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Expand Down Expand Up @@ -739,7 +739,7 @@ <h2 class="anchored" data-anchor-id="metadata"><code>Metadata</code></h2>
<li><code>md.idcs</code> is the dictionary that maps each <code>VarName</code> instance to its index in <code>md.vns</code>, <code>md.ranges</code>, <code>md.dists</code>, <code>md.orders</code> and <code>md.flags</code>.</li>
<li><code>md.vns[md.idcs[vn]] == vn</code>.</li>
<li><code>md.dists[md.idcs[vn]]</code> is the distribution of <code>vn</code>.</li>
<li><code>md.gids[md.idcs[vn]]</code> is the set of algorithms used to sample <code>vn</code>. This is used in the Gibbs sampling process.</li>
<li><code>md.gids[md.idcs[vn]]</code> is the set of algorithms used to sample <code>vn</code>. This was used by the Gibbs sampler. Since Turing v0.36 it is unused and will eventually be deleted.</li>
<li><code>md.orders[md.idcs[vn]]</code> is the number of <code>observe</code> statements before <code>vn</code> is sampled.</li>
<li><code>md.ranges[md.idcs[vn]]</code> is the index range of <code>vn</code> in <code>md.vals</code>.</li>
<li><code>md.vals[md.ranges[md.idcs[vn]]]</code> is the linearized vector of values of corresponding to <code>vn</code>.</li>
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Expand Down Expand Up @@ -761,16 +761,16 @@ <h1>Consider a probabilistic model defined by</h1>
parameters mean std mcse ess_bulk ess_tail rh ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float ⋯

a 0.9771 0.8977 0.0031 83138.7250 121471.3060 1.00 ⋯
b 2.8808 0.4890 0.0012 171437.4763 214293.9000 1.00 ⋯
a 0.9811 0.8994 0.0032 79824.8778 119042.5191 1.00 ⋯
b 2.8813 0.4881 0.0012 171981.0044 210069.8438 1.00 ⋯
2 columns omitted

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parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64

a -0.7765 0.3724 0.9756 1.5817 2.7425
b 1.9206 2.5516 2.8812 3.2103 3.8394</code></pre>
a -0.7862 0.3763 0.9809 1.5870 2.7400
b 1.9244 2.5526 2.8825 3.2092 3.8371</code></pre>
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<p>We compare these results with Turing.</p>
Expand All @@ -792,25 +792,25 @@ <h1>Consider a probabilistic model defined by</h1>
Iterations = 1:1:1000000
Number of chains = 1
Samples per chain = 1000000
Wall duration = 30.9 seconds
Compute duration = 30.9 seconds
Wall duration = 31.49 seconds
Compute duration = 31.49 seconds
parameters = a, b
internals = lp

Summary Statistics
parameters mean std mcse ess_bulk ess_tail rh ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float ⋯

a 0.9747 0.8991 0.0032 80991.8672 124482.5382 1.00 ⋯
b 2.8809 0.4883 0.0012 171713.6363 217545.2969 1.00 ⋯
a 0.9743 0.9040 0.0032 81189.5129 121269.8584 1.00 ⋯
b 2.8811 0.4891 0.0012 174601.6868 216651.2495 1.00 ⋯
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parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64

a -0.7918 0.3694 0.9750 1.5803 2.7369
b 1.9255 2.5504 2.8809 3.2120 3.8349</code></pre>
a -0.8004 0.3640 0.9756 1.5839 2.7366
b 1.9225 2.5506 2.8809 3.2119 3.8368</code></pre>
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<p>As you can see, with our simple probabilistic programming language and custom samplers we get similar results as Turing.</p>
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parameters mean std mcse ess_bulk ess_tail rh ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float ⋯

a 0.9728 0.9011 0.0032 81189.5526 121519.1032 1.00 ⋯
b 2.8810 0.4880 0.0012 171599.0254 215249.0967 1.00 ⋯
a 0.9713 0.9014 0.0032 80387.2713 119459.5514 1.00 ⋯
b 2.8796 0.4892 0.0012 172170.0034 211488.7837 1.00 ⋯
2 columns omitted

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parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64

a -0.7901 0.3647 0.9706 1.5798 2.7445
b 1.9254 2.5512 2.8796 3.2096 3.8379</code></pre>
a -0.8075 0.3629 0.9732 1.5803 2.7342
b 1.9202 2.5498 2.8803 3.2090 3.8386</code></pre>
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<p>or we can choose to sample from the prior instead</p>
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parameters mean std mcse ess_bulk ess_tail rha ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float6 ⋯

a 0.4970 0.9983 0.0040 62973.7057 122183.9030 1.000 ⋯
b 0.4793 2.2238 0.0135 27032.5061 52975.4949 1.000 ⋯
a 0.4977 1.0017 0.0040 64312.4640 127157.3129 1.000 ⋯
b 0.4929 2.2363 0.0135 27644.6544 51960.2825 1.000 ⋯
2 columns omitted

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parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64

a -1.4634 -0.1761 0.4964 1.1686 2.4551
b -3.8746 -1.0195 0.4793 1.9764 4.8342</code></pre>
a -1.4669 -0.1768 0.4980 1.1717 2.4600
b -3.8853 -1.0153 0.4988 2.0051 4.8568</code></pre>
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<p>Of course, using an MCMC algorithm to sample from the prior is unnecessary and silly (<code>PriorSampler</code> exists, after all), but the point is to illustrate the flexibility of the context system. We could, for instance, use the same setup to implement an <em>Approximate Bayesian Computation</em> (ABC) algorithm.</p>
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Expand Down Expand Up @@ -553,33 +553,33 @@ <h1 class="title">Manually Defining a Model</h1>
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>chain <span class="op">=</span> <span class="fu">sample</span>(model2, <span class="fu">NUTS</span>(), <span class="fl">1000</span>; progress<span class="op">=</span><span class="cn">false</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>┌ Info: Found initial step size
└ ϵ = 2.25</code></pre>
└ ϵ = 1.6</code></pre>
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<div class="cell-output cell-output-display" data-execution_count="1">
<pre><code>Chains MCMC chain (1000×14×1 Array{Float64, 3}):

Iterations = 501:1:1500
Number of chains = 1
Samples per chain = 1000
Wall duration = 7.28 seconds
Compute duration = 7.28 seconds
Wall duration = 7.59 seconds
Compute duration = 7.59 seconds
parameters = s², m
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

Summary Statistics
parameters mean std mcse ess_bulk ess_tail rhat e ⋯
Symbol Float64 Float64 Float64 Float64 Float64 Float64 ⋯

s² 2.1230 1.8732 0.0953 542.9428 423.7377 1.0007
m 1.2087 0.8148 0.0374 500.8337 502.9597 1.0032
s² 2.3949 4.5577 0.2982 236.3625 313.1934 1.0074
m 1.1687 0.9203 0.0525 353.5452 198.7300 1.0057
1 column omitted

Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64

s² 0.5793 1.0428 1.5723 2.3921 7.2768
m -0.3750 0.7304 1.2244 1.6654 2.8921</code></pre>
s² 0.5984 1.0413 1.5606 2.4078 7.7236
m -0.9764 0.6446 1.2060 1.7290 3.0094</code></pre>
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<p>The subsequent pages in this section will show how the <code>@model</code> macro does this behind-the-scenes.</p>
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