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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8"/>
<title>An introduction to TensorFlow Probability (TFP)</title>
<meta name="author" content="Jeff Pollock"/>
<style type="text/css">
.underline { text-decoration: underline; }
</style>
<link rel="stylesheet" href="file:///home/jeff/workspace/reveal.js/css/reveal.css"/>
<link rel="stylesheet" href="file:///home/jeff/workspace/reveal.js/css/theme/black.css" id="theme"/>
<!-- If the query includes 'print-pdf', include the PDF print sheet -->
<script>
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var link = document.createElement( 'link' );
link.rel = 'stylesheet';
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<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
</head>
<body>
<div class="reveal">
<div class="slides">
<section id="sec-title-slide" data-background="plain_quantbet_background.png">
<h1>An introduction to TensorFlow Probability (TFP)</h1><p>Jeff Pollock, QuantBet</p><p><a href="https://github.com/jeffpollock9/tfp-intro">https://github.com/jeffpollock9/tfp-intro</a></p>
</section>
<section>
<section id="slide-org2849cc3" data-background="plain_quantbet_background.png">
<h2 id="org2849cc3">Who am I?</h2>
<ul>
<li>My name is Jeff</li>
<li>Minor contributor to TensorFlow Probability</li>
<li>At QuantBet I try to predict football outcomes so we can bet on them</li>
</ul>
</section>
</section>
<section>
<section id="slide-org10e8f22" data-background="plain_quantbet_background.png">
<h2 id="org10e8f22">Questions I hope this talk answers</h2>
<ol>
<li>What's probabilistic programming?</li>
<li>Why TensorFlow?</li>
<li>How do I use TensorFlow Probability?</li>
<li>How does Tensorflow Probability compare to Stan?</li>
</ol>
</section>
</section>
<section>
<section id="slide-orgfe2345c" data-background="plain_quantbet_background.png">
<h2 id="orgfe2345c">Probabilistic Programming</h2>
<div class="outline-text-2" id="text-orgfe2345c">
</div>
</section>
<section id="slide-org9412e4d" data-background="plain_quantbet_background.png">
<h3 id="org9412e4d">What is probabilistic programming?</h3>
<ol>
<li>Specify a model</li>
<li>Run an inference algorithm</li>
<li>Inspect the output</li>
</ol>
</section>
<section id="slide-org483dd01" data-background="plain_quantbet_background.png">
<h3 id="org483dd01">What exactly does specify a model mean?</h3>
<div class="outline-text-3" id="text-org483dd01">
</div>
</section>
<section id="slide-org73dbe60" data-background="plain_quantbet_background.png">
<h4 id="org73dbe60">The joint distribution of data and parameters</h4>
<p>
\[
P(Y, \theta) = P(Y | \theta) P(\theta) \propto P(\theta | Y)
\]
</p>
<p>
For observed data \(Y\) and model parameters \(\theta\).
</p>
</section>
<section id="slide-org1ae7472" data-background="plain_quantbet_background.png">
<h4 id="org1ae7472">Linear regression example</h4>
<div>
\begin{aligned}
\beta& \sim Normal(0, 1^2) \\
\sigma& \sim Exponential(1) \\
Y& \sim Normal(\beta X, \sigma^2) \\
\end{aligned}
</div>
<p>
\(\implies P(Y, \beta, \sigma) = P(Y | \beta, \sigma) P(\beta) P(\sigma)\)
</p>
<p>
For observation \(Y\) and model parameters \(\theta = (\beta, \sigma)\).
</p>
</section>
<section id="slide-org559717b" data-background="plain_quantbet_background.png">
<h4 id="org559717b">Linear regression example in Stan</h4>
<div>
\begin{aligned}
\beta& \sim Normal(0, 1^2) \\
\sigma& \sim Exponential(1) \\
Y& \sim Normal(\beta X, \sigma^2) \\
\end{aligned}
</div>
<div class="org-src-container">
<pre class="src src-stan"><span style="color: #F0DFAF; font-weight: bold;">model</span> {
coefficient <span style="color: #BFEBBF;">~</span> <span style="color: #93E0E3;">normal</span>(0.0, 1.0);
observation_noise_scale <span style="color: #BFEBBF;">~</span> <span style="color: #93E0E3;">exponential</span>(1.0);
observation <span style="color: #BFEBBF;">~</span> <span style="color: #93E0E3;">normal</span>(coefficient * covariate, observation_noise_scale);
}
</pre>
</div>
</section>
<section id="slide-orgae03f1d" data-background="plain_quantbet_background.png">
<h4 id="orgae03f1d">(Slight aside) What Stan does with this</h4>
<div class="org-src-container">
<pre class="src src-C++"><span style="color: #7CB8BB;">real</span> <span style="color: #93E0E3;">log_prob</span>(<span style="color: #7CB8BB;">vector</span> <span style="color: #DFAF8F;">parameters</span>) {
<span style="color: #5F7F5F;">// </span><span style="color: #7F9F7F;">model parameters</span>
<span style="color: #7CB8BB;">real</span> <span style="color: #DFAF8F;">coefficient</span> = parameters[0];
<span style="color: #7CB8BB;">real</span> <span style="color: #DFAF8F;">observation_noise_scale</span> = parameters[1];
<span style="color: #5F7F5F;">// </span><span style="color: #7F9F7F;">model body</span>
<span style="color: #7CB8BB;">real</span> <span style="color: #DFAF8F;">lp</span> = 0.0;
lp += normal_log(coefficient, 0.0, 1.0);
lp += exponential_log(observation_noise_scale, 1.0);
lp += normal_log(
observation, coefficient * covariate, observation_noise_scale);
<span style="color: #F0DFAF; font-weight: bold;">return</span> lp;
}
</pre>
</div>
<p>
Note: generated C++ heavily edited
</p>
</section>
<section id="slide-org32bbaea" data-background="plain_quantbet_background.png">
<h4 id="org32bbaea">Linear regression example in TensorFlow Probability</h4>
<div>
\begin{aligned}
\beta& \sim Normal(0, 1^2) \\
\sigma& \sim Exponential(1) \\
Y& \sim Normal(\beta X, \sigma^2) \\
\end{aligned}
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">model</span>():
<span style="color: #DFAF8F;">coefficient</span> = <span style="color: #F0DFAF; font-weight: bold;">yield</span> Root(tfd.Normal(loc=0.0, scale=1.0))
<span style="color: #DFAF8F;">observation_noise_scale</span> = <span style="color: #F0DFAF; font-weight: bold;">yield</span> Root(tfd.Exponential(rate=1.0))
<span style="color: #DFAF8F;">observation</span> = <span style="color: #F0DFAF; font-weight: bold;">yield</span> tfd.Normal(
loc=coefficient * covariate, scale=observation_noise_scale
)
<span style="color: #DFAF8F;">joint_dist</span> = tfd.JointDistributionCoroutine(model)
</pre>
</div>
</section>
<section id="slide-org2fd2a9b" data-background="plain_quantbet_background.png">
<h3 id="org2fd2a9b">Inference algorithms</h3>
<ol>
<li>MCMC (NUTS, HMC, …)</li>
<li>Variational inference (ADVI, …)</li>
<li>Optimisation (BFGS, …)</li>
</ol>
</section>
<section id="slide-org464fe83" data-background="plain_quantbet_background.png">
<h3 id="org464fe83">Other probabilistic programming languages</h3>
<ol>
<li><a href="https://pyro.ai/">Pyro</a></li>
<li><a href="https://github.com/pymc-devs/pymc3">PyMC3</a></li>
<li><a href="https://github.com/pymc-devs/pymc4">PyMC4</a> (based on TensorFlow Probability!)</li>
<li><a href="https://github.com/TuringLang/Turing.jl">Turing.jl</a></li>
</ol>
</section>
</section>
<section>
<section id="slide-orgcd03d4a" data-background="plain_quantbet_background.png">
<h2 id="orgcd03d4a">Tensorflow 2 basics</h2>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">import</span> tensorflow <span style="color: #F0DFAF; font-weight: bold;">as</span> tf
<span style="color: #F0DFAF; font-weight: bold;">import</span> tensorflow_probability <span style="color: #F0DFAF; font-weight: bold;">as</span> tfp
<span style="color: #F0DFAF; font-weight: bold;">import</span> arviz <span style="color: #F0DFAF; font-weight: bold;">as</span> az
</pre>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(tf.__version__)
</pre>
</div>
<pre class="example">
2.0.0
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(tfp.__version__)
</pre>
</div>
<pre class="example">
0.8.0
</pre>
</section>
<section id="slide-org74dd426" data-background="plain_quantbet_background.png">
<h3 id="org74dd426">Automatic Differentiation</h3>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">objective</span>(x):
<span style="color: #F0DFAF; font-weight: bold;">return</span> tf.sin(x) + tf.exp(x)
<span style="color: #DFAF8F;">x</span> = 3.14
<span style="color: #DFAF8F;">manual_grad</span> = tf.cos(x) + tf.exp(x)
<span style="color: #DFAF8F;">value</span>, <span style="color: #DFAF8F;">automatic_grad</span> = tfp.math.value_and_gradient(objective, x)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(f<span style="color: #CC9393;">"manual: {manual_grad}"</span>)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(f<span style="color: #CC9393;">"automatic: {automatic_grad.numpy()}"</span>)
</pre>
</div>
<pre class="example">
manual: 22.103872299194336
automatic: 22.103872299194336
</pre>
</section>
<section id="slide-orgbff4956" data-background="plain_quantbet_background.png">
<h3 id="orgbff4956">Easy to use accelerated hardware</h3>
<div style="font-size: 65%;">
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #7CB8BB;">@benchmark</span>(warmup=5, runs=100)
<span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">tf_matvec</span>(mat, vec):
<span style="color: #F0DFAF; font-weight: bold;">return</span> tf.linalg.matvec(mat, vec)
<span style="color: #7CB8BB;">@benchmark</span>(warmup=5, runs=100)
<span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">np_matvec</span>(mat, vec):
<span style="color: #F0DFAF; font-weight: bold;">return</span> np.dot(mat, vec)
<span style="color: #DFAF8F;">num_data</span> = 100_000
<span style="color: #DFAF8F;">num_coefficients</span> = 100
<span style="color: #DFAF8F;">design_matrix</span> = tf.random.normal([num_data, num_coefficients])
<span style="color: #DFAF8F;">coefficients</span> = tf.random.normal([num_coefficients])
np.testing.assert_allclose(
tf_matvec(design_matrix, coefficients),
np_matvec(design_matrix.numpy(), coefficients.numpy()),
atol=1e-5,
)
</pre>
</div>
<pre class="example">
100 runs of tf_matvec took 0.0141 seconds
100 runs of np_matvec took 0.1190 seconds
</pre>
</section>
<section id="slide-orgd251846" data-background="plain_quantbet_background.png">
<h3 id="orgd251846">Easy to switch between CPU and GPU</h3>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">with</span> tf.device(<span style="color: #CC9393;">"/CPU:0"</span>):
<span style="color: #DFAF8F;">design_matrix</span> = tf.random.normal([num_data, num_coefficients])
<span style="color: #DFAF8F;">coefficients</span> = tf.random.normal([num_coefficients])
<span style="color: #DFAF8F;">_</span> = tf_matvec(design_matrix, coefficients),
<span style="color: #F0DFAF; font-weight: bold;">with</span> tf.device(<span style="color: #CC9393;">"/GPU:0"</span>):
<span style="color: #DFAF8F;">design_matrix</span> = tf.random.normal([num_data, num_coefficients])
<span style="color: #DFAF8F;">coefficients</span> = tf.random.normal([num_coefficients])
<span style="color: #DFAF8F;">_</span> = tf_matvec(design_matrix, coefficients),
</pre>
</div>
<pre class="example">
100 runs of tf_matvec took 0.2639 seconds
100 runs of tf_matvec took 0.0073 seconds
</pre>
</section>
<section id="slide-orgd6d07fd" data-background="plain_quantbet_background.png">
<h3 id="orgd6d07fd">Deep learning</h3>
<div class="figure">
<p><img src="statistics_vs_deep_learning.jpg" alt="statistics_vs_deep_learning.jpg" />
</p>
</div>
</section>
</section>
<section>
<section id="slide-org693843e" data-background="plain_quantbet_background.png">
<h2 id="org693843e">Tensorflow Probability basics</h2>
<p>
Key ingredients for a probabilistic programming language:
</p>
<ul>
<li>Model specification: <code>tfp.distributions</code></li>
<li>Parameter inference: <code>tfp.mcmc</code>, <code>tfp.vi</code>, <code>tfp.optimizer</code></li>
</ul>
<p>
Some extra goodies: <code>tfp.bijectors</code>, <code>tfp.glm</code>, <code>tfp.sts</code>, <code>tfp.layers</code>
</p>
</section>
<section id="slide-org42ce3b7" data-background="plain_quantbet_background.png">
<h3 id="org42ce3b7">Distributions</h3>
<p>
The building blocks of our model specification.
</p>
</section>
<section id="slide-org5d52f65" data-background="plain_quantbet_background.png">
<h4 id="org5d52f65">Univariate Normal</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">dist</span> = tfd.Normal(loc=0.0, scale=1.0)
</pre>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">sample</span> = dist.sample(3)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(sample)
</pre>
</div>
<pre class="example">
tf.Tensor([-0.20061125 -1.2735859 0.48410097], shape=(3,), dtype=float32)
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(dist.log_prob(sample))
</pre>
</div>
<pre class="example">
tf.Tensor([-0.9390609 -1.729949 -1.0361154], shape=(3,), dtype=float32)
</pre>
</section>
<section id="slide-org8de825d" data-background="plain_quantbet_background.png">
<h4 id="org8de825d">"Batched" univariate Normal</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">dist</span> = tfd.Normal(loc=[0.0, 10.0], scale=1.0)
</pre>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">sample</span> = dist.sample(3)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(sample)
</pre>
</div>
<pre class="example">
tf.Tensor(
[[ 1.1442095 8.391022 ]
[-1.347222 9.383066 ]
[ 0.8454657 10.157462 ]], shape=(3, 2), dtype=float32)
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(dist.log_prob(sample))
</pre>
</div>
<pre class="example">
tf.Tensor(
[[-1.5735462 -2.213344 ]
[-1.826442 -1.1092422]
[-1.2763447 -0.9313357]], shape=(3, 2), dtype=float32)
</pre>
</section>
<section id="slide-org557f5e0" data-background="plain_quantbet_background.png">
<h4 id="org557f5e0"><code>tfd.Independent</code></h4>
<blockquote>
<p>
This distribution is useful for regarding a collection of independent, non-identical
distributions as a single random variable.
</p>
</blockquote>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">batch_dist</span> = tfd.Normal(loc=[0.0, 1.0], scale=1.0)
print_shapes(batch_dist)
</pre>
</div>
<pre class="example">
batch: (2,), event: ()
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">event_dist</span> = tfd.Independent(batch_dist, reinterpreted_batch_ndims=1)
print_shapes(event_dist)
</pre>
</div>
<pre class="example">
batch: (), event: (2,)
</pre>
</section>
<section id="slide-org3a327a4" data-background="plain_quantbet_background.png">
<h4 id="org3a327a4"><code>tfd.Independent</code></h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">y</span> = [1.5, 4.2]
</pre>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(batch_dist.log_prob(y))
</pre>
</div>
<pre class="example">
tf.Tensor([-2.0439386 -6.038938 ], shape=(2,), dtype=float32)
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(event_dist.log_prob(y))
</pre>
</div>
<pre class="example">
tf.Tensor(-8.082876, shape=(), dtype=float32)
</pre>
</section>
<section id="slide-org4e261a5" data-background="plain_quantbet_background.png">
<h4 id="org4e261a5">Scotland Vs England</h4>
<div>
\begin{aligned}
Sco & \sim Poisson(2) \\
Eng & \sim Poisson(0.5)
\end{aligned}
</div>
<p>
\(P(Sco=x, Eng=y) = P(Sco=x) P(Eng=y)\)
</p>
</section>
<section id="slide-orgdcfa240" data-background="plain_quantbet_background.png">
<h4 id="orgdcfa240">Scotland Vs England</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">scotland_rate</span> = 2.0
<span style="color: #DFAF8F;">england_rate</span> = 0.5
<span style="color: #DFAF8F;">batched_dist</span> = tfd.Poisson(rate=[scotland_rate, england_rate])
<span style="color: #DFAF8F;">bivariate_dist</span> = tfd.Independent(batched_dist, reinterpreted_batch_ndims=1)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(bivariate_dist.sample(3))
</pre>
</div>
<pre class="example">
tf.Tensor(
[[6. 0.]
[3. 0.]
[1. 0.]], shape=(3, 2), dtype=float32)
</pre>
</section>
<section id="slide-org2421eae" data-background="plain_quantbet_background.png">
<h4 id="org2421eae">Scotland Vs England</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">scores</span> = [[[0, 0], [0, 1], [0, 2]],
[[1, 0], [1, 1], [1, 2]],
[[2, 0], [2, 1], [2, 2]]]
<span style="color: #DFAF8F;">probs</span> = bivariate_dist.prob(scores)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(probs)
</pre>
</div>
<pre class="example">
tf.Tensor(
[[0.08208499 0.0410425 0.01026062]
[0.16416998 0.08208499 0.02052125]
[0.16416998 0.08208499 0.02052125]], shape=(3, 3), dtype=float32)
</pre>
</section>
<section id="slide-orgf619f60" data-background="plain_quantbet_background.png">
<h4 id="orgf619f60">Scotland Vs England</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">max_goals</span> = 20
<span style="color: #DFAF8F;">scores</span> = [[[i, j] <span style="color: #F0DFAF; font-weight: bold;">for</span> j <span style="color: #F0DFAF; font-weight: bold;">in</span> <span style="color: #DCDCCC; font-weight: bold;">range</span>(max_goals)] <span style="color: #F0DFAF; font-weight: bold;">for</span> i <span style="color: #F0DFAF; font-weight: bold;">in</span> <span style="color: #DCDCCC; font-weight: bold;">range</span>(max_goals)]
<span style="color: #DFAF8F;">probs</span> = bivariate_dist.prob(scores)
<span style="color: #DFAF8F;">scotland</span> = np.tril(probs, k=-1).<span style="color: #DCDCCC; font-weight: bold;">sum</span>()
<span style="color: #DFAF8F;">draw</span> = np.diag(probs).<span style="color: #DCDCCC; font-weight: bold;">sum</span>()
<span style="color: #DFAF8F;">england</span> = np.triu(probs, k=1).<span style="color: #DCDCCC; font-weight: bold;">sum</span>()
<span style="color: #F0DFAF; font-weight: bold;">print</span>(f<span style="color: #CC9393;">"Scotland: {scotland:.4f}, Draw: {draw:.4f}, England: {england:.4f}"</span>)
</pre>
</div>
<pre class="example">
Scotland: 0.7310, Draw: 0.1871, England: 0.0819
</pre>
</section>
<section id="slide-org72c035f" data-background="plain_quantbet_background.png">
<h4 id="org72c035f">Joint distributions</h4>
<div>
\begin{aligned}
\log(\lambda_{Sco}) & \sim Normal(2, 1^2) \\
\log(\lambda_{Eng}) & \sim Normal(0.5, 1^2) \\
Sco & \sim Poisson(\lambda_{Sco}) \\
Eng & \sim Poisson(\lambda_{Eng})
\end{aligned}
</div>
</section>
<section id="slide-org3d3f248" data-background="plain_quantbet_background.png">
<h4 id="org3d3f248">Joint distributions</h4>
<div>
\begin{aligned}
P(\lambda_{Sco}, \lambda_{Eng}, Sco, Eng) = &P(\lambda_{Sco}) P(\lambda_{Eng}) \times \\
&P(Sco | \lambda_{Sco}) P(Eng | \lambda_{Eng})
\end{aligned}
</div>
</section>
<section id="slide-orga526427" data-background="plain_quantbet_background.png">
<h4 id="orga526427">Joint distributions</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">Root</span> = tfd.JointDistributionCoroutine.Root
<span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">model</span>():
<span style="color: #DFAF8F;">scotland_log_rate</span> = <span style="color: #F0DFAF; font-weight: bold;">yield</span> Root(tfd.Normal(loc=2.0, scale=1.0))
<span style="color: #DFAF8F;">england_log_rate</span> = <span style="color: #F0DFAF; font-weight: bold;">yield</span> Root(tfd.Normal(loc=0.5, scale=1.0))
<span style="color: #DFAF8F;">log_rates</span> = tf.stack([scotland_log_rate, england_log_rate], axis=-1)
<span style="color: #DFAF8F;">goals</span> = <span style="color: #F0DFAF; font-weight: bold;">yield</span> tfd.Independent(
tfd.Poisson(log_rate=log_rates), reinterpreted_batch_ndims=1
)
<span style="color: #DFAF8F;">joint_dist</span> = tfd.JointDistributionCoroutine(model)
</pre>
</div>
</section>
<section id="slide-orga6cf69f" data-background="plain_quantbet_background.png">
<h4 id="orga6cf69f">Joint distributions</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">scotland_log_rates</span>, <span style="color: #DFAF8F;">england_log_rates</span>, <span style="color: #DFAF8F;">goals</span> = joint_dist.sample(3)
</pre>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(scotland_log_rates)
</pre>
</div>
<pre class="example">
tf.Tensor([2.0298522 2.3707705 2.102074 ], shape=(3,), dtype=float32)
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(england_log_rates)
</pre>
</div>
<pre class="example">
tf.Tensor([-0.9017575 0.35339326 3.388393 ], shape=(3,), dtype=float32)
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(goals)
</pre>
</div>
<pre class="example">
tf.Tensor(
[[10. 0.]
[12. 0.]
[ 8. 31.]], shape=(3, 2), dtype=float32)
</pre>
</section>
<section id="slide-org357d867" data-background="plain_quantbet_background.png">
<h3 id="org357d867">Bijectors</h3>
<blockquote>
<p>
A bijective function is a one-to-one (injective) and onto (surjective) mapping
</p>
</blockquote>
<div class="figure">
<p><img src="function-mapping.png" alt="function-mapping.png" />
</p>
</div>
</section>
<section id="slide-org519cc2a" data-background="plain_quantbet_background.png">
<h4 id="org519cc2a">How to transform random variables</h4>
<p>
\(f\) is a bijective function.
</p>
<div>
\begin{aligned}
Y &= f(X) \\
P_Y(y) &= P_X(f^{-1}(y)) \left| \frac{d}{dy} f^{-1}(y) \right|
\end{aligned}
</div>
</section>
<section id="slide-orgf76f389" data-background="plain_quantbet_background.png">
<h4 id="orgf76f389">Main bijector operations</h4>
<ol>
<li><code>forward</code>: \(f(X)\)</li>
<li><code>inverse</code>: \(f^{-1}(X)\)</li>
<li><code>inverse_log_det_jacobian</code>: \(\log \left( \left| \frac{d}{dy} f^{-1}(y) \right| \right)\)</li>
</ol>
</section>
<section id="slide-org744784a" data-background="plain_quantbet_background.png">
<h4 id="org744784a">The Log-Normal distribution</h4>
<p>
\(X \sim Normal(\mu, \sigma^{2}), \quad Y = f(X)\)
</p>
<p>
\(f(x) = \exp(x), \quad f^{-1}(x) = \log(x)\)
</p>
<div>
\begin{aligned}
P_Y(y) &= P_X(f^{-1}(y)) \left| \frac{d}{dy} f^{-1}(y) \right| \\
&= P_X(\log(y)) \frac{1}{y}
\end{aligned}
</div>
</section>
<section id="slide-orgb23cb83" data-background="plain_quantbet_background.png">
<h4 id="orgb23cb83">Correlation distribution</h4>
<div>
\begin{aligned}
X &\sim Beta(2, 2) \\
Y &= f(X) \\
&= 2X - 1
\end{aligned}
</div>
<p>
Taken from <a href="https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations">Stan's Prior Choice Recommendations</a>
</p>
</section>
<section id="slide-org80d584e" data-background="plain_quantbet_background.png">
<h4 id="org80d584e">Correlation distribution</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">f</span> = tfb.AffineScalar(shift=-1.0, scale=2.0)
<span style="color: #DFAF8F;">beta</span> = tfd.Beta(concentration0=2.0, concentration1=2.0)
<span style="color: #DFAF8F;">correlation_dist</span> = f(beta)
<span style="color: #DFAF8F;">y</span> = tf.<span style="color: #DCDCCC; font-weight: bold;">range</span>(start=-1.0, limit=1.0, delta=1e-4)
plt.hist(correlation_dist.sample(100_000), bins=50, density=<span style="color: #BFEBBF;">True</span>)
plt.plot(y, correlation_dist.prob(y), linewidth=2.5)
</pre>
</div>
<div class="figure">
<p><object type="image/svg+xml" data="correlation_samples.svg" class="org-svg">
Sorry, your browser does not support SVG.</object>
</p>
</div>
</section>
<section id="slide-org02a50b5" data-background="plain_quantbet_background.png">
<h3 id="org02a50b5">MCMC</h3>
<p>
Going NUTS with Stan and TensorFlow Probability
</p>
</section>
<section id="slide-org475463e" data-background="plain_quantbet_background.png">
<h4 id="org475463e">MVN example</h4>
<div>
\begin{aligned}
\begin{bmatrix} X \\ Y \end{bmatrix}
\sim
MVN
\left(
\begin{bmatrix} 0 \\ 10 \end{bmatrix},
\begin{bmatrix} 16, 24 \\ 24, 64 \end{bmatrix}
\right)
\end{aligned}
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">scale_x</span>, <span style="color: #DFAF8F;">scale_y</span>, <span style="color: #DFAF8F;">correlation</span> = 4.0, 8.0, 0.75
<span style="color: #DFAF8F;">mvn_dist</span> = tfd.MultivariateNormalFullCovariance(
loc=[0.0, 10.0],
covariance_matrix=[
[scale_x**2, scale_x * scale_y * correlation],
[scale_x * scale_y * correlation, scale_y**2],
],
)
</pre>
</div>
</section>
<section id="slide-org525e9f9" data-background="plain_quantbet_background.png">
<h4 id="org525e9f9">MVN example</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python">print_shapes(mvn_dist)
</pre>
</div>
<pre class="example">
batch: (), event: (2,)
</pre>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(mvn_dist.sample(3))
</pre>
</div>
<pre class="example">
tf.Tensor(
[[-0.9434234 10.352297 ]
[ 2.6398509 20.04413 ]
[ 3.2340534 25.358845 ]], shape=(3, 2), dtype=float32)
</pre>
</section>
<section id="slide-org66346f8" data-background="plain_quantbet_background.png">
<h4 id="org66346f8">MVN example</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">samples</span> = mvn_dist.sample(10_000)
az.plot_joint({<span style="color: #CC9393;">"x"</span>: samples[:, 0], <span style="color: #CC9393;">"y"</span>: samples[:, 1]}, kind=<span style="color: #CC9393;">"kde"</span>)
</pre>
</div>
<div class="figure">
<p><object type="image/svg+xml" data="mvn_samples.svg" class="org-svg">
Sorry, your browser does not support SVG.</object>
</p>
</div>
</section>
<section id="slide-org4380b63" data-background="plain_quantbet_background.png">
<h4 id="org4380b63">MVN example Stan</h4>
<div class="org-src-container">
<pre class="src src-stan" id="mvn-stan"><span style="color: #F0DFAF; font-weight: bold;">data</span> {
<span style="color: #7CB8BB;">vector</span>[2] <span style="color: #DFAF8F;">location</span>;
<span style="color: #7CB8BB;">cov_matrix</span>[2] <span style="color: #DFAF8F;">covariance</span>;
}
<span style="color: #F0DFAF; font-weight: bold;">transformed data</span> {
<span style="color: #7CB8BB;">cholesky_factor_cov</span>[2] covariance_tril <span style="color: #BFEBBF;">=</span> <span style="color: #93E0E3;">cholesky_decompose</span>(covariance);
}
<span style="color: #F0DFAF; font-weight: bold;">parameters</span> {
<span style="color: #7CB8BB;">vector</span>[2] <span style="color: #DFAF8F;">x</span>;
}
<span style="color: #F0DFAF; font-weight: bold;">model</span> {
x <span style="color: #BFEBBF;">~</span> <span style="color: #93E0E3;">multi_normal_cholesky</span>(location, covariance_tril);
}
</pre>
</div>
</section>
<section id="slide-orgd604c68" data-background="plain_quantbet_background.png">
<h4 id="orgd604c68">MVN example Stan</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">stan_model</span> = pystan.StanModel(
stan_mvn, extra_compile_args=[<span style="color: #CC9393;">"-O3"</span>, <span style="color: #CC9393;">"-march=native"</span>, <span style="color: #CC9393;">"-ffast-math"</span>]
)
<span style="color: #DFAF8F;">stan_data</span> = {
<span style="color: #CC9393;">"location"</span>: mvn_dist.mean().numpy(),
<span style="color: #CC9393;">"covariance"</span>: mvn_dist.covariance().numpy(),
}
<span style="color: #F0DFAF; font-weight: bold;">with</span> Timer() <span style="color: #F0DFAF; font-weight: bold;">as</span> stan_timer:
<span style="color: #DFAF8F;">stan_fit</span> = stan_model.sampling(data=stan_data, chains=6)
</pre>
</div>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">print</span>(f<span style="color: #CC9393;">"Stan took {stan_timer.duration:.4f} seconds"</span>)
</pre>
</div>
<pre class="example">
Stan took 0.4984 seconds
</pre>
</section>
<section id="slide-orgcd5c217" data-background="plain_quantbet_background.png">
<h4 id="orgcd5c217">(Slight aside) Fast math</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">foo</span>(x):
<span style="color: #F0DFAF; font-weight: bold;">return</span> x * x * x * x
<span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">bar</span>(x):
<span style="color: #DFAF8F;">x2</span> = x * x
<span style="color: #F0DFAF; font-weight: bold;">return</span> x2 * x2
<span style="color: #DFAF8F;">x</span> = 1.123456789123456789
<span style="color: #F0DFAF; font-weight: bold;">print</span>(f<span style="color: #CC9393;">"foo(x)={foo(x):.15f}"</span>)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(f<span style="color: #CC9393;">"bar(x)={bar(x):.15f}"</span>)
</pre>
</div>
<pre class="example">
foo(x)=1.593035640411334
bar(x)=1.593035640411333
</pre>
<p>
see <a href="https://godbolt.org/z/a3KzfG">godbolt.org</a>
</p>
</section>
<section id="slide-org6bf9ebc" data-background="plain_quantbet_background.png">
<h4 id="org6bf9ebc">MVN example TFP</h4>
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">step_size_setter_fn</span>(pkr, new_step_size):
<span style="color: #F0DFAF; font-weight: bold;">return</span> pkr._replace(step_size=new_step_size)
<span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">step_size_getter_fn</span>(pkr):
<span style="color: #F0DFAF; font-weight: bold;">return</span> pkr.step_size
<span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">log_accept_prob_getter_fn</span>(pkr):
<span style="color: #F0DFAF; font-weight: bold;">return</span> pkr.log_accept_ratio
</pre>
</div>
</section>
<section id="slide-org23f82de" data-background="plain_quantbet_background.png">
<h4 id="org23f82de">MVN example TFP</h4>
<div style="font-size: 75%;">
<div class="org-src-container">
<pre class="src src-jupyter-python"><span style="color: #DFAF8F;">initial_state</span> = tf.random.uniform([2**15, 2], minval=-2.0, maxval=2.0)
<span style="color: #7CB8BB;">@tf.function</span>(autograph=<span style="color: #BFEBBF;">False</span>)
<span style="color: #F0DFAF; font-weight: bold;">def</span> <span style="color: #93E0E3;">run_mcmc</span>():
<span style="color: #DFAF8F;">nuts</span> = tfp.mcmc.NoUTurnSampler(mvn_dist.log_prob, step_size=[[1.0, 1.0]])
<span style="color: #DFAF8F;">adaptive_nuts</span> = tfp.mcmc.DualAveragingStepSizeAdaptation(
inner_kernel=nuts,
num_adaptation_steps=800,
target_accept_prob=0.8,
step_size_setter_fn=step_size_setter_fn,
step_size_getter_fn=step_size_getter_fn,
log_accept_prob_getter_fn=log_accept_prob_getter_fn,
)
<span style="color: #F0DFAF; font-weight: bold;">return</span> tfp.mcmc.sample_chain(
num_results=1_000,
current_state=initial_state,
num_burnin_steps=1_000,
kernel=adaptive_nuts,
trace_fn=<span style="color: #BFEBBF;">None</span>,
)
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<h4 id="org775519e">MVN example TFP</h4>
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<pre class="src src-jupyter-python"><span style="color: #F0DFAF; font-weight: bold;">with</span> Timer() <span style="color: #F0DFAF; font-weight: bold;">as</span> tfp_timer:
[tfp_samples] = tf.xla.experimental.<span style="color: #DCDCCC; font-weight: bold;">compile</span>(run_mcmc)
<span style="color: #F0DFAF; font-weight: bold;">print</span>(f<span style="color: #CC9393;">"TFP took {tfp_timer.duration:.4f} seconds"</span>)
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