Prior, Likelihood, Posterior: The Three Ingredients

Every Bayesian update — the medical test, the courtroom verdict, the “is my flight delayed?” guess — is cooked from exactly three ingredients. Learn their names once and you stop seeing a fog of probabilities and start seeing a recipe: what you believed before, how well the new evidence fits, and the belief you’re left with after. That’s it. Three pieces, one move, infinite uses.

The intro lesson promised these three by name — prior, likelihood, posterior — and waved at the idea that you combine them. This lesson slows down and hands you each one, because almost every reasoning error in the whole course is really just grabbing the wrong ingredient or forgetting one entirely. The panicking patient didn’t fail at arithmetic; he dropped the prior. Name the ingredients clearly enough and his mistake becomes obvious before you compute a single number.

A quick promise about the math: there is an exact formula that snaps these three together, and it gets a whole lesson of its own (lesson 3, “Bayes’ Theorem”). Here we stay in plain language. Your only job today is to recognize the three pieces in the wild and to feel — in words — how they combine.

The prior: where you start

The prior, written $P(H)$, is your belief in a hypothesis before the new evidence arrives. It’s the answer to “how likely was this, going in?” The key word is before — the prior is your starting line, and you are never, ever allowed to start from a blank slate. Every update begins from a prior, even when you didn’t consciously pick one.

Where does the prior come from? Usually straight from the base rates lesson: it’s the outside-view anchor, how common the thing is across the relevant crowd. The disease’s 1-in-1,000 was a prior. “Most new restaurants close within five years” is a prior. The prior is the base rate, now wearing a Bayesian hat and waiting for evidence to push on it.

In our coin example, the prior couldn’t be cleaner. Your friend pulls a coin at random from a bag of two, so before any flip, the trick coin and the fair coin are equally likely: the prior is 50/50. $P(\text{trick}) = 0.5$. That’s the belief the flip is about to push against.

The likelihood: how well the evidence fits

Now the evidence arrives. The likelihood, written $P(E \mid H)$, asks: if the hypothesis were true, how expected would this evidence be? Read that bar $\mid$ out loud as the word “given.” $P(E \mid H)$ is “the probability of the Evidence, given the Hypothesis is true.” It runs forward: assume the hypothesis, then ask how natural the evidence looks from inside that world.

There’s a twin you can’t skip: $P(E \mid \lnot H)$, the likelihood under the hypothesis being false ($\lnot$ means “not”). “If the hypothesis were false, how expected would this same evidence be?” Evidence is only useful when these two differ — when the clue is much more expected in one world than the other. If a clue is equally likely whether or not the hypothesis is true, it tells you nothing, no matter how dramatic it feels.

Back to the coins. The evidence is one heads. How expected is heads in each world?

• If they’re holding the trick coin: heads is guaranteed. $P(\text{heads} \mid \text{trick}) = 1$.
• If they’re holding the fair coin: heads half the time. $P(\text{heads} \mid \text{fair}) = 0.5$.

So heads is twice as likely to come from the trick coin as from the fair one. That ratio — 1 versus 0.5 — is the entire informational content of the flip. It’s what will pull your 50/50 prior off-center.

The posterior: where you land

The posterior, written $P(H \mid E)$, is your updated belief in the hypothesis after taking the evidence into account. It’s the payoff — the sharper belief you get by combining the prior with the likelihood. Note it has the same “given” bar as the likelihood, but flipped: $P(H \mid E)$ is “probability of the Hypothesis, given the Evidence.” That’s the number you actually wanted all along.

How do the two ingredients combine? In one informal sentence that the whole course rests on:

Posterior is proportional to likelihood × prior.

In words: take how plausible each hypothesis was to begin with (prior), multiply by how well it predicts the evidence you saw (likelihood), and the hypothesis that scores highest on both wins the most belief. A hypothesis needs to have been reasonably likely and to explain the evidence well. (The exact equation — what turns “proportional to” into “equals” — is lesson 3. Resist the urge to compute it yet; the intuition is the point.)

Let’s land the coin. Each world gets a score of likelihood × prior, then we normalize so the two scores add to 1:

The two scores are 0.50 and 0.25, totalling 0.75. The trick coin’s share is $0.50 / 0.75 = 2/3$. So after one heads, you should believe about 67% it’s the trick coin — up from 50%, but nowhere near certain. The flip was real evidence (it moved you from 50% to 67%), just not conclusive evidence, because the fair coin can fake a heads.

And posteriors chain. Suppose your friend flips the same coin again and it’s heads a second time. Your belief is no longer 50/50 — it’s the 2/3 you just earned. That 2/3 becomes your new prior, and you update again on the second heads. Run the same machinery and you land at 4/5 (80%). Each posterior becomes the next prior; evidence stacks. This is the same “yesterday’s posterior is today’s prior” engine you’ll watch power the second medical test next lesson.

Spotting the three pieces in the wild

The coins are clean on purpose, but the ingredients are everywhere. Train the reflex on an ordinary situation: a friend is 20 minutes late. Are they stuck in traffic, or did they just forget?

• Prior $P(H)$ — going in, how often does this friend get genuinely stuck in traffic versus flake? If they’re reliable but drive a congested route, maybe you’d have said 70% “traffic, not flaking” before tonight. That’s your starting belief.
• Likelihood $P(E \mid H)$ — the evidence is “20 minutes late, no text.” How expected is that if they’re stuck in traffic? Pretty expected — driving, can’t text safely. And the twin $P(E \mid \lnot H)$: how expected is the same silence if they forgot entirely? Also fairly expected — someone who forgot also sends no text. The clue barely distinguishes the two worlds, so it’s weak evidence and shouldn’t move you much.
• Posterior $P(H \mid E)$ — combine them: a high prior on “traffic” plus weak, non-distinguishing evidence leaves you roughly where you started — still leaning “traffic,” only slightly dented. Now imagine the evidence changes to “they texted I’m so sorry, totally spaced on this.” That clue is far more expected if they forgot than if they’re driving — a strong likelihood ratio — and it should swing your posterior hard toward “flaked.”

Same recipe, no coins, no medicine: name the prior, weigh how differently the evidence fits each world, combine into a posterior. Once you can do this on a late friend, you can do it on a job offer, a diagnosis, or a headline.

When to reach for it

Reach for these three names the instant new evidence lands on any belief you hold — a test result, an interview, a late friend, a market rumor, a scary headline. The trigger is the feeling “this changes things.” Before you let it change anything, run the checklist: What was my prior? (say it out loud), How much more expected is this evidence if my hypothesis is true than if it’s false? (the likelihood gap), and So where does that leave me? (the posterior). Naming the pieces costs five seconds and instantly exposes the two classic blunders — dropping the prior, and mistaking the forward likelihood for the backward answer.

Where this goes next

You now hold the whole vocabulary the rest of the course speaks: prior (where you start, usually the base rate), likelihood (how well the evidence fits each world — forward, given the hypothesis, and emphatically not the answer), and posterior (where you land, proportional to likelihood × prior). You watched all three combine on a coin, chain across two flips, and reappear on a late friend with no math at all.

Next, in “The Medical Test,” we take these three ingredients to the most famous Bayesian puzzle of all — a rare disease, a 99%-accurate test, a positive result — and find a true chance of only about 9%. You’ll build it with a natural-frequency tree that makes the prior impossible to forget, drive an interactive posterior calculator, and watch the inverse fallacy (likelihood mistaken for posterior) get dismantled for good. The vocabulary you just learned is exactly what turns that “trick” into something obvious.