\(\begin{split} P(y = 1 \mid \mathbf{x}, \mathbf{w}) &= \sigma( \mathbf{w}^T\mathbf{x} + b ) = \frac{1}{1 + e^{-(\mathbf{w}^T\mathbf{x }+ b)}}\\ P(y = 0 \mid \mathbf{x}, \mathbf{w}) &= 1 - \sigma( \mathbf{w}^T\mathbf{x} + b ) \end{split}\)
\(\mathcal{L}(\mathbf{w}) = \prod_{i=1}^n \begin{cases} P(y = 1 \mid \mathbf{x_i}, \mathbf{w}), \text{if } y_i = 1 \\ P(y = 0 \mid \mathbf{x_i}, \mathbf{w}), \text{if } y_i = 0 \\ \end{cases}\)
\(\mathcal{L}(\mathbf{w})= \prod_{i=1}^n P(y = 1 \mid \mathbf{x_i}, \mathbf{w})^{y_i} \times P(y = 0 \mid \mathbf{x_i}, \mathbf{w})^{1-y_i}\)
\(\mathcal{LL}(\mathbf{w}) = \sum_{i=1}^n y_i \log P(y = 1 \mid \mathbf{x_i}, \mathbf{w}) + (1 - y_i) \log P(y = 0 \mid \mathbf{x_i}, \mathbf{w})\)
Calculate the likelihood and the log likelihood for the following data set:
| \(\sigma( \mathbf{w}^T\mathbf{x_i} + b )\) | \(y_i\) |
| .7 | 1 |
| .25 | 0 |
| .9 | 1 |