Likelihood ratios

What you will learn

Convert prior probability to odds, apply a likelihood ratio and convert back.

Prerequisites: Bayes with a frequency table · Odds versus probability

Odds reveal the update structure

For two exhaustive states S and not-S with positive relevant probabilities, prior odds are p/(1−p). The likelihood ratio for evidence E is P(E|S)/P(E|not-S). Posterior odds equal prior odds multiplied by that ratio.

A ratio above one favors S, below one favors the alternative, and exactly one leaves the prior unchanged. This is a comparison of the same observed evidence under two states, not a comparison of P(S|E) with P(E|S).

Multiply odds, then normalize

If posterior odds are r, the posterior probability is r/(1+r). Multiplying the prior probability directly by a likelihood ratio generally gives the wrong answer and can even exceed one.

An infinite ratio caused by zero alternative likelihood needs careful support handling; if the evidence is possible under the other state and that state had positive prior, the two-state posterior can become one. A zero-over-zero ratio is undefined rather than decisive. Prefer the joint-probability calculation at boundaries.

Common mistakes

• Multiplying a probability by the likelihood ratio without renormalizing.
• Comparing likelihoods for different observed events.
• Calling an undefined boundary ratio infinite evidence.

Keep these ideas

• Evidence multiplies odds.
• Convert odds back through r/(1+r).
• Check zero-probability boundaries directly.