Softmax Regression
From Ufldl
(→Optimizing Softmax Regression) |
m (Added missing brackets to log-likelihood.) |
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&= \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> |