Exercise 5.2 - Optimal threshold on classification probability

Answers

For question (a), the posterior loss expectation for choosing action $ŷ$ , given $x$ , is:

\begin{aligned} ρ (ŷ | x) & = 𝔼_{p (y | x)} [L (y, ŷ)] \\ = p_{0} \cdot L (ŷ, 0) + p_{1} \cdot L (ŷ, 1) \\ = p_{0} \cdot L (ŷ, 0) + (1 - p_{0}) \cdot L (ŷ, 1) \\ = L (ŷ, 1) + p_{0} (L (ŷ, 0) - L (ŷ, 1)) . \end{aligned}

Thus:

ρ (0 | x) = λ_{01} - p_{0} \cdot λ_{01},

ρ (1 | x) = p_{0} \cdot λ_{10} .

As both $ρ (0 | x)$ and $ρ (1 | x)$ are linear functions of $p_{0}$ , hence the unique optimal threshold is where $ρ (0 | x) = ρ (1 | x)$ , i.e.,

p_{0} = \frac{λ_{01}}{λ_{01} + λ_{10}},

p_{1} = \frac{λ_{10}}{λ_{01} + λ_{10}} .

For question (b), let $λ_{10} = 1$ , $λ_{01} = 9$ will do.

solour_lfq

2021-03-24 13:42