# Imputing binary variable when no 0s, only 1s are available

I'm trying to impute missing values for a binary variable (values 0 and 1) with some challenging data (of about 1 million observations).

The data can be divided into two groups:

• in group 1, we know that all individuals take 1 as a value. This amounts to about 1% of the data.

• in group 2, we don't know any of the values. We only know that the proportion of 1s is small (about as many as in group 1, i.e. 10,000 observations).

The fact that group 1 has only 1s and no 0s makes it impossible to train a model just on group 1, to then use it and get predictions for group 2. Still my thinking is that observations taking 1 as a value in group 2 should be closer to group 1 in their characteristics (i.e. other variables available in the data) than observations taking 0 as a value in group 2.

So how can I use the information given by group 1, to try and find the 1s in group 2 ? If someone has an idea, I'd be happy to hear their thoughts.

Thank you very much!