Will Multiple Imputation (MICE) work on dataset with missing data on only one feature?

Question

Based on this article, it is apparent that MICE works with the following logic:

1. Fill missing values in every column apart from the column in question with either random or the mean of the given values
2. Use these feature sets as dependent values to regress given values in the column in question (dropping missing values)
3. Use the model trained on given values to predict missing values in the column in question
4. Cycle through the other columns using these given values and the predictions a set number of iterations (minimum of however many columns with missing data plus a certain factor)

It seems that this method for imputing missing values requires at least 2 variables with missing values, as the iteration is done with predictions for each column with missing values to introduce error into the imputation process.

Say you only have one feature with missing values (ie 10 features; 1 feature is missing 10% of rows; 9 features are 100% filled). Will MICE work in this scenario, or will it be reduced to simple/singular imputation? In other words, does MICE require datasets with more than one feature to impute missing values on?

Answer 1

Multiple imputation certainly works fine with just one incomplete variable/feature. The difference is that with only one variable with missing values there is no need for the procedure to be iterative and no need for the 'chained equations' aspect of mice. The imputation model for the incomplete variable is fitted to those rows with the variable observed, posterior draws of the imputation model parameters are taken, and the multiple imputations are drawn conditional on these and the predictor variables used.