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<ul>
<li><a class="reference internal" href="#">1.9. Naive Bayes</a><ul>
<li><a class="reference internal" href="#gaussian-naive-bayes">1.9.1. Gaussian Naive Bayes</a></li>
<li><a class="reference internal" href="#multinomial-naive-bayes">1.9.2. Multinomial Naive Bayes</a></li>
<li><a class="reference internal" href="#bernoulli-naive-bayes">1.9.3. Bernoulli Naive Bayes</a></li>
<li><a class="reference internal" href="#out-of-core-naive-bayes-model-fitting">1.9.4. Out-of-core naive Bayes model fitting</a></li>
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<div class="section" id="naive-bayes">
<span id="id1"></span><h1>1.9. Naive Bayes<a class="headerlink" href="#naive-bayes" title="Permalink to this headline">¶</a></h1>
<p>Naive Bayes methods are a set of supervised learning algorithms
based on applying Bayes’ theorem with the “naive” assumption of independence
between every pair of features. Given a class variable <img class="math" src="../_images/math/b124ff74afb0914bb434e8fb849eb56d734412f8.png" alt="y"/> and a
dependent feature vector <img class="math" src="../_images/math/b31b1ea8e4fe3f618dedfb0e2d8c69a3df7a391d.png" alt="x_1"/> through <img class="math" src="../_images/math/a8211bc2dab4ab052a0ebe1ef9065fadcdedee1c.png" alt="x_n"/>,
Bayes’ theorem states the following relationship:</p>
<div class="math">
<p><img src="../_images/math/42e4f1acceccb3e96c3dfd90baa3af862d076de1.png" alt="P(y \mid x_1, \dots, x_n) = \frac{P(y) P(x_1, \dots x_n \mid y)}
{P(x_1, \dots, x_n)}"/></p>
</div><p>Using the naive independence assumption that</p>
<div class="math">
<p><img src="../_images/math/c3781294d5c6e5d98a8962119c79f072d498d31f.png" alt="P(x_i | y, x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n) = P(x_i | y),"/></p>
</div><p>for all <img class="math" src="../_images/math/a581f053bbfa5115f42c13094857cdd12a37ec49.png" alt="i"/>, this relationship is simplified to</p>
<div class="math">
<p><img src="../_images/math/45bc8d8261a9d85f886a998420ac78d02c660aa4.png" alt="P(y \mid x_1, \dots, x_n) = \frac{P(y) \prod_{i=1}^{n} P(x_i \mid y)}
{P(x_1, \dots, x_n)}"/></p>
</div><p>Since <img class="math" src="../_images/math/c03af9de0e9abcf81b8a4a530020c46f4db8174b.png" alt="P(x_1, \dots, x_n)"/> is constant given the input,
we can use the following classification rule:</p>
<div class="math">
<p><img src="../_images/math/1ec6f3fe56afecec546113525110fb384766829e.png" alt="P(y \mid x_1, \dots, x_n) \propto P(y) \prod_{i=1}^{n} P(x_i \mid y)
\Downarrow
\hat{y} = \arg\max_y P(y) \prod_{i=1}^{n} P(x_i \mid y),"/></p>
</div><p>and we can use Maximum A Posteriori (MAP) estimation to estimate
<img class="math" src="../_images/math/6d48612122b8ba03b4f704429e4e7488197f328f.png" alt="P(y)"/> and <img class="math" src="../_images/math/dd27ab59b712f5f48b6cd30c93b6a39511643b62.png" alt="P(x_i \mid y)"/>;
the former is then the relative frequency of class <img class="math" src="../_images/math/b124ff74afb0914bb434e8fb849eb56d734412f8.png" alt="y"/>
in the training set.</p>
<p>The different naive Bayes classifiers differ mainly by the assumptions they
make regarding the distribution of <img class="math" src="../_images/math/dd27ab59b712f5f48b6cd30c93b6a39511643b62.png" alt="P(x_i \mid y)"/>.</p>
<p>In spite of their apparently over-simplified assumptions, naive Bayes
classifiers have worked quite well in many real-world situations, famously
document classification and spam filtering. They require a small amount
of training data to estimate the necessary parameters. (For theoretical
reasons why naive Bayes works well, and on which types of data it does, see
the references below.)</p>
<p>Naive Bayes learners and classifiers can be extremely fast compared to more
sophisticated methods.
The decoupling of the class conditional feature distributions means that each
distribution can be independently estimated as a one dimensional distribution.
This in turn helps to alleviate problems stemming from the curse of
dimensionality.</p>
<p>On the flip side, although naive Bayes is known as a decent classifier,
it is known to be a bad estimator, so the probability outputs from
<tt class="docutils literal"><span class="pre">predict_proba</span></tt> are not to be taken too seriously.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li>H. Zhang (2004). <a class="reference external" href="http://www.cs.unb.ca/profs/hzhang/publications/FLAIRS04ZhangH.pdf">The optimality of Naive Bayes.</a>
Proc. FLAIRS.</li>
</ul>
</div>
<div class="section" id="gaussian-naive-bayes">
<h2>1.9.1. Gaussian Naive Bayes<a class="headerlink" href="#gaussian-naive-bayes" title="Permalink to this headline">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB" title="sklearn.naive_bayes.GaussianNB"><tt class="xref py py-class docutils literal"><span class="pre">GaussianNB</span></tt></a> implements the Gaussian Naive Bayes algorithm for
classification. The likelihood of the features is assumed to be Gaussian:</p>
<div class="math">
<p><img src="../_images/math/196f9e31dfd9403804db63b03de06cc584586a8c.png" alt="P(x_i \mid y) &= \frac{1}{\sqrt{2\pi\sigma^2_y}} \exp\left(-\frac{(x_i - \mu_y)^2}{2\sigma^2_y}\right)"/></p>
</div><p>The parameters <img class="math" src="../_images/math/eafb759b3682cc78f395f0f31b2d1c453c61b662.png" alt="\sigma_y"/> and <img class="math" src="../_images/math/e85e9172fdbaf56c03dced3ea4be9eb4787d0514.png" alt="\mu_y"/>
are estimated using maximum likelihood.</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">datasets</span>
<span class="gp">>>> </span><span class="n">iris</span> <span class="o">=</span> <span class="n">datasets</span><span class="o">.</span><span class="n">load_iris</span><span class="p">()</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.naive_bayes</span> <span class="kn">import</span> <span class="n">GaussianNB</span>
<span class="gp">>>> </span><span class="n">gnb</span> <span class="o">=</span> <span class="n">GaussianNB</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">y_pred</span> <span class="o">=</span> <span class="n">gnb</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">iris</span><span class="o">.</span><span class="n">target</span><span class="p">)</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="p">)</span>
<span class="gp">>>> </span><span class="k">print</span><span class="p">(</span><span class="s">"Number of mislabeled points out of a total </span><span class="si">%d</span><span class="s"> points : </span><span class="si">%d</span><span class="s">"</span>
<span class="gp">... </span> <span class="o">%</span> <span class="p">(</span><span class="n">iris</span><span class="o">.</span><span class="n">data</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],(</span><span class="n">iris</span><span class="o">.</span><span class="n">target</span> <span class="o">!=</span> <span class="n">y_pred</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()))</span>
<span class="go">Number of mislabeled points out of a total 150 points : 6</span>
</pre></div>
</div>
</div>
<div class="section" id="multinomial-naive-bayes">
<span id="id2"></span><h2>1.9.2. Multinomial Naive Bayes<a class="headerlink" href="#multinomial-naive-bayes" title="Permalink to this headline">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.naive_bayes.MultinomialNB.html#sklearn.naive_bayes.MultinomialNB" title="sklearn.naive_bayes.MultinomialNB"><tt class="xref py py-class docutils literal"><span class="pre">MultinomialNB</span></tt></a> implements the naive Bayes algorithm for multinomially
distributed data, and is one of the two classic naive Bayes variants used in
text classification (where the data are typically represented as word vector
counts, although tf-idf vectors are also known to work well in practice).
The distribution is parametrized by vectors
<img class="math" src="../_images/math/6eedd22a03d5c334159634dbc99dff67fe4e0d4a.png" alt="\theta_y = (\theta_{y1},\ldots,\theta_{yn})"/>
for each class <img class="math" src="../_images/math/b124ff74afb0914bb434e8fb849eb56d734412f8.png" alt="y"/>, where <img class="math" src="../_images/math/413f8a8e40062a9090d9d50b88bc7b551b314c26.png" alt="n"/> is the number of features
(in text classification, the size of the vocabulary)
and <img class="math" src="../_images/math/da203f970e5bbff0c0652f37b320fba12937ca36.png" alt="\theta_{yi}"/> is the probability <img class="math" src="../_images/math/dd27ab59b712f5f48b6cd30c93b6a39511643b62.png" alt="P(x_i \mid y)"/>
of feature <img class="math" src="../_images/math/a581f053bbfa5115f42c13094857cdd12a37ec49.png" alt="i"/> appearing in a sample belonging to class <img class="math" src="../_images/math/b124ff74afb0914bb434e8fb849eb56d734412f8.png" alt="y"/>.</p>
<p>The parameters <img class="math" src="../_images/math/8264a126c36284f782d9bc3a1b2f761cda0c2ea8.png" alt="\theta_y"/> is estimated by a smoothed
version of maximum likelihood, i.e. relative frequency counting:</p>
<div class="math">
<p><img src="../_images/math/bb1961efcad95a40ed49cc64a9b9ca681e06720a.png" alt="\hat{\theta}_{yi} = \frac{ N_{yi} + \alpha}{N_y + \alpha n}"/></p>
</div><p>where <img class="math" src="../_images/math/d95f1d719cb6b6e373cb6fc65fdcf66ca50ec589.png" alt="N_{yi} = \sum_{x \in T} x_i"/> is
the number of times feature <img class="math" src="../_images/math/a581f053bbfa5115f42c13094857cdd12a37ec49.png" alt="i"/> appears in a sample of class <img class="math" src="../_images/math/b124ff74afb0914bb434e8fb849eb56d734412f8.png" alt="y"/>
in the training set <img class="math" src="../_images/math/6d42c88506b8da39a2a23653aecbfb7c29728063.png" alt="T"/>,
and <img class="math" src="../_images/math/39d979d28ea13da6cd6fe64c3a8ee5005e5cabce.png" alt="N_{y} = \sum_{i=1}^{|T|} N_{yi}"/> is the total count of
all features for class <img class="math" src="../_images/math/b124ff74afb0914bb434e8fb849eb56d734412f8.png" alt="y"/>.</p>
<p>The smoothing priors <img class="math" src="../_images/math/268201c9ebe3275932cf0c0287de32f65d316a99.png" alt="\alpha \ge 0"/> accounts for
features not present in the learning samples and prevents zero probabilities
in further computations.
Setting <img class="math" src="../_images/math/7eab7af9a483d7c6e4e628ba496fb65c5c56faac.png" alt="\alpha = 1"/> is called Laplace smoothing,
while <img class="math" src="../_images/math/f053df68f37ffa28944fe80c99adea2c99593e38.png" alt="\alpha < 1"/> is called Lidstone smoothing.</p>
</div>
<div class="section" id="bernoulli-naive-bayes">
<span id="id3"></span><h2>1.9.3. Bernoulli Naive Bayes<a class="headerlink" href="#bernoulli-naive-bayes" title="Permalink to this headline">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.naive_bayes.BernoulliNB.html#sklearn.naive_bayes.BernoulliNB" title="sklearn.naive_bayes.BernoulliNB"><tt class="xref py py-class docutils literal"><span class="pre">BernoulliNB</span></tt></a> implements the naive Bayes training and classification
algorithms for data that is distributed according to multivariate Bernoulli
distributions; i.e., there may be multiple features but each one is assumed
to be a binary-valued (Bernoulli, boolean) variable.
Therefore, this class requires samples to be represented as binary-valued
feature vectors; if handed any other kind of data, a <tt class="docutils literal"><span class="pre">BernoulliNB</span></tt> instance
may binarize its input (depending on the <tt class="docutils literal"><span class="pre">binarize</span></tt> parameter).</p>
<p>The decision rule for Bernoulli naive Bayes is based on</p>
<div class="math">
<p><img src="../_images/math/1f41dea8d6bdb5d736a645ff2c6179fd733c0a27.png" alt="P(x_i \mid y) = P(i \mid y) x_i + (1 - P(i \mid y)) (1 - x_i)"/></p>
</div><p>which differs from multinomial NB’s rule
in that it explicitly penalizes the non-occurrence of a feature <img class="math" src="../_images/math/a581f053bbfa5115f42c13094857cdd12a37ec49.png" alt="i"/>
that is an indicator for class <img class="math" src="../_images/math/b124ff74afb0914bb434e8fb849eb56d734412f8.png" alt="y"/>,
where the multinomial variant would simply ignore a non-occurring feature.</p>
<p>In the case of text classification, word occurrence vectors (rather than word
count vectors) may be used to train and use this classifier. <tt class="docutils literal"><span class="pre">BernoulliNB</span></tt>
might perform better on some datasets, especially those with shorter documents.
It is advisable to evaluate both models, if time permits.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li>C.D. Manning, P. Raghavan and H. Schütze (2008). Introduction to
Information Retrieval. Cambridge University Press, pp. 234-265.</li>
<li>A. McCallum and K. Nigam (1998).
<a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.1529">A comparison of event models for Naive Bayes text classification.</a>
Proc. AAAI/ICML-98 Workshop on Learning for Text Categorization, pp. 41-48.</li>
<li>V. Metsis, I. Androutsopoulos and G. Paliouras (2006).
<a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.5542">Spam filtering with Naive Bayes – Which Naive Bayes?</a>
3rd Conf. on Email and Anti-Spam (CEAS).</li>
</ul>
</div>
</div>
<div class="section" id="out-of-core-naive-bayes-model-fitting">
<h2>1.9.4. Out-of-core naive Bayes model fitting<a class="headerlink" href="#out-of-core-naive-bayes-model-fitting" title="Permalink to this headline">¶</a></h2>
<p>Naive Bayes models can be used to tackle large scale classification problems
for which the full training set might not fit in memory. To handle this case,
<a class="reference internal" href="generated/sklearn.naive_bayes.MultinomialNB.html#sklearn.naive_bayes.MultinomialNB" title="sklearn.naive_bayes.MultinomialNB"><tt class="xref py py-class docutils literal"><span class="pre">MultinomialNB</span></tt></a>, <a class="reference internal" href="generated/sklearn.naive_bayes.BernoulliNB.html#sklearn.naive_bayes.BernoulliNB" title="sklearn.naive_bayes.BernoulliNB"><tt class="xref py py-class docutils literal"><span class="pre">BernoulliNB</span></tt></a>, and <a class="reference internal" href="generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB" title="sklearn.naive_bayes.GaussianNB"><tt class="xref py py-class docutils literal"><span class="pre">GaussianNB</span></tt></a>
expose a <tt class="docutils literal"><span class="pre">partial_fit</span></tt> method that can be used
incrementally as done with other classifiers as demonstrated in
<a class="reference internal" href="../auto_examples/applications/plot_out_of_core_classification.html#example-applications-plot-out-of-core-classification-py"><em>Out-of-core classification of text documents</em></a>. Both discrete
classifiers support sample weighting; <a class="reference internal" href="generated/sklearn.naive_bayes.GaussianNB.html#sklearn.naive_bayes.GaussianNB" title="sklearn.naive_bayes.GaussianNB"><tt class="xref py py-class docutils literal"><span class="pre">GaussianNB</span></tt></a> does not.</p>
<p>Contrary to the <tt class="docutils literal"><span class="pre">fit</span></tt> method, the first call to <tt class="docutils literal"><span class="pre">partial_fit</span></tt> needs to be
passed the list of all the expected class labels.</p>
<p>For an overview of available strategies in scikit-learn, see also the
<a class="reference internal" href="scaling_strategies.html#scaling-strategies"><em>out-of-core learning</em></a> documentation.</p>
<p>note:</p>
<div class="highlight-python"><div class="highlight"><pre>The ``partial_fit`` method call of naive Bayes models introduces some
computational overhead. It is recommended to use data chunk sizes that are as
large as possible, that is as the available RAM allows.
</pre></div>
</div>
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