# ML Wiki

## Cost Matrix

Used for comparing two different models

• A cost matrix is a matrix of the following form:
$y = +$ $y = -$
$h_\theta(x) = +$ +)$-)$
$h_\theta(x) = -$ +)$-)$

In general case:

• $C(i | j)$
• a cost of classifying an example of class $j$ as class $i$
• this way we can express that some mispredictions are very costly

### Example

$y = +$ $y = -$
$h_\theta(x) = +$ +) = -1$-) = 1$
$h_\theta(x) = -$ +) = 100$-) = 0$
• we put $C(- | +) = 100$ because in this example false negatives are very costly

And assume we're comparing two classifiers $C_1$ and $C_2$

stats of $C_1$
$y = +$ $y = -$
$h_{C_1}(x) = +$ 150 60
$h_{C_1}(x) = -$ 40 250
• $\text{acc}(C_1) = \cfrac{150+250}{150+40+60+250} = 80\%$
• $\text{cost}(C_1) = -1 \cdot 150 + 1 \cdot 60 + 100 \cdot 40 + 0 \cdot 250 = 3910$
stats of $C_2$
$y = +$ $y = -$
$h_{C_2}(x) = +$ 250 5
$h_{C_2}(x) = -$ 45 200
• $\text{acc}(C_2) = \cfrac{250+200}{250+45+5+200} = 90\%$
• $\text{cost}(C_2) = -1 \cdot 250 + 1 \cdot 5 + 100 \cdot 45 + 0 \cdot 200 = 4255$

Selecting $C_1$

• because $C_1$ has lower cost: $\text{cost}(C_1) < \text{cost}(C_2)$
• even though $C_2$ has better accuracy: $\text{acc}(C_2) > \text{acc}(C_1)$