\mathbb{P}\{q = f \,|\, W = w\}

= \frac{\mathbb{P}\{W = w \,|\, q = f\}\mathbb{P}\{q = f\}}

{\mathbb{P}\{W = w\}}

\quad \text{and} \quad

$$

and

$$

\mathbb{P}\{W = w\} = \sum_{\omega \in \{f, g\}} \mathbb{P}\{W = w \,|\, q = \omega\} \mathbb{P}\{q = \omega\}

$$

$$

q_{\pi}(w) := \pi f(w) + (1 - \pi) g(w)

\quad \text{and} \quad

$$

and

$$

\kappa(w, \pi) := \frac{\pi f(w)}{\pi f(w) + (1 - \pi) g(w)}

$$