  Missing time-series data is a prevalent practical problem. Imputation methods
in time-series data often are applied to the full panel data with the purpose
of training a model for a downstream out-of-sample task. For example, in
finance, imputation of missing returns may be applied prior to training a
portfolio optimization model. Unfortunately, this practice may result in a
look-ahead-bias in the future performance on the downstream task. There is an
inherent trade-off between the look-ahead-bias of using the full data set for
imputation and the larger variance in the imputation from using only the
training data. By connecting layers of information revealed in time, we propose
a Bayesian posterior consensus distribution which optimally controls the
variance and look-ahead-bias trade-off in the imputation. We demonstrate the
benefit of our methodology both in synthetic and real financial data.
