A fast, non-MCMC based algorithm, called Iterative Conditional Expectation (ICE), was derived for the calculation of GW-EBV using dense SNP genotype and phenotypic data. The speed of improvement is due to the analytical integration of the integrals involved in the calculation of E [g|Y]. The Bayesian estimation model used here has very similar characteristics to BayesB as described by [1] and denoted MBayesB here: the prior distribution is a mixture of a heavy-tailed distribution (fBayesB: exponential distribution; MBayesB: a normal distribution whose variance is sampled from an inverse chi-squared) and a distribution with zero effects. The latter mixture distribution is also called a 'spike and slab' mixture [5]. The QTL effects had a Gamma distribution, and thus differed from both that of fBayesB and MBayesB. It may be noted that the prior distribution of fBayesB and MBayesB does not apply to the QTL effects but to the marker effects, for which the true prior distribution will be hard to derive. Although we cannot rule out that the slightly better accuracy of MBayesB compared to fBayesB (Table 1) is due to the prior distribution of MBayesB being better than that of fBayesB, the non-linear regressions of MBayesB and fBayesB seemed very similar since they resulted in a linear regression of the fBayesB-EBV on the MBayesB-EBV (Figure 4). The difference in accuracy seems to be due to that the ICE algorithm ignores higher order derivatives of E(g|Y) function to Y (Eqn. 5). Relaxing this assumption requires (1) taking second order derivatives of the E(g|Y) function, (2) calculation of prediction error variances of
, and more research is needed to perform these calculations computationally efficiently.

The justification for the 'spike and slab' prior distribution is that many of the SNPs will not be in LD with a QTL and thus have no effect, whereas the SNPs that are in LD with a QTL have a distribution of effects that is similar to that of the QTL, albeit smaller in magnitude due to the need for several markers to predict the effect of the true QTL genotypes. The true distribution of QTL is often reported to be exponential or gamma [6]. Hayes and Goddard [6] found a shape parameter for the gamma distribution of 0.4, *i.e*. a leptokurtic shape similar to that of the exponential distribution. Where the marker is not in perfect LD with the gene, it will pick up only a fraction of the gene effect and the impact of this on the distribution of marker effects is included within the assumptions concerning their prior distribution.

The non-linear regression curve, resulting from the choice of the prior distribution, is rather flat for values of Y close to 0, but approaches a ratio of E [g|Y]/Y = 1 for Y of large magnitude, so E [g|Y] ≈ Y albeit for very large, and hence rare, deviations. Thus large values of Y are assumed to represent true marker effects, whereas small values are regressed back substantially, *i.e*. are unlikely to represent a true effect. In contrast, if Best Linear Unbiased Prediction (BLUP) of marker effects is used, *i.e*. a normal prior distribution of the marker effects, the regression does not depend on Y and is a constant equal to *σ*
^{2}/(*σ*
^{2} + *σ*
_{m}
^{2}), *i.e*. E(g|Y) = Y* *σ*
^{2}/(*σ*
^{2} + *σ*
_{m}
^{2}), where *σ*
_{m}
^{2} is the variance of the marker effects, which will be *σ*
_{a}
^{2}/m. This distinction is due to the use of the normal prior instead of the exponential and, as a consequence, the heavier tails giving credence to large marker effects. Nevertheless the high value of E [g|Y]/Y when using the exponential prior may not be a desirable effect, if outlier data points are encountered.

The variance due to a marker is Var("Marker") = Var(b_{i})Var(g). Here we standardised the variance of the genotypes to Var(b_{i}) = 1, *i.e*. the prior variance assumed for the marker effects, g, applies directly for the variance due to the marker, and thus does not depend on marker frequency. We prefer this parameterisation, assuming that the variance of the marker effects is frequency dependent, because (1) QTL with large effects are expected to be at rare frequencies, which implies that the variance of the QTL is roughly constant (at least considerably more constant than when QTL effects were not frequency dependent); (2) if we assume that the QTL variance needs to be above a certain threshold before the markers pick up its effect, these QTL will have a much more constant variance than randomly picked QTL. The algorithm is equally capable of handling the coding of b_{i} as 0, 1, and 2, (or after subtracting the mean -2p, (1–2p), and 2(1-p)) for the genotypes mm, Mm, and MM, respectively. The accuracies of the GW-EBV were virtually the same as those in Table 1 when the latter parameterisation of the b_{i} was used (result not shown).

The computational advantage of our fast algorithm for the BayesB approach to GW-EBV will not outweigh the reduced accuracy observed, if confirmed for typical trait architecture, when used in practical breeding schemes and the computational time and effort can be afforded. If, in practice, breeding schemes wish to select upon GW-EBV that require frequent updating, then a more appropriate comparison is between frequently updated fBayesB estimates of marker effects and the use of 'old' MCMC based estimates of marker effects, where the GW-EBV of animals are calculated without updating the estimates of marker effects, because of computational constraints. In the latter case there will be a loss of accuracy. It will depend on the amount of new information coming into the breeding value evaluation system, which of these alternatives should be favoured. In simulations of breeding schemes and in cross-validation testing of GW-EBV, the large number of EBV evaluations required may make our fast algorithm the only means to implement BayesB type genome-wide breeding value estimation.