Approximate Empirical Bayes Estimation of the Regularization Parameter in ℓ1 Trend Filtering

Trend filtering is often used in economics and other fields. ℓ 1 trend filtering was proposed as a derivative of Hodrick-Prescott filtering based on the sparsity in the changes of trends. Although it has a regularization parameter, which needs to be set in advance, the non-conjugacy arising from the ℓ 1 regularization term prohibits a tractable Bayesian treatment including the sequence-dependent estimation of the regularization parameter. In this study, we consider the empirical Bayes estimation of the regularization parameter by approximating the non-conjugate prior distribution in ℓ 1 trend filtering by variational approximation.