Five Losses, No Probabilities

Two models on a team's dashboard log only their per-example loss. The probabilities behind each loss are thrown away.

A spam filter reports one number, the probability that an email is spam; not spam gets whatever is left. It is trained with binary cross-entropy.
An animal classifier gives a probability to each of four animals (cat, dog, bird, fish) through softmax, and is trained with categorical cross-entropy.

For both models, the loss on one example is

$\text{loss} = -\ln p$

where $p$ is the probability the model gave the true answer and $\ln$ is the natural logarithm. A model's verdict on an example is whichever option it gave the highest probability; there are no ties.

Five entries from the log:

example	model	loss
E1	spam filter	$0.51$
E2	spam filter	$0.92$
P1	animal classifier	$0.51$
P2	animal classifier	$0.92$
P3	animal classifier	$1.61$

Using nothing but this table, for which examples can you say for certain whether the model's verdict was right or wrong?

Five Losses, No Probabilities

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