Ugg smell rain before rain come

Long ago tribe argue two way about the word "likely." Ugg settle it by fire.

Two tribe, two idea of "likely"

Old tribe — the frequentist — say likely mean: do same hunt many many time, count how often thing happen. To them the true thing (how many mammoth hide in valley) is one fixed number, buried, never move. Ugg only guess it from long counting.

New tribe — the Bayesian — say likely mean degree of belief: how much Ugg trust an idea, given all Ugg know right now. And belief is not stone. Belief MOVE when Ugg see fresh footprint. This idea older! Grandfather Bayes carve it in 1763. Ugg like it — it how brain truly work.

The one magic rock — Bayes rule

All of it is ONE equation. Ugg call it magic rock. This is Bayes' theorem:

$P(H\mid D)=\frac{P(D\mid H)\cdot P(H)}{P(D)}$

In caveman: Posterior = (Likelihood × Prior) ÷ Evidence. Four word. Learn them:

• Prior $P(H)$ — what Ugg believe BEFORE footprint. Old hunt memory, elder wisdom.
• Likelihood $P(D\mid H)$ — the model: IF this idea true, how likely Ugg see THIS footprint?
• Posterior $P(H\mid D)$ — new belief AFTER footprint. The update.
• Evidence $P(D)$ — total chance of footprint over every idea. Only a number to make belief add to one whole.

Big secret: Evidence no care which H, so it no change the SHAPE of belief, only shrink to fit. So Ugg mostly use:

$P(H\mid D)\propto P(D\mid H)\cdot P(H)$

"Posterior grow from Prior times Likelihood." Ugg carve on cave wall. That the whole framework. Evidence itself come from adding every world: $P(D)={\sum }_{H}P(D\mid H)\cdot P(H)$.

Umbrella tell Ugg about rain

Maya walk up carry umbrella. Rain today? Ugg start with prior: in this land in spring, rain only 15% of day. $P(\text{rain})=0.15$.

Maya forgetful. When rain, she carry umbrella only 30%; when dry, she carry 5% "just case." Those the likelihoods. Turn magic rock:

$P(\text{rain}\mid \text{umb})=\frac{0.30\cdot 0.15}{0.30\cdot 0.15+0.05\cdot 0.85}=\frac{0.045}{0.0875}\approx 0.514$

Umbrella push belief from 15% up to 51%! Big jump — but NOT sure-thing, because Maya not always smart. That is Bayes: footprint move belief, no slam it to certain.

Same trick guard Ugg on sick-test. Sick rare, 1% of tribe. Test catch 90% of sick, but false-alarm 7%. A "positive" test give only $\frac{0.90\cdot 0.01}{0.90\cdot 0.01+0.07\cdot 0.99}\approx 11.5\%$ true-sick. When prior tiny, one loud test still weak. Ugg no panic.

Two idea fight — Bayes Factor

Old tribe throw p-value. New tribe fight two idea head to head with Bayes Factor:

${\text{BF}}_{10}=\frac{P(D\mid H_1)}{P(D\mid H_0)}$

It ask: which idea predict the footprint better? BF₁₀ = 10 mean footprint 10× more likely under H₁. BF₁₀ = 0.1 (same as BF₀₁ = 10) mean 10× more likely under H₀ — footprint actively CHEER for the null. p-value can never do that; it only "fail to reject." BF is smooth number, no magic line.

Read strength on the Jeffreys scale: 1–3 tiny (anecdotal), 3–10 moderate, 10–30 strong, 30–100 very strong, over 100 decisive. Same scale on BF₀₁ for evidence toward null.

And BF plug into the great update law:

$\frac{P(H_1\mid D)}{P(H_0\mid D)}={\text{BF}}_{10}\cdot \frac{P(H_1)}{P(H_0)}$

"Posterior odds = Prior odds × Bayes Factor." Belief-odds after footprint = belief-odds before × footprint-ratio. Ugg report only BF because it data-only — every reader agree on it, then each bring own prior. Flat prior? Then posterior odds = BF straight. Today posterior become tomorrow prior; evidence pile up by multiply.

Why old tribe stumble in the dark

Ugg warn about old-tribe traps:

• p-value twist tongue: it means P(footprint this wild | H₀ true), NOT P(idea true). Even elder misread it.
• A credible interval (Bayesian) is the "95% chance true number sit in here" that everyone WISH they had. Old confidence interval is NOT that — it only mean "the procedure trap the number 95% of many hunts."
• Optional stopping — the big one. Old tribe MUST fix hunt-size first. If Ugg peek at p after every subject and stop when p < 0.05, then even with NO real effect Ugg cross the line by luck sooner or later. One extra peek push false-alarm 5% → 8%; endless peek → almost sure fool. Bayes Factor no care — the likelihood principle say all footprint-info about the thing lives in the likelihood; the stop-rule is fact about the hunter, not the thing. So peek all Ugg want, stop when evidence enough. This the strongest reason to join new tribe.
• α = 0.05 is a made-up line. BF let Ugg pick the evidence strength that matter for OWN hunt. And only BF can show evidence FOR the null; a not-significant p just mean "we no catch it," silent.

Handy stone tools

• Conjugate prior — some prior+likelihood pair give posterior in the SAME family, so math easy. Classic: Beta + binomial → Beta. Start $\theta \sim \text{Beta}(\alpha ,\beta )$, see k win in n try, posterior $=\text{Beta}(\alpha +k,\beta +n-k)$. Flip coin 10 time, 8 head, start flat Beta(1,1) → Beta(9,3), mean 9/12 = 0.75.
• MCMC (Markov Chain Monte Carlo) — when posterior too twisty for clean formula, computer sample many stone from it. Tools: Stan, JAGS, PyMC. Need only likelihood + prior.
• R BayesFactor pack: ttestBF, anovaBF, regressionBF, contingencyTableBF — Bayesian twin of every old test, default weak Cauchy prior.
• Lindley's paradox — with HUGE n, p scream "reject!" while BF calm say "look just like the null." No contradiction — different question. Report both p AND BF for full story.

Ugg warn: no do this!

• BF is NOT the posterior probability. BF is a likelihood RATIO; posterior come from BF × prior odds.
• Prior not "bad because personal." Prior make Ugg guess OPEN and arguable — strength, not weak. Old tribe hide their prior.
• BF = 0.5 is NOT "fail." It means BF₀₁ = 2, a mild lean to H₀. Read every BF as ratio, not yes/no.
• A non-significant p is NOT proof of the null. Want to show absence? Use BF₀₁.
• Likelihood IS the model. Bayes no rescue a bad model.

Ugg remember

• One rock rule all: posterior ∝ prior × likelihood. Every analysis is a belief update; P(D) only normalise.
• Bayes Factor $=P(D\mid H_1)/P(D\mid H_0)$ — smooth evidence, can favor the null, read on the Jeffreys scale (3 moderate, 10 strong, 100 decisive).
• Posterior odds = prior odds × BF. Today posterior become tomorrow prior.
• Bayes survive peeking (optional stopping); old tribe does not. The biggest practical win.
• Credible interval ≠ confidence interval. Beta+binomial → Beta for easy math. Report both p and BF.