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@@ -1000,73 +1000,41 @@ <h1>Consider a probabilistic model defined by</h1>
10001000
<div id="26" class="cell" data-execution_count="1">
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<div class="sourceCode cell-code" id="cb12"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sample</span>(<span class="fu">MiniModel</span>(m, (x<span class="op">=</span><span class="fl">3.0</span>,)), <span class="fu">MHSampler</span>(), <span class="fl">1_000_000</span>; chain_type<span class="op">=</span>Chains, 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>
10021002
<div class="cell-output cell-output-display" data-execution_count="1">
1003-
<div class="ansi-escaped-output">
1004-
<pre>Chains MCMC chain (1000000×2×1 Array{Float64, 3}):
1003+
<pre><code>Chains MCMC chain (1000000×2×1 Array{Float64, 3}):
10051004

10061005
Iterations = 1:1:1000000
10071006
Number of chains = 1
10081007
Samples per chain = 1000000
10091008
parameters = a, b
10101009

1011-
Summary Statistics
1012-
<span class="ansi-bold"> parameters </span><span class="ansi-bold"> mean </span><span class="ansi-bold"> std </span><span class="ansi-bold"> mcse </span><span class="ansi-bold"> ess_bulk </span><span class="ansi-bold"> ess_tail </span><span class="ansi-bold"> rh ⋯
1013-
</span><span class="ansi-bright-black-fg ansi-bold"> Symbol </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float ⋯
1014-
1015-
a 0.9761 0.9003 0.0032 80996.1504 120761.1548 1.00 ⋯
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b 2.8807 0.4892 0.0012 169532.7931 212343.5210 1.00 ⋯
1017-
</span><span class="ansi-cyan-fg ansi-bold"> 2 columns omitted
1018-
1019-
Quantiles
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</span><span class="ansi-cyan-fg ansi-bold"> parameters </span><span class="ansi-cyan-fg ansi-bold"> 2.5% </span><span class="ansi-cyan-fg ansi-bold"> 25.0% </span><span class="ansi-cyan-fg ansi-bold"> 50.0% </span><span class="ansi-cyan-fg ansi-bold"> 75.0% </span><span class="ansi-cyan-fg ansi-bold"> 97.5%
1021-
</span><span class="ansi-bright-black-fg ansi-bold"> Symbol </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64
1022-
1023-
a -0.7900 0.3698 0.9736 1.5849 2.7370
1024-
b 1.9201 2.5500 2.8813 3.2103 3.8383
1025-
</span></pre>
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</div>
1010+
Use `describe(chains)` for summary statistics and quantiles.</code></pre>
10271011
</div>
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</div>
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<p>We compare these results with Turing.</p>
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<div id="28" class="cell" data-execution_count="1">
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<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Turing</span></span>
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<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">PDMats</span></span>
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<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a></span>
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<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a><span class="pp">@model</span> <span class="kw">function</span> <span class="fu">turing_m</span>(x)</span>
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<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a> a <span class="op">~</span> <span class="fu">Normal</span>(<span class="fl">0.5</span>, <span class="fl">1</span>)</span>
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<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a> b <span class="op">~</span> <span class="fu">Normal</span>(a, <span class="fl">2</span>)</span>
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<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a> x <span class="op">~</span> <span class="fu">Normal</span>(b, <span class="fl">0.5</span>)</span>
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<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> <span class="cn">nothing</span></span>
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<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
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<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a></span>
1041-
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a><span class="fu">sample</span>(<span class="fu">turing_m</span>(<span class="fl">3.0</span>), <span class="fu">MH</span>(<span class="fu">ScalMat</span>(<span class="fl">2</span>, <span class="fl">1.0</span>)), <span class="fl">1_000_000</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>
1015+
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Turing</span></span>
1016+
<span id="cb14-2"><a href="#cb14-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">PDMats</span></span>
1017+
<span id="cb14-3"><a href="#cb14-3" aria-hidden="true" tabindex="-1"></a></span>
1018+
<span id="cb14-4"><a href="#cb14-4" aria-hidden="true" tabindex="-1"></a><span class="pp">@model</span> <span class="kw">function</span> <span class="fu">turing_m</span>(x)</span>
1019+
<span id="cb14-5"><a href="#cb14-5" aria-hidden="true" tabindex="-1"></a> a <span class="op">~</span> <span class="fu">Normal</span>(<span class="fl">0.5</span>, <span class="fl">1</span>)</span>
1020+
<span id="cb14-6"><a href="#cb14-6" aria-hidden="true" tabindex="-1"></a> b <span class="op">~</span> <span class="fu">Normal</span>(a, <span class="fl">2</span>)</span>
1021+
<span id="cb14-7"><a href="#cb14-7" aria-hidden="true" tabindex="-1"></a> x <span class="op">~</span> <span class="fu">Normal</span>(b, <span class="fl">0.5</span>)</span>
1022+
<span id="cb14-8"><a href="#cb14-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> <span class="cn">nothing</span></span>
1023+
<span id="cb14-9"><a href="#cb14-9" aria-hidden="true" tabindex="-1"></a><span class="kw">end</span></span>
1024+
<span id="cb14-10"><a href="#cb14-10" aria-hidden="true" tabindex="-1"></a></span>
1025+
<span id="cb14-11"><a href="#cb14-11" aria-hidden="true" tabindex="-1"></a><span class="fu">sample</span>(<span class="fu">turing_m</span>(<span class="fl">3.0</span>), <span class="fu">MH</span>(<span class="fu">ScalMat</span>(<span class="fl">2</span>, <span class="fl">1.0</span>)), <span class="fl">1_000_000</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>
10421026
<div class="cell-output cell-output-display" data-execution_count="1">
1043-
<div class="ansi-escaped-output">
1044-
<pre>Chains MCMC chain (1000000×3×1 Array{Float64, 3}):
1027+
<pre><code>Chains MCMC chain (1000000×3×1 Array{Float64, 3}):
10451028

10461029
Iterations = 1:1:1000000
10471030
Number of chains = 1
10481031
Samples per chain = 1000000
1049-
Wall duration = 24.45 seconds
1050-
Compute duration = 24.45 seconds
1032+
Wall duration = 21.37 seconds
1033+
Compute duration = 21.37 seconds
10511034
parameters = a, b
10521035
internals = lp
10531036

1054-
Summary Statistics
1055-
<span class="ansi-bold"> parameters </span><span class="ansi-bold"> mean </span><span class="ansi-bold"> std </span><span class="ansi-bold"> mcse </span><span class="ansi-bold"> ess_bulk </span><span class="ansi-bold"> ess_tail </span><span class="ansi-bold"> rh ⋯
1056-
</span><span class="ansi-bright-black-fg ansi-bold"> Symbol </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float ⋯
1057-
1058-
a 0.9780 0.9016 0.0031 82020.3602 121725.2524 1.00 ⋯
1059-
b 2.8810 0.4897 0.0012 175647.1391 216521.0287 1.00 ⋯
1060-
</span><span class="ansi-cyan-fg ansi-bold"> 2 columns omitted
1061-
1062-
Quantiles
1063-
</span><span class="ansi-cyan-fg ansi-bold"> parameters </span><span class="ansi-cyan-fg ansi-bold"> 2.5% </span><span class="ansi-cyan-fg ansi-bold"> 25.0% </span><span class="ansi-cyan-fg ansi-bold"> 50.0% </span><span class="ansi-cyan-fg ansi-bold"> 75.0% </span><span class="ansi-cyan-fg ansi-bold"> 97.5%
1064-
</span><span class="ansi-bright-black-fg ansi-bold"> Symbol </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64 </span><span class="ansi-bright-black-fg ansi-bold"> Float64
1065-
1066-
a -0.7965 0.3693 0.9783 1.5882 2.7384
1067-
b 1.9215 2.5505 2.8800 3.2120 3.8379
1068-
</span></pre>
1069-
</div>
1037+
Use `describe(chains)` for summary statistics and quantiles.</code></pre>
10701038
</div>
10711039
</div>
10721040
<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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