Updating in sequence

Someone hands you a bent, suspicious coin. What’s the chance it lands heads? So far this tutorial’s questions have been yes-or-no — sick or healthy, blue or green, butler or someone else. This one is different: the answer is a number between 0 and 1, and your belief has to spread itself over all the candidates at once. Maybe the bias is 0.5. Maybe 0.6. Maybe 0.93. Before any flips, an honest belief treats every value as equally credible — drawn as a picture, a flat line.

Then you flip, and the line bends. A head makes high biases a little more credible and low ones a little less; the curve tilts right. Each further flip multiplies the whole curve by how well every candidate bias explains that flip — Module IV’s multiplication, performed everywhere along the axis at once. Heads push weight rightward, tails push it back, and gradually the curve gathers itself into a hump that narrows around the truth.

The hump has a name — a beta distribution — and its mechanics are almost embarrassingly simple: keep two tallies, heads plus one and tails plus one, and the curve redraws itself. (Multiplication doesn’t care about order, either: the flips could arrive in any sequence and leave you the same curve. Evidence has no choreography, only a total.) But the shape matters less than what you can read off it. The peak is your best single guess. The width is your honesty about that guess. The 90 percent credible interval printed under the chart is the range you’d give 9 : 1 odds on — a belief with error bars, which is what a belief about a quantity ought to be.

Two things are worth noticing as you flip. First, certainty runs on diminishing returns: the opening ten flips reshape the curve dramatically, while flips 90 through 100 barely tighten it. Precision gets expensive as you approach it, which is why “how sure do I actually need to be?” is a budgeting question, not a philosophical one. Second, the prior washes out. Had you started mildly convinced the coin was fair, a hundred flips would drag you to nearly the same hump — two people with different starting curves, sharing data, converge. Disagreement about priors is temporary wherever evidence keeps arriving; the permanent disagreements are the ones starved of data.

There’s a practical translation of all this that has nothing to do with coins. Any time you track a rate — a conversion rate, a defect rate, a free-throw percentage — your knowledge of it is a curve like this one, wide after ten observations, narrow after a thousand. The person who says “20 clicks, 2 sales, so 10 percent conversion” has reported the peak and hidden the width; after twenty observations that curve is still wide enough to drive a truck through.

One module remains: the same arithmetic, loose in the world.