Publish docs @ c722e96

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core-functionality/index.html

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search.json

@@ -859,7 +859,7 @@ <h2 class="anchored" data-anchor-id="the-lotkavolterra-model">The Lotka–Volter
 <h2 class="anchored" data-anchor-id="direct-handling-of-bayesian-estimation-with-turing">Direct Handling of Bayesian Estimation with Turing</h2>
 <p><a href="https://docs.sciml.ai/DiffEqDocs/stable/">DifferentialEquations.jl</a> is the main Julia package for numerically solving differential equations. Its functionality is completely interoperable with Turing.jl, which means that we can directly simulate differential equations inside a Turing <code>@model</code>.</p>
 <p>For the purposes of this tutorial, we choose priors for the parameters that are quite close to the ground truth. As justification, we can imagine we have preexisting estimates for the biological rates. Practically, this helps us to illustrate the results without needing to run overly long MCMC chains.</p>
-<p>Note we also have to take special care with the ODE solver. For certain parameter combinations, the numerical solver may predict animal densities that are just barely below zero. This causes errors with the Poisson distribution, which needs a non-negative mean <span class="math inline">\(\lambda\)</span>. To avoid this happening, we tell the solver to aim for small abolute and relative errors (<code>abstol=1e-6, reltol=1e-6</code>). We also add a fudge factor <code>ϵ = 1e-5</code> to the predicted data. Since <code>ϵ</code> is greater than the solver’s tolerance, it should overcome any remaining numerical error, making sure all predicted values are positive. At the same time, it is so small compared to the data that it should have a negligible effect on inference. If this approach doesn’t work, there are some more ideas to try <a href="https://docs.sciml.ai/DiffEqDocs/stable/basics/faq/#My-ODE-goes-negative-but-should-stay-positive,-what-tools-can-help?">here</a>.</p>
+<p>Note we also have to take special care with the ODE solver. For certain parameter combinations, the numerical solver may predict animal densities that are just barely below zero. This causes errors with the Poisson distribution, which needs a non-negative mean <span class="math inline">\(\lambda\)</span>. To avoid this happening, we tell the solver to aim for small absolute and relative errors (<code>abstol=1e-6, reltol=1e-6</code>). We also add a fudge factor <code>ϵ = 1e-5</code> to the predicted data. Since <code>ϵ</code> is greater than the solver’s tolerance, it should overcome any remaining numerical error, making sure all predicted values are positive. At the same time, it is so small compared to the data that it should have a negligible effect on inference. If this approach doesn’t work, there are some more ideas to try <a href="https://docs.sciml.ai/DiffEqDocs/stable/basics/faq/#My-ODE-goes-negative-but-should-stay-positive,-what-tools-can-help?">here</a>. In the case of continuous observations (e.g.&nbsp;data derived from modelling chemical reactions), it is sufficient to use a normal distribution with the mean as the data point and an appropriately chosen variance (which can itself also be a parameter with a prior distribution).</p>
 <div id="10" class="cell" data-execution_count="1">
 <div class="code-copy-outer-scaffold"><div class="sourceCode cell-code" id="cb4"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="pp">@model</span> <span class="kw">function</span> <span class="fu">fitlv</span>(data, prob)</span>
 <span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>    <span class="co"># Prior distributions.</span></span>
@@ -903,8 +903,8 @@ <h2 class="anchored" data-anchor-id="direct-handling-of-bayesian-estimation-with
 Iterations        = 501:1:1500
 Number of chains  = 3
 Samples per chain = 1000
-Wall duration     = 51.56 seconds
-Compute duration  = 47.9 seconds
+Wall duration     = 52.06 seconds
+Compute duration  = 48.05 seconds
 parameters        = α, β, γ, δ, q
 internals         = 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, lp, logprior, loglikelihood
 
@@ -983,8 +983,8 @@ <h2 class="anchored" data-anchor-id="lotkavolterra-model-without-data-of-prey">L
 Iterations        = 1001:1:6000
 Number of chains  = 3
 Samples per chain = 5000
-Wall duration     = 28.73 seconds
-Compute duration  = 28.28 seconds
+Wall duration     = 29.65 seconds
+Compute duration  = 29.15 seconds
 parameters        = α, β, γ, δ, q
 internals         = 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, lp, logprior, loglikelihood
 
@@ -1095,8 +1095,8 @@ <h2 class="anchored" data-anchor-id="inference-of-delay-differential-equations">
 Iterations        = 151:1:450
 Number of chains  = 3
 Samples per chain = 300
-Wall duration     = 12.55 seconds
-Compute duration  = 12.32 seconds
+Wall duration     = 12.98 seconds
+Compute duration  = 12.73 seconds
 parameters        = α, β, γ, δ, q
 internals         = 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, lp, logprior, loglikelihood
 
@@ -1154,8 +1154,8 @@ <h2 class="anchored" data-anchor-id="scaling-to-large-models-adjoint-sensitiviti
 Iterations        = 501:1:1500
 Number of chains  = 1
 Samples per chain = 1000
-Wall duration     = 264.41 seconds
-Compute duration  = 264.41 seconds
+Wall duration     = 278.7 seconds
+Compute duration  = 278.7 seconds
 parameters        = α, β, γ, δ, q
 internals         = 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, lp, logprior, loglikelihood