Note that we could think of inferring on the basis of
each newly observed datum . After the one observation:

We have written Eq. (47) in a way that the dependence between observables can be accommodated. From the product rule in Tab. 1, we can rewrite Eq. (47) as

(48) |

Comparing this equation with (47) we see that the sequential inference gives exactly the same result of a single inference that properly takes into account all available information. This is an important result of the Bayesian approach.

The extension to many variables is straightforward,
obtaining

(49) |

Furthermore, when the are independent, we get for the likelihood

that is, the combined likelihood is given by the product of the individual likelihoods.