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    $\begingroup$ Nice derivation. One extra point: the averaging over ${\mathcal{M}}$ is done only for "numerical stability" (this is mentioned in 2006 Bishop's book Page 330), if the optimization process is exact (which is not), $b$ could be calculated using only one point from ${\mathcal{M}}$ without averaging. $\endgroup$ Commented May 3, 2019 at 14:35