mhjensen
171 lines
6.2 KiB
HTML
171 lines
6.2 KiB
HTML
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<title>Data Analysis and Machine Learning: Logistic Regression</title>
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{'highest level': 2,
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'sections': [('Logistic Regression', 2, None, '___sec0'),
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('Basics', 2, None, '___sec1'),
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('Linear classifier', 2, None, '___sec2'),
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('Some selected properties', 2, None, '___sec3'),
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('The cross-entropy as a cost function for logistic regression',
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2,
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None,
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('Maximum likelihood', 2, None, '___sec5'),
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('Minimizing the cross entropy', 2, None, '___sec6')]}
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<a class="navbar-brand" href="LogReg-bs.html">Data Analysis and Machine Learning: Logistic Regression</a>
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<!-- navigation toc: --> <li><a href="._LogReg-bs001.html#___sec0" style="font-size: 80%;">Logistic Regression</a></li>
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<h2 id="___sec4" class="anchor">The cross-entropy as a cost function for logistic regression </h2>
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<p>
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The perceptron is an example of a ``hard classification": each datapoint is deterministically assigned to a category (i.e \( y_i=0 \) or \( y_i=1 \)). In many cases, it is favorable to have a "soft" classifier that outputs the probability of a given category rather than a single value. For example, given \( \mathbf{x}_i \), the classifier outputs the probability of being in category \( m \).
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Logistic regression is the most canonical example of a soft classifier. In logistic regression, the probability that a data point \( \boldsymbol{x}_i \) belongs to a category \( y_i=\{0,1\} \) is is given by
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$$
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\begin{eqnarray}
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P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& \frac{1}{1+\mathrm{e}^{-\mathbf{x}^T_i\mathbf{w}}},\nonumber\\
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P(y_i=0|\boldsymbol{x}_i,\boldsymbol{\theta)} &=& 1 - P(y_i=1|\boldsymbol{x}_i,\boldsymbol{\theta)},
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\end{eqnarray}
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$$
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where \( \boldsymbol{\theta}=\mathbf{w} \) are the weights we wish to learn from the data.
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<p>
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Notice that in terms of the logistic function, we can write
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$$
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P(y_i=1) =f(\mathbf{x}_i^T\mathbf{w})=1-P(y_i=0).
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$$
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<li><a href="._LogReg-bs001.html">2</a></li>
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<li><a href="._LogReg-bs002.html">3</a></li>
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<li><a href="._LogReg-bs007.html">8</a></li>
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<li><a href="._LogReg-bs008.html">9</a></li>
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