175 lines
5.9 KiB
HTML
175 lines
5.9 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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'___sec4'),
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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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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></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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<a name="part0006"></a>
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<!-- !split -->
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<h2 id="___sec5" class="anchor">Maximum likelihood </h2>
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<p>
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We now define the cost function for logistic regression using Maximum
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Likelihood Estimation (MLE). Recall, that in MLE we choose parameters
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to maximize the probability of seeing the observed data. Consider a
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dataset \( \mathcal{D}=\{(y_i,\boldsymbol{x}_i)\} \) with binary labels
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\( y_i\in\{0,1\} \) where the data points are drawn independently. The
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likelihood of the seeing the data under our model is just:
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$$
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\begin{align}
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P(\mathcal{D}|\mathbf{w})& = \prod_{i=1}^n \left[f(\mathbf{x}_i^T\mathbf{w})\right]^{y_i}\left[1-f(\mathbf{x}_i^T\mathbf{w})\right]^{1-y_i}\nonumber \\
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\tag{3}
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\end{align}
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$$
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from which we can readily compute the log-likelihood:
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$$
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\begin{equation}
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l(\mathbf{w}) = \sum_{i=1}^n y_i\log f(\mathbf{x}_i^T\mathbf{w}) + (1-y_i)\log\left[1-f(\mathbf{x}_i^T\mathbf{w})\right].
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\tag{4}
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\end{equation}
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$$
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