Lesson 15 gave you the beta-binomial update rule: start with Beta(α, β), observe k successes out
of n trials, and the posterior is Beta(α + k, β + (n − k)) — you just add counts. So far every
lesson has updated on one batch of data at a time. Real data rarely arrives that tidily: it comes in
chunks, on different days, from different sources.
Setup. You start with a uniform prior, Beta(1, 1) (genuine ignorance — every value of p is
equally plausible). Two batches of evidence arrive, in some order:
- Batch A: 3 successes, 1 failure (4 trials)
- Batch B: 2 successes, 2 failures (4 trials)
Part 1 — Process A, then B. Starting from Beta(1, 1), update on Batch A to get an intermediate
posterior. Then treat that posterior as your new prior, and update on Batch B. Write down the final
Beta(α, β) and its mean.
Part 2 — Process B, then A. Now start over from Beta(1, 1) again, but update on Batch B first,
then treat that result as your prior and update on Batch A. Write down this final Beta(α, β) and
its mean.
Part 3 — Compare. Are the two final posteriors the same? Give the shared posterior mean as a decimal (this is the numeric answer above). Then, in a sentence, say why — what property of “just add counts” guarantees the order can’t matter here.