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.

The Tikhonov regularization theory converts ill-posed inverse problems into well-posed problems by putting penalty on the solution sought. Instead of solving an inverse problem, the regularization theory minimizes a weighted sum of "data error" and "penalty" function, and it has been successfully applied to a variety of problems including tomography, inverse scattering, detection of radiation sources and early vision algorithms. Since the function to be minimized is a weighted sum of functions, one should estimate appropriate weights. This is a problem of hyperparameter estimation and a vast literature exists. Another problem is how one should compare a particular penalty function (regularizer) with another. This is a special class of model comparison problems which are generally difficult. A Hierarchical Bayesian scheme is proposed with multiple hyperparameters in order to cope with data containing subsets which consist of different degree of smoothness. The scheme outperforms the previous scheme with single hyperparameter.