How does evidence change Bayesian odds?

Bayesian updating combines an initial model with evidence. A prior probability describes a hypothesis before the stated observation. Odds compare probabilities by division. Prior odds A:B of one to three correspond to probabilities one quarter for A and three quarters for B when these two hypotheses cover every possibility.

In this fictional example, exactly one of two generators produced a marker. Generator A produces blue with probability three quarters; generator B produces blue with probability one quarter. These are likelihoods: probabilities of the same evidence under different hypotheses. They are not probabilities that a hypothesis is true after observing the evidence.

A blue observation has likelihood ratio (3/4)/(1/4) = 3 in favor of A over B. Bayes’ rule in odds form multiplies prior odds by that ratio. Starting from one to three, the updated odds become three to three, equivalent to one to one. Dividing A’s weight by the sum of both weights gives posterior probability one half.

The evidence raises A’s probability without making A certain. An amber observation instead has likelihood ratio one third and moves the odds toward B. The result depends on both the likelihood ratio and the starting odds; the likelihood alone cannot replace the prior.

All probabilities here belong to the explicitly stated two-generator model. They describe a coherent update under its assumptions, not a cause of the marker or a universal measure of confidence. Using the same observation twice would count evidence twice. New observations require likelihoods that reflect what is already known.