; then update ; and so on. When the true distribution is sharply peaked, the integral determining may be not much changed by replacing the probability distribution over with a point estimate representing the distribution's peak (or, alternatively, its mean), With this approximation, the above iterative scheme becomes the EM algorithm. The term "Empirical Bayes" can cover a wide variety of methods, but most can be regarded as an early truncation of either the above scheme or something quite like it. Point estimates, rather than the whole distribution, are typically used for the parameter(s) . The estimates for are typically made from the first approximation to without subsequent refinement. These estimates for are usually made without considering an appropriate prior distribution for . Point estimation Robbins' method: non-parametric empirical Bayes (NPEB) Robbins considered a case of sampling from a mixed distribution, where probability for each (conditional on ) is specified by a Poisson distribution, while the prior on θ is unspecified except that it is also i.i.d. from an unknown distribution, with cumulative distribution function . Compound sampling arises in a variety of statistical estimation problems, such as accident rates and clinical trials. We simply seek a point prediction of given all the observed data. Because the prior is unspecified, we seek to do this without knowledge of G. Under squared error loss (SEL), the conditional expectation E(θi | Yi = yi) is a reasonable quantity to use for prediction. For the Poisson compound sampling model, this quantity is This can be simplified by multiplying both the numerator and denominator by , yielding where pG is the marginal distribution obtained by integrating out θ over G. To take advantage of this, Robbins suggested estimating the marginals with their empirical frequencies (), yielding the fully non-parametric estimate as: where denotes "number of". (See also Good–Turing frequency estimation.) Example – Accident rates Suppose each customer of an insurance company has an "accident rate" Θ and is insured against accidents; the probability distribution of Θ is the underlying distribution, and is unknown. The number of accidents suffered by each customer in a specified time period has a Poisson distribution with expected value equal to the particular customer's accident rate. The actual number of accidents experienced by a customer is the observable quantity. A crude way to estimate the underlying probability distribution of the accident rate Θ is to estimate the proportion of members