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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?

afreelunch
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  • 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. –  Jul 09 '20 at 14:12
  • See https://en.wikipedia.org/wiki/Recursive_Bayesian_estimation – kjetil b halvorsen Jul 09 '20 at 20:07
  • Related: https://stats.stackexchange.com/questions/122960/bayes-rule-and-multiple-conditioning, https://math.stackexchange.com/questions/408774/bayes-rule-with-multiple-conditions, https://stats.stackexchange.com/questions/384883/iterating-bayes-rule-over-time, – kjetil b halvorsen Jul 09 '20 at 20:17

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