Suppose that I have a prior on a parameter $\theta$ and update this prior in light of the realisation of $n$ random variables. It seems plausible that it is equivalent to update the prior $n$ times, once for every data point, or instead update it once using the $n$ data points. For example, suppose that I start with a beta prior (with parameters $\alpha > 0$, $\beta > 0$) on the fraction of balls in an urn that are black. I observe two black balls. If I update all at once, then my posterior is immediately beta distributed with parameters $\alpha + 2$, $\beta$. If I instead update twice, my posterior becomes first beta with parameters $\alpha + 1$, $\beta$, and then (after the second update) beta with parameters $\alpha + 2$, $\beta$. So in this case, it is equivalent to update all at once or one at a time.

Can it be proven that this is true in general?



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