# Bayesian inference: all at once vs one at a time [duplicate]

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?

• Yes by simply using the independence between the observations order (exchangeability) and the different ways you can write their joint distribution in terms of conditional distributions.
– user289381
Jul 9, 2020 at 14:12
• Jul 9, 2020 at 20:07
• Jul 9, 2020 at 20:17