Iteratively reweighted least squares in Machine Learning Probability perspective

I am studying Machine Learning Probability perspective and I have a question with Algorithm 8.2, which is

$$\mathbf{w} = 0_{D}$$ $$w_{0} = \log (\bar{y}/(1-\bar{y}))$$

Repeat:
$$\eta_{i} = w_{0}+\mathbf{w}^{T} \mathbf{x}_{i}$$
$$\mu_{i} = \text{sigm}(\eta_{i})$$
$$s_{i} = \mu_{i}(1-\mu_{i})$$
$$z_{i} = \eta_{i}+\frac{y_{i}-\mu_{i}}{s_{i}}$$
$$\mathbf{S} = \text{diag}(s_{1:N})$$
$$\mathbf{w} = (\mathbf{X}^{T}\mathbf{S}\mathbf{X})^{-1}\mathbf{X}^{T}\mathbf{S}\mathbf{z}$$
until converged

I think $$w_{0}$$ means the coefficient of intercept, but why it is not updated in the loop?
It seems like $$w_{0}$$ is fixed like a constant?

$w_{0} = \log (\bar{y}/(1-\bar{y}))$