Clarification for $\beta = {\{\beta_{10},\beta_1}\} $ when fitting logistic regression and the number of classes is k=2

Question

I was learning from Elements of statistics p.120 under section 4.4.1 Fitting Logistics Regression Models

The log likelihood function was given as

$l(\beta) = \sum_{i=1}^N {y_i\log p(x_i;\beta) + (1-y_i)log(1-p(x_i;\beta))}$

Here

$$\beta = {\{\beta_{10},\beta_1}\} \qquad (1)$$

and we assume that the vector of inputs $x_i$ includes the constant term 1 to accommodate the intercept.

Please my question is :

Assuming we have only two inputs $X$ = $X_1$ + $X_2$ and adding the intercept or constant term ($X_0)$ that contains only 1's, we will have $X$ = $X_1 + X_2 + X_0$. When we find $\beta$ using linear regression, it will be a vector in $R^3$ or the vector will contain three elements i.e $\beta = \{ b_1,b_2,b_3 \}$

How did they get $\beta_{10}$ in $(1)$ and also I want to know if $\beta_{10}$ and $\beta_1$ in $(1)$ are scalars or their vectors

Answer 1

To address your first question, assuming that you have two features, $X_1$ and $X_2$, and you choose to include an intercept, the number of parameters that you would expect to estimate is 3, i.e. $\beta = \{\beta_0, \beta_1, \beta_2\}$. However, if you define a new variable $X = X_{1}+X_{2}$, we could treat this as a single feature and therefore model using $\beta = \{\beta_0, \beta_1\}$

To address your second question regarding $\beta_{10}$ and $\beta_1$. If you look at the next statement on page 120

\begin{equation} l(\beta) = \sum_{i=1}^N {y_i\log p(x_i;\beta) + (1-y_i)log(1-p(x_i;\beta))} \\ = \sum_{i=1}^N {y_i\beta^Tx_i - log(1+e^{\beta^Tx_i})} \end{equation}

We see that $\beta$ is treated as vector which suggest that $\beta_{10}$ and $\beta_{1}$ are the components of $\beta$. Based on this, I would assume that the authors are implying these are scalar values.