  Modern large scale datasets are often plagued with missing entries. For
tabular data with missing values, a flurry of imputation algorithms solve for a
complete matrix which minimizes some penalized reconstruction error. However,
almost none of them can estimate the uncertainty of its imputations. This paper
proposes a probabilistic and scalable framework for missing value imputation
with quantified uncertainty. Our model, the Low Rank Gaussian Copula, augments
a standard probabilistic model, Probabilistic Principal Component Analysis,
with marginal transformations for each column that allow the model to better
match the distribution of the data. It naturally handles Boolean, ordinal, and
real-valued observations and quantifies the uncertainty in each imputation. The
time required to fit the model scales linearly with the number of rows and the
number of columns in the dataset. Empirical results show the method yields
state-of-the-art imputation accuracy across a wide range of data types,
including those with high rank. Our uncertainty measure predicts imputation
error well: entries with lower uncertainty do have lower imputation error (on
average). Moreover, for real-valued data, the resulting confidence intervals
are well-calibrated.
