Bayesian Odds, Likelihood Ratios, and Base-Rate Traps

A test catches 99% of a disease and falsely flags only 5% of healthy people. You test positive. Most people guess you almost certainly have it. The actual answer is about 16.7%, because the disease is rare. The odds form of Bayes makes that result one quick multiplication.

Standard-form Bayes (covered in 1g) divides by the total probability of the evidence. The odds form skips that denominator. You compare the two hypotheses head to head, so the shared normalizing constant cancels. What is left is a clean product.

$$\underset{\text{posterior odds}}{\underbrace{\frac{P(H\mid E)}{P(H^c\mid E)}}}=\underset{\text{prior odds}}{\underbrace{\frac{P(H)}{P(H^c)}}}\times \underset{\text{likelihood ratio}}{\underbrace{\frac{P(E\mid H)}{P(E\mid H^c)}}}$$

The likelihood ratio measures how much more the evidence favors $H$ over its complement. An $LR$ above 1 pushes the odds up; an $LR$ below 1 pulls them down; an $LR$ of exactly 1 means the evidence is useless.

In testing language, $H$ is "has the disease" and $E$ is "tests positive." Then $P(E\mid H)$ is the sensitivity (the true-positive rate) and $P(E\mid H^c)$ is the false-positive rate, which equals $1$...

Common mistakes

• Reading the posterior off the test accuracy. The base-rate trap. A 99% sensitive, 95% specific test on a 1% disease leaves only 16.7% after one positive, not 99%. Trap value: 0.99 (or 0.95).
• Using $1-\text{sensitivity}$ as the false-positive rate. The false-positive rate is $1-\text{specificity}$, not $1-\text{sensitivity}$. Here $1-0.99=0.01$ is the false-NEGATIVE rate. The $LR$ denominator is $1-0.95=0.05$.
• Confusing odds with probability in the final step. Posterior odds of 0.2 are NOT a probability of 0.2. Convert: $p=0.2/1.2=0.1667$. Reporting 0.2 is the trap; it overstates the answer by 20%.

Bottom line

• Odds form of Bayes: posterior odds = prior odds $\times$ likelihood ratio. One multiplication updates a belief, with no total-probability denominator.
• Likelihood ratio: $LR=P(E\mid H)/P(E\mid H^c)$. For a positive test, that is sensitivity over $1-\text{specificity}$.
• Negative result: carries its own $LR=(1-\text{sensitivity})/\text{specificity}$, usually well below 1, dragging the posterior below the base rate.
• Odds to probability: if the odds are $o$, then $p=o/(1+o)$. Probability to odds: $o=p/(1-p)$. Odds of $1/99$ are a probability of $0.01$.

Exam shortcut

Spot the base-rate trap: when a stem gives sensitivity, specificity, and a small prevalence, the answer choice equal to the sensitivity is the bait. Always run the multiplication, then convert odds to probability. Memory aid: "prior times ratio, then $o$ over one-plus-$o$." For a positive test, the ratio is "sens over one-minus-spec"; for a negative test, "one-minus-sens over spec." Two independent positives means square the ratio; one positive plus one...

The full lesson (about 2,447 words, 16 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1a
• 1b
• 1c
• 1d
• 1e
• 1f
• 1g

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