The odds form

Idea: Keep a running score

Odds turn each new clue into $1$ multiplication instead of a new crowd.

• Prior odds come from the counts
• Convert to a probability only at the end

Formula: The odds form of Bayes

$O(H\mid E)=O(H)\times LR$

Posterior odds are the prior odds times the likelihood ratio of the evidence.

Formula: The likelihood ratio

$LR=\frac{P(E\mid H)}{P(E\mid \neg H)}$

Above 1 it supports H, below 1 it counts against H, and exactly 1 says nothing.

Example: 3 heads from a bag of coins

$7$ fair coins and $1$ double-headed. Draw one and flip it: heads $3$ times. Chance it is double-headed?

1. Prior odds $\frac{1}{7}$; each head has ratio $\frac{1}{\frac{1}{2}}=2$
2. After $3$ heads: $\frac{1}{7}\times 2^3=\frac{8}{7}$
3. Probability: $\frac{8}{8+7}=\frac{8}{15}$

Common mistake: Multiplying probabilities

$\frac{1}{8}\times 2^3=1$ would make the coin certain after $3$ heads.

Instead Multiply odds, convert at the end: odds of $a$ to $b$ give $\frac{a}{a+b}$.

Idea: Several clues

Independent clues multiply in any order, given H and given ¬H.

• On: $\frac{0.9}{0.3}=3$; off: $\frac{0.1}{0.7}=\frac{1}{7}$
• Prior $\frac{1}{2}$, one on, one off: $\frac{1}{2}\times 3\times \frac{1}{7}=\frac{3}{14}$
• Signals built on the same data are one clue

Quick check: The next flip

After those $3$ heads, what is the chance that the next flip lands heads?

Idea: Strong evidence adds up fast

Repeated ratios compound.

• $1$ double-headed coin among $1000$: prior odds $\frac{1}{999}$
• $10$ heads: $\frac{2^{10}}{999}=\frac{1024}{999}$, just past even
• A prior near $1$ in $2^n$ needs about n heads

Idea: Run it out loud

State the prior odds, name each clue's ratio, and keep the running product aloud.

• Say which independence you assume
• Convert to a probability only at the end