Comment by wosk

The \pi_i in the paper is not the estimate of a latent parameter. It is the predictive probability of the event, which is a single number by necessity in a binary challenge. It's the integration of a distribution function which can contains very complex distributions: in my example something_you_believe can be a probability distribution.

So everything in the paper is distribution and when you forecast for a binary event, you give a number which is the expectation of that distribution. This is a probabilistic forecast.

If you were to give a probabilistic forecast for a continuous quantity, then yes you would give in a distribution, as in section 4.2