So the posterior density is also a gamma distribution , where , and . Also notice that the marginal is simply the integral of the posterior over all , which turns out to be a negative binomial distribution. To apply empirical Bayes, we will approximate the marginal using the maximum likelihood estimate (MLE). But since the posterior is a gamma distribution, the MLE of the marginal turns out to be just the mean of the posterior, which is the point estimate we need. Recalling that the mean of a gamma distribution is simply , we have To obtain the values of and , empirical Bayes prescribes estimating mean and variance using the complete set of empirical data. The resulting point estimate is therefore like a weighted average of the sample mean and the prior mean . This turns out to be a general feature of empirical Bayes; the point estimates for the prior (i.e. mean) will look like a weighted averages of the sample estimate and the prior estimate (likewise for estimates of the variance). See also Bayes estimator Bayesian network Hyperparameter Hyperprior Best linear unbiased prediction Robbins lemma Spike-and-slab variable selection References Further reading External links Use of empirical Bayes Method in estimating road safety (North America) Empirical Bayes methods for missing data analysis Using the Beta-Binomial distribution to assess performance of a biometric identification device A Hierarchical Naive Bayes Classifiers (for continuous and discrete variables). Nonparametric Bayesian statistics