Priors and Posteriors

Bob texted Tau a screenshot of the shaded curve on his Results page and asked what the hell it was. Tau told him to come to the taquería.

Bob slid into the booth. The waiter was already approaching with the combo menu. Tau waved him off without looking up.

Bob: OK so there's this bell-curve thing on my Results page. Wider on some players, narrower on others. Shaded area, tick marks, the whole thing. What is it?

Uncle Tau: It's a posterior.

Bob: Right. Cool. What's a posterior?

Uncle Tau: A belief.

Bob: A belief about what?

Uncle Tau: About you. About your ROI in whatever buy-in and format that page is filtering on. The curve is the app's best belief about where your real ROI lives, given your data. The centre is its best guess. The width is its admission of ignorance.

Bob: OK but "belief" is weird. It's either my ROI or it isn't.

Uncle Tau: Sure, there's a true number. You just don't know what it is, and neither does the app. What the app has is your results and a prior. Run those through Bayes and you get a posterior — a distribution over the candidates for your real ROI, weighted by how well each one fits your sample.

Bob: Hold on. Prior?

Uncle Tau: What the app believed about your ROI before it looked at a single hand of yours.

Bob: You can have a belief before the data?

Uncle Tau: You can't not have a belief before the data. If I told you a random person was sitting in a €55 turbo right now, what would you guess their ROI is?

Bob: I don't know. Zero? Maybe slightly negative after rake.

Uncle Tau: That's a prior. You just used it. You looked at "random person," "€55 turbo," "online," "has to beat rake" — and you landed on "probably near zero." That's your belief before seeing any of their data. Everybody does this. The app just does it with more structure.

Bob: So where does its prior come from?

Uncle Tau: From the regs who already play that format. It looks at the field — buy-in, speed, site — and builds a distribution over "people who sit in games like this." That's the prior. Then it sees your results and updates.

Bob: Updates how?

Uncle Tau: Bayes. Posterior is proportional to prior times likelihood. The likelihood is "how well does each candidate ROI explain the results I actually saw." Multiply, renormalise, done. The curve on your screen is what comes out.

Bob: And when I have more tournaments?

Uncle Tau: The likelihood gets sharper, the prior gets outweighed, the posterior narrows around what your data is saying. Run 500 tournaments and the app barely listens to the prior anymore — your results speak for themselves. Run 10 and the app basically quotes the prior back at you with a tiny nudge. In between it's a compromise, weighted by how informative your sample is.

Bob: So the width of the curve is telling me how much the app is actually learning from me versus just assuming.

Uncle Tau: That's exactly what it's telling you. And it's the most important thing on the page. People stare at the mean and ignore the width. A +15% ROI with the curve going from −5% to +35% is not the same number as +15% with the curve from +12% to +18%. Same centre. Totally different information.

Bob: I've definitely shown people +15% and let them assume it was the tight version.

Uncle Tau: Everybody has. The app stops you from lying to yourself by drawing the width.

Where you see this in the app

Bob: OK so this curve is only on the Results tab?

Uncle Tau: Every page the app gives you a range instead of a single number, that's a posterior talking.

Results tab, obviously — that's where you see the curve directly. Strategy and Stable Strategy show it implicitly: when SALSA runs, it doesn't pick one ROI and simulate. It samples from your posterior a few thousand times and simulates each sample. That's why every output has a range. The range width is telling you how uncertain the underlying posterior is.

Package Builder uses the posterior to price action. Narrow posterior, tight markup band. Wide posterior, wider band — because the seller and the buyer are both negotiating about a belief, not a fact.

Scout does it for other players. If you're looking up a reg with three thousand tournaments, the confidence band on their ROI is going to be tight. If you're looking up a kid with sixty tournaments, it's going to be hilariously wide. The app isn't being lazy. It's being honest.

Bob: So the whole app is built on this.

Uncle Tau: Every number you trust in this app came out of a prior, some data, and a Bayes update. If you don't know what those three words mean, you can't read anything.

The one-line math, for people who want it

Bob: Give me the formula so I can say I saw it.

Uncle Tau:

$$\text{posterior} \propto \text{prior} \times \text{likelihood}$$

That's it. "Belief after data" equals "belief before data" times "how well the data fits each hypothesis," then renormalise so the total probability is one. Thomas Bayes, 1763, posthumously. Literally older than the United States.

Bob: That's the whole thing?

Uncle Tau: That's the whole thing. The reason this app exists is not the formula — the formula is in every first-year stats book. The reason this app exists is that nobody else in poker is actually running it on your tournaments.

Bob: So what do I actually do with this?

Uncle Tau: Two things. One: when you look at any output in this app, look at the width before you look at the centre. The width tells you how much the app is echoing its prior versus learning from your data. Two: stop quoting point estimates to strangers. ROI without a posterior width is just a vibe.

Bob: Got it. Thanks, Uncle Tau.

Uncle Tau: Go estimate your shapes, kid. Next time we'll talk about where the prior actually comes from — why a kid with 20 tournaments and a kid with 2000 get treated the way they do. It's called shrinkage. It's what's saving you from yourself every time you look at your own results.

What's next

Shrinkage and Empirical Bayes — where the prior comes from, why small samples get pulled toward it, and why this is the single most important thing keeping your ROI estimate honest.
Why Bayesian beats point estimates — what actually goes wrong when you just take the mean of your results and call it a day.
Reading the Strategy tab — applying priors and posteriors to a real output you'll see tomorrow at the tables.

Priors and Posteriors

Where you see this in the app

The one-line math, for people who want it

What's next

Further reading