Softmax Regression

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(Optimizing Softmax Regression)
m (Added missing brackets to log-likelihood.)
Line 56: Line 56:
&= \ln \prod_{i=1}^{m}{ P(y^{(i)} | x^{(i)}) } \\
&= \ln \prod_{i=1}^{m}{ P(y^{(i)} | x^{(i)}) } \\
&= \sum_{i=1}^{m}{ \ln \frac{ e^{ \theta^T_{y^{(i)}} x^{(i)} } }{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} } } \\
&= \sum_{i=1}^{m}{ \ln \frac{ e^{ \theta^T_{y^{(i)}} x^{(i)} } }{ \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }} } } \\
-
&= \sum_{i=1}^{m}{\theta^T_{y^{(i)}} x^{(i)} - \ln \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }}}
+
&= \sum_{i=1}^{m}{\left[ \theta^T_{y^{(i)}} x^{(i)} - \ln \sum_{j=1}^{n}{e^{ \theta_j^T x^{(i)} }}\right]}
\end{align}
\end{align}
</math>
</math>

Revision as of 04:02, 9 May 2011

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