Continuing the discussion from Multiple data groups with custom log-likelihood function:

In Bayesian updating, everything boils down to evaluating Bayes’ theorem written in its unnormalized form as

with the posterior distribution \pi(x\vert\mathcal{Y}), the prior distribution \pi(x) and the likelihood function \mathcal{L}(x;\mathcal{Y}). The data set is denoted as \mathcal{Y}=\{y_1,\dots,y_N\}. This corresponds to updating in *bulk* because all data enters into the likelihood function *at once*.

Opposed to this you can imagine a scenario where the data arrives *sequentially* and you need to update the distributions multiple times. For simplicity, assume we do two such updating steps by splitting the data set into two disjoint sets \mathcal{Y}_1 \cup \mathcal{Y}_2 = \mathcal{Y}. You can then update the initial prior distribution in two steps by

where you need to be careful to use the **posterior** of the first updating step as the **prior** of the second updating step. If this is the case and the data were collected independently, it is clear that the last equation is **identical** to the bulk updating equation.

So there should be **no difference** between updating sequentially and in bulk if the conditions mentioned are met. Is this the case?