Confidence interval of probability estimator of Laplace smoothing

Sometimes, we do not use a maximum likelihood estimator of a probability but it's a smoothed estimator in order to cope with the zero frequency problem. This is often the case when we use the Naive Bayes classifier. Laplace smoothing is a popular choice with the value of Laplace smoothing estimator being the expected value of posterior distribution of the probability where we assume that the prior is uniform distribution. In this paper, we investigate the confidence intervals of the estimator of Laplace smoothing. We show that the likelihood function for this confidence interval is the same as the likelihood of a maximum likelihood estimated value of a probability of Bernoulli trials. Although the confidence interval of the maximum likelihood estimator of the Bernoulli trial probability has been studied well, and although the approximate formulas for the confidence interval are well known, we cannot use the interval of maximum likelihood estimator since the interval contains the value 0, which is not suitable for the Naive Bayes classifier. We are also interested in the accuracy of existing approximation methods since these approximation methods are frequently used but their accuracy is not well discussed. Thus, we obtain the confidence interval by numerically integrating the likelihood function. In this paper, we report the difference between the confidence interval that we computed and the confidence interval by approximate formulas. Finally, we include a URL, where all of the intervals that we computed are available.