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<center><h1>Probabilistic Programming in Python</h1><br>
<h2>Dr. Thomas Wiecki</h2>
<h2><a href="http://twitter.com/twiecki">@twiecki</a></h2>
<img width=40% src="http://i2.wp.com/stuffled.com/wp-content/uploads/2014/09/Quantopian-Logo-EPS-vector-image.png?resize=1020%2C680">
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<div class=" highlight hl-ipython2"><pre><span class="o">%</span><span class="k">pyplot</span> inline
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">Image</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">scipy</span> <span class="kn">as</span> <span class="nn">sp</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="kn">as</span> <span class="nn">sns</span>
<span class="n">sns</span><span class="o">.</span><span class="n">set_context</span><span class="p">(</span><span class="s">'talk'</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">matplotlib</span> <span class="kn">import</span> <span class="n">rc</span>
<span class="n">rc</span><span class="p">(</span><span class="s">'font'</span><span class="p">,</span><span class="o">**</span><span class="p">{</span><span class="s">'family'</span><span class="p">:</span><span class="s">'sans-serif'</span><span class="p">,</span><span class="s">'sans-serif'</span><span class="p">:[</span><span class="s">'Helvetica'</span><span class="p">],</span> <span class="s">'size'</span><span class="p">:</span> <span class="mi">22</span><span class="p">})</span>
<span class="n">rc</span><span class="p">(</span><span class="s">'xtick'</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
<span class="n">rc</span><span class="p">(</span><span class="s">'ytick'</span><span class="p">,</span> <span class="n">labelsize</span><span class="o">=</span><span class="mi">12</span><span class="p">)</span>
<span class="c">## for Palatino and other serif fonts use:</span>
<span class="c">#rc('font',**{'family':'serif','serif':['Palatino']})</span>
<span class="n">rc</span><span class="p">(</span><span class="s">'text'</span><span class="p">,</span> <span class="n">usetex</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="kn">as</span> <span class="nn">pd</span>
<span class="n">figsize</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">8</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">IPython.html</span> <span class="kn">import</span> <span class="n">widgets</span> <span class="c"># Widget definitions</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span> <span class="c"># Used to display widgets in the notebook</span>
<span class="kn">from</span> <span class="nn">IPython.html.widgets</span> <span class="kn">import</span> <span class="n">interact</span><span class="p">,</span> <span class="n">interactive</span>
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">clear_output</span><span class="p">,</span> <span class="n">display</span><span class="p">,</span> <span class="n">HTML</span>
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<h1 id="About-me">About me<a class="anchor-link" href="#About-me">¶</a></h1><ul>
<li>Lead Data Scientist at <a href="https://www.quantopian.com">Quantopian Inc</a>: Building a crowd sourced hedge fund.</li>
<li>PhD from Brown University -- research on computational neuroscience and machine learning using Bayesian modeling.</li>
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<h1 id="Develop-trading-strategies-in-your-browser">Develop trading strategies in your browser<a class="anchor-link" href="#Develop-trading-strategies-in-your-browser">¶</a></h1><p><img src="https://raw.githubusercontent.com/twiecki/pymc3_talk/master/quantopian.png"></p>
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<h2 id="Live-Trading">Live Trading<a class="anchor-link" href="#Live-Trading">¶</a></h2><p><img width=70% src='live_trading.png'></img></p>
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<h2 id="Crowd-sourced-hedge-fund">Crowd-sourced hedge fund<a class="anchor-link" href="#Crowd-sourced-hedge-fund">¶</a></h2><p><img width=100% src='quant_fund.png'></img></p>
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<h1 id="Why-should-you-care-about-Probabilistic-Programming?">Why should you care about Probabilistic Programming?<a class="anchor-link" href="#Why-should-you-care-about-Probabilistic-Programming?">¶</a></h1>
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<p><img src='https://raw.githubusercontent.com/twiecki/pymc3_talk/master/blackbox.png'>
Source: <a href='https://speakerdeck.com/ogrisel/trends-in-machine-learning'>Olivier Grisel's talk on ML</a></p>
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<p><img src='https://raw.githubusercontent.com/twiecki/pymc3_talk/master/openbox.png'>
Source: <a href='https://speakerdeck.com/ogrisel/trends-in-machine-learning'>Olivier Grisel's talk on ML</a></p>
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<h1 id="Identify-trading-algorithms-with-highest-chance-of-beating-the-market">Identify trading algorithms with highest chance of beating the market<a class="anchor-link" href="#Identify-trading-algorithms-with-highest-chance-of-beating-the-market">¶</a></h1>
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<div class=" highlight hl-ipython2"><pre><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">10</span><span class="p">)</span>
<span class="n">algos</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">algos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="o">.</span><span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">200</span><span class="p">))</span>
<span class="n">algos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="o">.</span><span class="mi">4</span><span class="p">)</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">50</span><span class="p">))</span>
<span class="n">algos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="o">.</span><span class="mi">6</span><span class="p">)</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">150</span><span class="p">))</span>
<span class="n">algos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="o">.</span><span class="mi">5</span><span class="p">)</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">25</span><span class="p">))</span>
<span class="n">algos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="o">.</span><span class="mi">45</span><span class="p">)</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">5</span><span class="p">))</span>
<span class="n">algos</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="o">.</span><span class="mi">45</span><span class="p">)</span><span class="o">.</span><span class="n">rvs</span><span class="p">(</span><span class="mi">10</span><span class="p">))</span>
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<h1 id="Identify-trading-algorithms-with-highest-chance-of-beating-the-market">Identify trading algorithms with highest chance of beating the market<a class="anchor-link" href="#Identify-trading-algorithms-with-highest-chance-of-beating-the-market">¶</a></h1>
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<div class=" highlight hl-ipython2"><pre><span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">algo</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">algos</span><span class="p">):</span>
<span class="k">print</span> <span class="s">'Success algo </span><span class="si">%i</span><span class="s">: '</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span> <span class="n">algo</span>
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<pre>Success algo 0: [1 0 1 1 0 0 0 1 0 0 1 1 0 1 1 1 1 0 1 1 1 0 0 1 0 0 1 1 1 1 1 1 1 0 0 0 0
1 0 1 1 1 0 1 0 1 0 1 0 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 1
0 0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 1 1 0 0 1 1 1 0 0 1 0
1 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 1 1 0 0
1 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 1 1 1 0 1 0 0 1 0 1 0 0 1
1 0 0 0 0 0 1 0 0 1 0 1 1 0 0]
Success algo 1: [1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 0 0
0 0 1 0 0 1 0 0 1 0 0 1 0]
Success algo 2: [0 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 0 1 1 1 0 1 0 1 1 1 0 0 1 1 0
1 1 1 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 1 1 0
1 1 1 0 1 1 0 1 1 0 1 0 1 1 0 0 1 0 1 0 1 1 1 1 0 0 0 0 1 0 1 1 1 0 0 1 1
1 0 0 1 1 0 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0
1 1]
Success algo 3: [0 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 0 1]
Success algo 4: [1 1 0 1 1]
Success algo 5: [0 1 1 1 1 1 1 0 0 1]
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<h1 id="Very-general-problem----reddit-comment-ratings">Very general problem -- reddit comment ratings<a class="anchor-link" href="#Very-general-problem----reddit-comment-ratings">¶</a></h1><p><img src="http://i.imgur.com/QjtUF3x.png"></p>
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<li>Simple question: How to rank them?</li>
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<h1 id="The-simplest-answer">The simplest answer<a class="anchor-link" href="#The-simplest-answer">¶</a></h1><ul>
<li>Take the mean.</li>
<li>This is equivalent to the maximum likelihood estimate, so statistically not a bad idea.</li>
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<div class=" highlight hl-ipython2"><pre><span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">algo</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">algos</span><span class="p">):</span>
<span class="k">print</span> <span class="s">'Prob of success algo </span><span class="si">%i</span><span class="s">: '</span> <span class="o">%</span> <span class="n">i</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">algo</span><span class="p">)</span> <span class="o">*</span> <span class="mi">100</span>
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<pre>Prob of success algo 0: 49.0
Prob of success algo 1: 38.0
Prob of success algo 2: 58.6666666667
Prob of success algo 3: 60.0
Prob of success algo 4: 80.0
Prob of success algo 5: 30.0
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<h1 id="Sort-and-display">Sort and display<a class="anchor-link" href="#Sort-and-display">¶</a></h1>
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<div class=" highlight hl-ipython2"><pre><span class="n">probs</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">Series</span><span class="p">({</span><span class="n">i</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">algo</span><span class="p">)</span> <span class="o">*</span> <span class="mi">100</span> <span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">algo</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">algos</span><span class="p">)})</span>
<span class="n">probs</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">ascending</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">probs</span><span class="o">.</span><span class="n">values</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">probs</span><span class="o">.</span><span class="n">index</span><span class="p">)),</span> <span class="s">'o'</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s">'P(success)'</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s">'Strategy'</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">100.</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">(</span><span class="o">-.</span><span class="mi">1</span><span class="p">,</span> <span class="mf">5.2</span><span class="p">);</span> <span class="n">plt</span><span class="o">.</span><span class="n">yticks</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">probs</span><span class="o">.</span><span class="n">index</span><span class="p">)),</span> <span class="n">probs</span><span class="o">.</span><span class="n">index</span><span class="p">);</span>
</pre></div>
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</div>
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<h1 id="How-to-capture-uncertainty?">How to capture uncertainty?<a class="anchor-link" href="#How-to-capture-uncertainty?">¶</a></h1>
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<h1 id="Bayesian-statistics">Bayesian statistics<a class="anchor-link" href="#Bayesian-statistics">¶</a></h1><ul>
<li><strong>Flexible statistical framework</strong> that allows <strong>uncertainty quantification</strong>.</li>
<li>Uses <strong>probability distributions</strong> to reason about unknowns.</li>
<li><strong>Not</strong> consider a <strong>single mean</strong> but rather assign <strong>every possible mean</strong> a <strong>probability</strong>.</li>
<li>Probability distribution: maps <strong>parameter values</strong> to <strong>probabilities</strong>.</li>
<li>Bayesian statistics updates <strong>probiability distributions</strong> after observing <strong>data</strong>.</li>
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<p><img src="https://pbs.twimg.com/profile_images/531163997022605313/b8jvpynB.png"></p>
<ul>
<li>"When I bet on horses, I never lose. Why? I bet on all the horses." Tom Haverford</li>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">gen_plot</span><span class="p">(</span><span class="n">successes</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">),</span> <span class="n">failures</span><span class="o">=</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">100</span><span class="p">,</span> <span class="mi">1</span><span class="p">)):</span>
<span class="n">alpha_prior</span><span class="o">=</span><span class="mi">5</span>
<span class="n">beta_prior</span><span class="o">=</span><span class="mi">5</span>
<span class="k">if</span> <span class="n">successes</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">failures</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">title</span> <span class="o">=</span> <span class="s">r'Prior probability distribution on $\theta$'</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">title</span> <span class="o">=</span> <span class="s">r'Posterior probability distribution of $\theta$ after having seen data'</span>
<span class="n">alpha</span> <span class="o">=</span> <span class="n">alpha_prior</span> <span class="o">+</span> <span class="n">successes</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">beta_prior</span> <span class="o">+</span> <span class="n">failures</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">201</span><span class="p">)</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="s">r'Chance of success (profit or conversion)'</span><span class="p">,</span>
<span class="n">ylabel</span><span class="o">=</span><span class="s">r'Probability'</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="n">title</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">beta</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">3.</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s">r'0\%'</span><span class="p">,</span> <span class="s">r'20\%'</span><span class="p">,</span> <span class="s">r'40\%'</span><span class="p">,</span> <span class="s">r'60\%'</span><span class="p">,</span> <span class="s">r'80\%'</span><span class="p">,</span> <span class="s">r'100\%'</span><span class="p">]);</span>
<span class="k">return</span>
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<div class=" highlight hl-ipython2"><pre><span class="k">def</span> <span class="nf">gen_plot</span><span class="p">(</span><span class="n">success</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">),</span> <span class="n">failure</span><span class="o">=</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)):</span>
<span class="n">alpha</span> <span class="o">=</span> <span class="n">success</span>
<span class="n">beta</span> <span class="o">=</span> <span class="n">failure</span>
<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">6</span><span class="p">))</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">,</span> <span class="n">xlabel</span><span class="o">=</span><span class="s">'Mean (probability of beating the market)'</span><span class="p">,</span>
<span class="n">ylabel</span><span class="o">=</span><span class="s">'Belief'</span><span class="p">,</span>
<span class="n">title</span><span class="o">=</span><span class="s">r'Probability distribution'</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">sp</span><span class="o">.</span><span class="n">stats</span><span class="o">.</span><span class="n">beta</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">),</span> <span class="n">linewidth</span><span class="o">=</span><span class="mf">3.</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s">'0\%'</span><span class="p">,</span> <span class="s">'20\%'</span><span class="p">,</span> <span class="s">'40\%'</span><span class="p">,</span> <span class="s">'60\%'</span><span class="p">,</span> <span class="s">'80\%'</span><span class="p">,</span> <span class="s">'100\%'</span><span class="p">]);</span>
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<h1 id="Probability-of-the-mean-without-having-seen-any-data.">Probability of the mean without having seen any data.<a class="anchor-link" href="#Probability-of-the-mean-without-having-seen-any-data.">¶</a></h1>
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<div class=" highlight hl-ipython2"><pre><span class="n">gen_plot</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
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