I'm dealing with a Naive Bayes approach to a Multiclass Classification problem with 9 different classes in the target variable. Let's assume the following: I've fitted a model to my training data and want to apply it to the remaining observations used as test data.
I'm wondering about the interpretation of the calculated probabilities after including information after the estimation of the classifier.
Let's assume the models returns the following hypothetical class probabilities for a single observation in my data: $$ [p_1,p_2,p_3,\ldots,p_9]=[0.3,0.2,0.4,0,\ldots,0,0.1] $$ This would imply a classification to Class 3 without any additional knowledge. Now I get an additional piece of information which tells me that the instance to classify belongs either to Class 1 OR to Class 2.
Other than assigning an observation to the most probable class: Are the probabilities meaningful and interpretable in any way?
Is it possible to assign the instance to Class 1 because I know that it holds the highest probability among the remaining classes if my information is right? The question might be stupid and I have to admit that I haven't actually tried to figure out the math behind this myself but does anyone have an intuition for this?