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KL Divergence
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redoules 2018-12-16 11:44:13 +01:00
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blog/Statistics_10days-day0.html
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@ -185,7 +185,7 @@ Mean : 43117.75, Median : 44627.5, Mode : 4978
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@ -219,7 +219,7 @@ Q1 : 6.0, Q2 : 12, Q3 : 16.0
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@ -429,7 +429,7 @@
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@ -256,7 +256,7 @@ we call B the event "a blue ball is drawn" and R the event "a red ball is drawn"
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@ -311,7 +311,7 @@
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@ -331,7 +331,7 @@ Between 20 and 22 hours?</p>
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@ -270,7 +270,7 @@ B = 515.68
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@ -277,7 +277,7 @@ $$</div>
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cryptocurrencies/Installing_bitcoind_on_raspberry_pi.html
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@ -286,7 +286,7 @@ maxuploadtarget=5000
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cryptocurrencies/mayer_multiple.html
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@ -365,7 +365,7 @@ Mayer multiple average 1.3789102045356179
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@ -279,7 +279,7 @@ Epoch 5/5
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@ -161,7 +161,7 @@ _________________________________________________________________
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@ -154,7 +154,7 @@
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@ -235,7 +235,7 @@ Epoch 5/5
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@ -280,7 +280,7 @@ Epoch 5/5
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index.html
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@ -644,6 +644,7 @@
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<li><a href="./mathematics/KLD.html">Kullback Leibler Divergence</a></li>
<li><a href="./mathematics/spearman.html">Spearman's Rank Correlation Coefficient</a></li>
<li><a href="./mathematics/pearson.html">Pearson correlation coefficient</a></li>
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linux/Reloading_.bashrc.html
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@ -144,7 +144,7 @@ However, you can use the following command to do it </p>
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linux/free_disk_space_linux.html
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@ -167,7 +167,7 @@ none 0 0 0 - /config
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linux/get_ip_linux.html
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@ -256,7 +256,7 @@ wifi0 Link encap:UNSPEC HWaddr 5C-51-4F-41-7A-AD-00-00-00-00-00-00-00-00-00
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linux/linux_process_information.html
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@ -172,7 +172,7 @@ USER;PID;%CPU;%MEM;VSZ;RSS;TTY;STAT;START;TIME;COMMAND</p>
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@ -158,7 +158,7 @@
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linux/simple_bash_function.html
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@ -208,7 +208,7 @@ The secret number is 126
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machine-learning/LeastSquareRegressionLine.html
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@ -234,7 +234,7 @@ Where Y=a+b*X
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@ -248,7 +248,7 @@ b_2\\
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<h1>
Kullback Leibler Divergence
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<time class="published" datetime="2018-12-16T10:54:00+01:00">
16 décembre 2018
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<p>The Kullback Leibler Divergence also famously called KL Divergence. The KL divergence actually measures the difference between any two distributions. </p>
<p>The KL divergence is defined as :
</p>
<div class="math">$$D_{KL}(p||q)=\int_{X} p(x)log\frac{p(x)}{q(x)} dx$$</div>
<p>And it is always non-negative and zero only when P is equal to Q. Indeed, when P is equal to Q, the log of 1 is zero, and that's why the distance is zero. Otherwise, it is always some non-negative quantity. </p>
<p>It is can be view as a distance measure but in reality it is not because it isn't a symertic operator and because it doesn't follow the triangle law.</p>
<div class="math">$$D_{KL}(p||q)\neq D_{KL}(q||p)$$</div>
<h2>Where to use the KL-Divergence</h2>
<p>In supervised learning you are always trying to model our data to a particular distribution. So in that case our <span class="math">\(P\)</span> can be the unknown distribution. We usually want to build an estimated probability distribution <span class="math">\(Q\)</span> based on the sample samples <span class="math">\(X\)</span>. </p>
<p>When the estimator is perfect, <span class="math">\(P\)</span> and <span class="math">\(Q\)</span> are the same hence <span class="math">\(D_{KL}(p||q) = 0\)</span>. This means that the KL-Divergence can be used as a mesure of the error.</p>
<p><a href="https://arxiv.org/pdf/1404.2000v1.pdf">Jonathon Shlens explains</a> that the KL-Divergence can be interpreted as measuring the likelihood that samples form an empirical distribution <span class="math">\(p\)</span> were generated by a fixed distribution <span class="math">\(q\)</span>. </p>
<div class="math">$$D_{KL}(p||q)=\int_{X} p(x)log\frac{p(x)}{q(x)}dx$$</div>
<div class="math">$$D_{KL}(p||q)=\int_{X}\left( -p(x)log q(x) +p(x) log p(x) \right) dx$$</div>
<p>The entropy of p is defined as <span class="math">\(H(p)=-\int_{X}p(x) log(p(x)) dx\)</span></p>
<p>The cross entropy between p and q is defined as $H(p,q)=-\int_{X} p(x)log q(x) $</p>
<p>Hence:</p>
<div class="math">$$D_{KL}(p||q)=H(p,q) - H(p)$$</div>
<p>In many machine learning algorithm, in particular deep learning, the optimization problem revolves around minimizing the cross-entropy. Taking into account what we learnt above : </p>
<div class="math">$$H(p,q) = H(p) + D_{KL}(p||q) $$</div>
<p>The cross entropy between two probability distributions <span class="math">\(p\)</span> and <span class="math">\(q\)</span> over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set, if a coding scheme is used that is optimized for an "artificial" probability distribution <span class="math">\(q\)</span> , rather than the "true" distribution <span class="math">\(p\)</span>. </p>
<p>This means that, the cross entropy corresponds to the number of bits needed to encode the distribution <span class="math">\(p\)</span> given by <span class="math">\(H(p)\)</span> the entropy of <span class="math">\(p\)</span> plus the number of bits needed to encode the directed divergence between the two distributions <span class="math">\(p\)</span> and <span class="math">\(q\)</span> given by <span class="math">\(D_{KL}(p||q)\)</span>, the Kullback Leibler Divergence</p>
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python/ensure_dir.html
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@ -177,7 +177,7 @@
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<center>This project contains 77 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
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python/list_files_directory.html
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@ -213,7 +213,7 @@
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python/logplot.html
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python/quandl.html
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@ -237,7 +237,7 @@
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<center>This project contains 77 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
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python/randint.html
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@ -156,7 +156,7 @@
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python/reverse.html
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@ -173,7 +173,7 @@ Reversed array : [4 3 2 1 0]
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python/sorting.html
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@ -192,7 +192,7 @@ Sorted array : [0.13876208 0.19939047 0.40021786 0.43734158 0.46015169]
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python/stock_pandas.html
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@ -381,7 +381,7 @@
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python/unique.html
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@ -171,7 +171,7 @@ Numpy array with unique values : [10 20 30 40 50 60]
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python/updating_all_python_package_with_anaconda.html
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@ -151,7 +151,7 @@
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search.html
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@ -112,7 +112,7 @@
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sitemap.xml
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@ -5,7 +5,7 @@ xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
<url>
<loc>redoules.github.io/</loc>
<lastmod>2018-12-16T09:52:23-00:00</lastmod>
<lastmod>2018-12-16T11:43:13-00:00</lastmod>
<changefreq>daily</changefreq>
<priority>0.5</priority>
</url>
@ -17,6 +17,13 @@ xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
<priority>0.5</priority>
</url>
<url>
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<lastmod>2018-12-16T10:54:00+01:00</lastmod>
<changefreq>monthly</changefreq>
<priority>0.5</priority>
</url>
<url>
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2
sql/Sorting_results.html
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@ -344,7 +344,7 @@ Done.
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sql/WHERE_SQL_keywords.html
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sql/display_table.html
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sql/display_table_filter.html
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tipuesearch_content.json
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