Part 1 — A then B. Beta(1,1) + Batch A (3 succ, 1 fail) → Beta(4, 2). Then Beta(4,2) +
Batch B (2 succ, 2 fail) → Beta(6, 4). Mean = 6/10 = 0.6.
Part 2 — B then A. Beta(1,1) + Batch B (2 succ, 2 fail) → Beta(3, 3). Then Beta(3,3) +
Batch A (3 succ, 1 fail) → Beta(6, 4). Mean = 6/10 = 0.6.
Part 3 — Identical. Both orders land on exactly Beta(6, 4), mean 0.6.
Why. The beta-binomial update is nothing but addition: α_new = α_old + successes,
β_new = β_old + failures. Whether you add “3 successes, then 2 successes” or “2 successes, then 3
successes” to α, you get the same total — 1 + 3 + 2 = 6 either way. Addition doesn’t care what
order its terms arrive in. The posterior only depends on the total counts accumulated, never on
the sequence they arrived in. This is why you can equivalently describe the process as: yesterday’s
posterior is today’s prior — every update folds all prior evidence into one distribution, and that
distribution is all the next update needs to know. It doesn’t need a log of how you got there.
Where this breaks — and it’s worth flagging now, before it bites you. Order-independence relies
on two things quietly holding: (1) each new batch’s likelihood only depends on its own counts (no
batch’s trials influence another’s outcomes), and (2) p itself isn’t secretly drifting between
batches. If Batch A came from before a website redesign and Batch B came from after, “combine the
counts” silently assumes both batches are evidence about the same fixed p — which may be false.
Order stops being irrelevant the moment the thing you’re estimating can itself change over time; at
that point you need a model that tracks when evidence arrived, not just how much of it there was.
File that away — it’s exactly the kind of assumption that’s easy to use correctly a hundred times and
then violate silently on the hundred-and-first.
Where this goes: next up is conjugacy — the beta-binomial update you’ve been hand-computing is one instance of a broader pattern (closed-form posterior updates), and there’s a whole other family (normal-normal) that works the same way for a different kind of data.