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I have problems with this exam problem (see image below) for machine learning.

Problem

For my answer i assumed that $\hat{y}^{}_{0}$ and $\hat{y}^{}_{1}$ are the predicted classes. That means that $\hat{y}^{}_{0}$ predicted class 0 which means $a$ and $\hat{y}^{}_{1}$ predicted class 1 which means $b$.

Furthermore, it is given that both $\hat{y}^{}_{0}$ and $\hat{y}^{}_{1}$ minimize the expected loss.

The exprected loss is:

$\sum^{K}_{k=1} e(\hat{y}, c_k) p_k$

Minimzing exprected loss means that $\hat{y}$ predicts the class $c_k$ with the highest probability $p_k$. Because both $\hat{y}^{}_{0}$ and $\hat{y}^{}_{1}$ minimize the expected loss means that the probability of $p_1$ and $p_2$ have to be the same (and bigger than $p_3$).

Resulting in $p_1 = p_2$ and $p_3 = 1 - (p_1 + p_2)$, $p_3 < p_2$.

Are my assumptions correct?

I was a bit confused about the naming. Why is it not called $\hat{y}^{}_{a}, \hat{y}^{}_{b}$ instead of $\hat{y}^{}_{0}, \hat{y}^{}_{1}$ and $(p_0, p_1, p_2)$ instead of $(p_1, p_2, p_3)$?

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I would have thought that

  • $\hat{y}^{}_{0}$ is the estimator which minimises the $0/1$ loss
    • as you say, this is class with the highest conditional probability $p_k$
  • $\hat{y}^{}_{1}$ is the estimator which minimises the $L_1$ loss
    • this should the median of the conditional distribution

As an example of where they might produce different results, consider $(p^{}_1, p^{}_2, p^{}_3) = (0.35, 0.25, 0.4)$, which would give $\hat{y}^{}_{0} = c$ but $\hat{y}^{}_{1} = b$.

As for notation, the subscripts in $\hat{y}^{}_{0}$ and $\hat{y}^{}_{1}$ are arbitrary though vaguely suggestive of the two loss functions, and they are nothing to do with $a,b,c$. Perhaps in another context you could similarly use $\hat{y}^{}_2$ for the estimator for the $L_2$ expected loss minimising estimator, which would be the mean of the conditional distribution

On the other hand, the subscripts in $p^{}_1, p^{}_2, p^{}_3$ (perhaps suggesting the ordering) could easily have been written as $p^{}_a, p^{}_b, p^{}_c$ here

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  • $\begingroup$ Oh wow. Thanks for pointing that out. That makes a lot more sense. $\endgroup$
    – Jens
    Commented Aug 12, 2018 at 11:03

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