Revisiting our Logistic Regression case

In our discussion on Logistic Regression we studied the case of two classes, with $y_i$ either $0$ or $1$. Furthermore we assumed also that we have only two parameters $\beta$ in our fitting, that is we defined probabilities

$$\begin{align*} p(y_i=1|x_i,\boldsymbol{\beta}) &= \frac{\exp{(\beta_0+\beta_1x_i)}}{1+\exp{(\beta_0+\beta_1x_i)}},\nonumber\\ p(y_i=0|x_i,\boldsymbol{\beta}) &= 1 - p(y_i=1|x_i,\boldsymbol{\beta}), \end{align*}$$

where $\boldsymbol{\beta}$ are the weights we wish to extract from data, in our case $\beta_0$ and $\beta_1$.

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