propagating uncertainty of fitted values among classes of categorical models

[Jonathan Judge]

I have a follow-up to a question Paul graciously answered earlier in another forum:

Assume we have a brms categorical model with three potential outcomes. 

After fitting the model, the predicted probabilities for an observation (obtained through the fitted function) for a given data point are .2, .3, and .5 respectively in each category. 

Assume further that each probability has an "Estimated Error" of ~ .02. 

The question then arises what the estimated errors "mean," as a practical matter, in determining the actual probable range of probabilities among the various outcomes for that observation.  

Presumably, for the first category, an Estimated Error of .02 around the posterior mean of .2 suggests that value could also easily be .018 or .022.  

But then how is that potential variance reassigned to (or taken from) the other categories, since the probabilities between the three outcomes must still add up to 1?  Is it reassigned to / taken away from the other two categories in equal measure?  Is the relationship between the variance for the first outcome prorated in some way over those other categories in some respect relative to their original assigned probability?

I am trying to get a sense for the actual probability range of each of the three possible outcomes, in light of their cumulative / respective uncertainties, but am having a difficult time finding resources that would explain how the prediction error for each outcome gets propagated among the various classes for prediction.

I'd appreciate any insight or references people can provide.

[Paul Buerkner]

The vector of probabilities is forming a simplex, so yes, the probability is taken away from the other categories.

I suggest, you closely investigate the output of fitted(..., summary = FALSE) to understand how the probabilities relate to each other.

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