# Probability of belonging

Suppose we have 3 groups/events (M1, M2 and M3) with their own features (A..H). And then this new guy comes along with one feature (F) for now.

What is the probability of this new guy belonging to each of the groups in the future?

Any help or even hint as to what branch of statistics this question belongs to is appreciated.

Assuming the features A, B, ... H are vectors that reside in a metric space, one simple approach is to :

1. Compute a representative for each group (e.g. pick one, compute centroid, e.t.c).
2. Compute the distance of the new guy from each representative
3. Normalize the distances to form a probability distribution (e.g. squashing with the softmax)

Edit: There is also this related question.

• Do you think the representation of a group can be the cardinality of the intersection of the new guy's features and each group? The reason I ask is that the features are actually categorical. – Auxiliary Jul 16 '17 at 0:03
• The group representation should not be dependent on the new guy, but the group members. I assume you mean the distance? If the number of the possible values for each categorical coordinate is fixed, one alternative is to encode them using one-hot encoding , or a histogram-based approach. – npit Jul 16 '17 at 0:14
• Either way, I'd run some experiments to evaluate each distance approach if no related work is available, if I were you. – npit Jul 16 '17 at 0:14
• Thanks, this seems very reasonable. Now that I have you one more follow-up. If I have many of these new guys getting added. Do you thinks it's reasonable to average their probability of belonging to each of the groups as a way to aggregate the information? AND or OR doesn't seem to be what I'm looking for, but I'm not sure if averaging probabilities is common in the stats community. Thanks again. – Auxiliary Jul 16 '17 at 0:22
• This makes sense If there is an entity that is represented by more than one vectors and you want to decide where the entity belongs, based on the probabilities of its vectors belonging to each group. In that case, then yes, averaging the probabilities of all the vectors into a single probability vector is very reasonable and is very common in ML : it's called late fusion. Alternatively, instead of averaging the probabilities, you could average the vectors, and compute the probabilities as usual : this is called early fusion (but it's not readily applicable to categorical data). – npit Jul 16 '17 at 1:10