of the whole population suffering 0, 1, 2, 3, ... accidents during the specified time period as the corresponding proportion in the observed random sample. Having done so, it is then desired to predict the accident rate of each customer in the sample. As above, one may use the conditional expected value of the accident rate Θ given the observed number of accidents during the baseline period. Thus, if a customer suffers six accidents during the baseline period, that customer's estimated accident rate is 7 × [the proportion of the sample who suffered 7 accidents] / [the proportion of the sample who suffered 6 accidents]. Note that if the proportion of people suffering k accidents is a decreasing function of k, the customer's predicted accident rate will often be lower than their observed number of accidents. This shrinkage effect is typical of empirical Bayes analyses. Parametric empirical Bayes If the likelihood and its prior take on simple parametric forms (such as 1- or 2-dimensional likelihood functions with simple conjugate priors), then the empirical Bayes problem is only to estimate the marginal and the hyperparameters using the complete set of empirical measurements. For example, one common approach, called parametric empirical Bayes point estimation, is to approximate the marginal using the maximum likelihood estimate (MLE), or a Moments expansion, which allows one to express the hyperparameters in terms of the empirical mean and variance. This simplified marginal allows one to plug in the empirical averages into a point estimate for the prior . The resulting equation for the prior is greatly simplified, as shown below. There are several common parametric empirical Bayes models, including the Poisson–gamma model (below), the Beta-binomial model, the Gaussian–Gaussian model, the Dirichlet-multinomial model, as well specific models for Bayesian linear regression (see below) and Bayesian multivariate linear regression. More advanced approaches include hierarchical Bayes models and Bayesian mixture models. Gaussian–Gaussian model For an example of empirical Bayes estimation using a Gaussian-Gaussian model, see Empirical Bayes estimators. Poisson–gamma model For example, in the example above, let the likelihood be a Poisson distribution, and let the prior now be specified by the conjugate prior, which is a gamma distribution () (where ): It is straightforward to show the posterior is also a gamma distribution. Write where the marginal distribution has been omitted since it does not depend explicitly on . Expanding terms which do depend on gives the posterior as: