usage of p0 and p1 ?

[lcrmorin]

I am not entirely sure what is supposed to be the difference between p0 and p1. By similarity to the sklearn framework I would expect something like p1 = 1 - p0. But this is not the case. From the paper I understand that the final prediction is a function of p0 and p1 depending on the target metric. wouldn't some other names for p0 and p1 be better ? wouldn't a predict_proba method be usefull ?

[ptocca]

Thanks for the comment and the suggestions.
I kept the names p0 and p1 to remain aligned with the notation used in the paper. Perhaps p_low and p_high might be a clearer alternative?
Let me try to clarify their meaning: p0 and p1 are the probability predictions for y=1 if we were to add to the training set a new hypothetical example z=(x_t, y_t) made up of the test object x_t and y_t=0 in one case and y_t=1 in the other.
So you could view the interval (p_0, p_1) as an indication of the sensitivity of the prediction to the data.
Just to reiterate: both p0 and p1 are predictions for the probability that the label y_t be 1. So they do not sum to 1.

[francescopisu]

Would it be correct to use this formula
p = p1 / (1 - p0 + p1)
to get a "point" prediction (single value prediction as per this presentation) and report it with p0, p1 acting as confidence bounds ? E.g., p [p0, p1]
I'm new to the topic so what I'm saying could potentially be wrong.

Thank you

[ptocca]

@francescopisu yes, that is correct.
Strictly speaking, the suggested way is optimal in some sense (see later) when you use log-loss as loss function.
However, if you use square loss, the optimality is achieved for a different way of combining the two predictions.
For this alternative way and for details on the precise meaning of optimality in this context, please refer to chapter 4 of the paper Venn-Abers Predictors by Vovk and Petej.