Lasso regression based on the L1 regularization is one of the most popular sparse estimation methods. It is often required to set appropriately in advance the regularization parameter that determines the degree of regularization. Although the empirical Bayes approach provides an effective method to estimate the regularization parameter, its solution has yet to be fully investigated in the lasso regression model. In this study, we analyze the empirical Bayes estimator of the one-parameter model of lasso regression and show its uniqueness and its properties. Furthermore, we compare this estimator with that of the variational approximation, and its accuracy is evaluated.

Bayesian methods are often applied for estimating the event rate from a series of event occurrences. However, the Bayesian posterior distribution requires the computation of the marginal likelihood which generally involves an analytically intractable integration. As an event rate is defined in a very high dimensional space, it is computationally demanding to obtain the Bayesian posterior distribution for the rate. We estimate the rate underlying a sequence of event counts by deriving an approximate Bayesian inference algorithm for the time-varying binomial process. This enables us to calculate the posterior distribution analytically. We also provide a method for estimating the prior hyperparameter, which determines the smoothness of the estimated event rate. Moreover, we provide an efficient method to compute the upper and lower bounds of the marginal likelihood, which evaluate the approximation accuracy. Numerical experiments demonstrate the effectiveness of the proposed method in terms of the estimation accuracy.