="9a45f11a.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../lib/internals/">« Internals</a><a class="docs-footer-nextpage" href="../initializing-hmc/">Initializing HMC »</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 1.8.0 on <span class="colophon-date" title="Saturday 18 January 2025 00:21">Saturday 18 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
plot!(; xlims=extrema(τ_range), ylims=extrema(β₁_range), xlabel=&quot;τ&quot;, ylabel=&quot;β₁&quot;, legend=false)</code></pre><img src="d0fa0c45.svg" alt="Example block output"/></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../../lib/internals/">« Internals</a><a class="docs-footer-nextpage" href="../initializing-hmc/">Initializing HMC »</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 1.8.0 on <span class="colophon-date" title="Sunday 19 January 2025 00:23">Sunday 19 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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Iterations = 501:1:1500
Number of chains = 8
Samples per chain = 1000
Wall duration = 6.3 seconds
Compute duration = 4.59 seconds
Wall duration = 6.23 seconds
Compute duration = 4.53 seconds
parameters = α, β, σ
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size

@@ -102,8 +102,8 @@
Iterations = 51:1:1050
Number of chains = 8
Samples per chain = 1000
Wall duration = 2.65 seconds
Compute duration = 2.27 seconds
Wall duration = 2.62 seconds
Compute duration = 2.24 seconds
parameters = α, β, σ
internals = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size, is_adapt

@@ -123,4 +123,4 @@
α 1.3887 1.4468 1.4742 1.5028 1.5598
β 1.9912 2.0014 2.0062 2.0110 2.0208
σ 0.1884 0.2059 0.2159 0.2267 0.2502
</code></pre><p>See <a href="../initializing-hmc/#Initializing-HMC-with-Pathfinder">Initializing HMC with Pathfinder</a> for further examples.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../initializing-hmc/">« Initializing HMC</a><a class="docs-footer-nextpage" href="../../references/">References »</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 1.8.0 on <span class="colophon-date" title="Saturday 18 January 2025 00:21">Saturday 18 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
</code></pre><p>See <a href="../initializing-hmc/#Initializing-HMC-with-Pathfinder">Initializing HMC with Pathfinder</a> for further examples.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../initializing-hmc/">« Initializing HMC</a><a class="docs-footer-nextpage" href="../../references/">References »</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 1.8.0 on <span class="colophon-date" title="Sunday 19 January 2025 00:23">Sunday 19 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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\end{align}\]</p><p><strong>References</strong></p><ul><li>[<a href="../../references/#Byrd1994">2</a>]: Byrd et al. Math. Program. 63, 1994.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/52fef889ec6b579e35d251c0fc5b74628214b98f/src/inverse_hessian.jl#L68-L97">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Pathfinder.lbfgs_inverse_hessians-Tuple{Any, Any}" href="#Pathfinder.lbfgs_inverse_hessians-Tuple{Any, Any}"><code>Pathfinder.lbfgs_inverse_hessians</code></a><span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">lbfgs_inverse_hessians(
θs, ∇logpθs; Hinit=gilbert_init, history_length=5, ϵ=1e-12
) -&gt; Tuple{Vector{WoodburyPDMat},Int}</code></pre><p>From an L-BFGS trajectory and gradients, compute the inverse Hessian approximations at each point.</p><p>Given positions <code>θs</code> with gradients <code>∇logpθs</code>, construct LBFGS inverse Hessian approximations with the provided <code>history_length</code>.</p><p>The 2nd returned value is the number of BFGS updates to the inverse Hessian matrices that were rejected due to keeping the inverse Hessian positive definite.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/52fef889ec6b579e35d251c0fc5b74628214b98f/src/inverse_hessian.jl#L12-L24">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Pathfinder.pdfactorize-Tuple{AbstractMatrix, AbstractMatrix, AbstractMatrix}" href="#Pathfinder.pdfactorize-Tuple{AbstractMatrix, AbstractMatrix, AbstractMatrix}"><code>Pathfinder.pdfactorize</code></a><span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">pdfactorize(A, B, D) -&gt; WoodburyPDFactorization</code></pre><p>Factorize the positive definite matrix <span>$W = A + B D B^\mathrm{T}$</span>.</p><p>The result is the &quot;square root&quot; factorization <code>(L, R)</code>, where <span>$W = L R$</span> and <span>$L = R^\mathrm{T}$</span>.</p><p>Let <span>$U^\mathrm{T} U = A$</span> be the Cholesky decomposition of <span>$A$</span>, and let <span>$Q X = U^{-\mathrm{T}} B$</span> be a thin QR decomposition. Define <span>$C = I + XDX^\mathrm{T}$</span>, with the Cholesky decomposition <span>$V^\mathrm{T} V = C$</span>. Then, <span>$W = R^\mathrm{T} R$</span>, where</p><p class="math-container">\[R = \begin{pmatrix} U &amp; 0 \\ 0 &amp; I \end{pmatrix} Q^\mathrm{T} V.\]</p><p>The positive definite requirement is equivalent to the requirement that both <span>$A$</span> and <span>$C$</span> are positive definite.</p><p>For a derivation of this decomposition for the special case of diagonal <span>$A$</span>, see <a href="../../references/#ZhangPathfinder2021">Zhang <em>et al.</em> [1]</a>, appendix A.</p><p>See <a href="#Pathfinder.pdunfactorize-Tuple{Pathfinder.WoodburyPDFactorization}"><code>pdunfactorize</code></a>, <a href="#Pathfinder.WoodburyPDFactorization"><code>WoodburyPDFactorization</code></a>, <a href="#Pathfinder.WoodburyPDMat"><code>WoodburyPDMat</code></a></p><p><strong>References</strong></p><ul><li>[<a href="../../references/#ZhangPathfinder2021">1</a>]: Zhang et al. JMLR 23(306), 2022.</li></ul></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/52fef889ec6b579e35d251c0fc5b74628214b98f/src/woodbury.jl#L173-L200">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Pathfinder.pdunfactorize-Tuple{Pathfinder.WoodburyPDFactorization}" href="#Pathfinder.pdunfactorize-Tuple{Pathfinder.WoodburyPDFactorization}"><code>Pathfinder.pdunfactorize</code></a><span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">pdunfactorize(F::WoodburyPDFactorization) -&gt; (A, B, D)</code></pre><p>Perform a reverse operation to <a href="#Pathfinder.pdfactorize-Tuple{AbstractMatrix, AbstractMatrix, AbstractMatrix}"><code>pdfactorize</code></a>.</p><p>Note that this function does not compute the inverse of <code>pdfactorize</code>. It only computes matrices that produce the same matrix <span>$W = A + B D B^\mathrm{T}$</span> as for the inputs to <code>pdfactorize</code>.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/52fef889ec6b579e35d251c0fc5b74628214b98f/src/woodbury.jl#L209-L217">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Pathfinder.resample-NTuple{4, Any}" href="#Pathfinder.resample-NTuple{4, Any}"><code>Pathfinder.resample</code></a><span class="docstring-category">Method</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">resample(rng, x, log_weights, ndraws) -&gt; (draws, psis_result)
resample(rng, x, ndraws) -&gt; draws</code></pre><p>Draw <code>ndraws</code> samples from <code>x</code>, with replacement.</p><p>If <code>log_weights</code> is provided, perform Pareto smoothed importance resampling.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/52fef889ec6b579e35d251c0fc5b74628214b98f/src/resample.jl#L1-L8">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../public/">« Public</a><a class="docs-footer-nextpage" href="../../examples/quickstart/">Quickstart »</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 1.8.0 on <span class="colophon-date" title="Saturday 18 January 2025 00:21">Saturday 18 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
resample(rng, x, ndraws) -&gt; draws</code></pre><p>Draw <code>ndraws</code> samples from <code>x</code>, with replacement.</p><p>If <code>log_weights</code> is provided, perform Pareto smoothed importance resampling.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/mlcolab/Pathfinder.jl/blob/52fef889ec6b579e35d251c0fc5b74628214b98f/src/resample.jl#L1-L8">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../public/">« Public</a><a class="docs-footer-nextpage" href="../../examples/quickstart/">Quickstart »</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 1.8.0 on <span class="colophon-date" title="Sunday 19 January 2025 00:23">Sunday 19 January 2025</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>References · Pathfinder.jl</title><meta name="title" content="References · Pathfinder.jl"/><meta property="og:title" content="References · Pathfinder.jl"/><meta property="twitter:title" content="References · Pathfinder.jl"/><meta name="description" content="Documentation for Pathfinder.jl."/><meta property="og:description" content="Documentation for Pathfinder.jl."/><meta property="twitter:description" content="Documentation for Pathfinder.jl."/><meta property="og:url" content="https://mlcolab.github.io/Pathfinder.jl/references/"/><meta property="twitter:url" content="https://mlcolab.github.io/Pathfinder.jl/references/"/><link rel="canonical" href="https://mlcolab.github.io/Pathfinder.jl/references/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" 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Journal of Machine Learning Research <strong>23</strong>, 1–49 (2022), <a href="https://arxiv.org/abs/2108.03782">arXiv:2108.03782</a>. Accessed on Dec 6, 2024.</div></dd><dt>[2]</dt><dd><div id="Byrd1994">R. H. Byrd, J. Nocedal and R. B. Schnabel. <em>Representations of Quasi-Newton Matrices and Their Use in Limited Memory Methods</em>. <a href="https://doi.org/10.1007/BF01582063">Mathematical Programming <strong>63</strong>, 129–156</a> (1994).</div></dd><dt>[3]</dt><dd><div id="StanHMCParameters"><a href="https://mc-stan.org/docs/reference-manual/mcmc.html#hmc-algorithm-parameters"><em>Stan Reference Manual: HMC algorithm parameters</em></a>. Accessed on Dec 6, 2024.</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../examples/turing/">« Turing usage</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 1.8.0 on <span class="colophon-date" title="Saturday 18 January 2025 00:21">Saturday 18 January 2025</span>. 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<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>References · Pathfinder.jl</title><meta name="title" content="References · Pathfinder.jl"/><meta property="og:title" content="References · Pathfinder.jl"/><meta property="twitter:title" content="References · Pathfinder.jl"/><meta name="description" content="Documentation for Pathfinder.jl."/><meta property="og:description" content="Documentation for Pathfinder.jl."/><meta property="twitter:description" content="Documentation for Pathfinder.jl."/><meta property="og:url" content="https://mlcolab.github.io/Pathfinder.jl/references/"/><meta property="twitter:url" content="https://mlcolab.github.io/Pathfinder.jl/references/"/><link rel="canonical" href="https://mlcolab.github.io/Pathfinder.jl/references/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" 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id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">Pathfinder.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li><span class="tocitem">Library</span><ul><li><a class="tocitem" href="../lib/public/">Public</a></li><li><a class="tocitem" href="../lib/internals/">Internals</a></li></ul></li><li><span class="tocitem">Examples</span><ul><li><a class="tocitem" href="../examples/quickstart/">Quickstart</a></li><li><a class="tocitem" href="../examples/initializing-hmc/">Initializing HMC</a></li><li><a class="tocitem" href="../examples/turing/">Turing usage</a></li></ul></li><li class="is-active"><a class="tocitem" href>References</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label 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Journal of Machine Learning Research <strong>23</strong>, 1–49 (2022), <a href="https://arxiv.org/abs/2108.03782">arXiv:2108.03782</a>. Accessed on Dec 6, 2024.</div></dd><dt>[2]</dt><dd><div id="Byrd1994">R. H. Byrd, J. Nocedal and R. B. Schnabel. <em>Representations of Quasi-Newton Matrices and Their Use in Limited Memory Methods</em>. <a href="https://doi.org/10.1007/BF01582063">Mathematical Programming <strong>63</strong>, 129–156</a> (1994).</div></dd><dt>[3]</dt><dd><div id="StanHMCParameters"><a href="https://mc-stan.org/docs/reference-manual/mcmc.html#hmc-algorithm-parameters"><em>Stan Reference Manual: HMC algorithm parameters</em></a>. Accessed on Dec 6, 2024.</div></dd></dl></div></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../examples/turing/">« Turing usage</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 1.8.0 on <span class="colophon-date" title="Sunday 19 January 2025 00:23">Sunday 19 January 2025</span>. 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