Bayes' rule updates posterior odds by the likelihood ratio

Bayes' rule in odds form — the update step of Bayesian model reduction.

Fix a finite hypothesis space and finite evidence space, a prior law over hypotheses, and a finite likelihood kernel from hypotheses to evidence. At evidence $e$ whose predictive mass under the model is positive, the posterior odds between a favored hypothesis and a reference hypothesis is the ratio of their exact finite Bayes posteriors.

Then the odds update multiplicatively:

provided the reference prior mass $P(h_r)$ and the reference likelihood $P(e \mid h_r)$ are strictly positive — these are not decoration but the exact division premises; a zero-prior hypothesis retains zero posterior mass rather than being recoverable by conditioning, and the substrate's totalized real division returns $0$ (not $\infty$) at a zero evidence denominator.

This is precisely the comparison step of Bayesian model reduction: when a model $m_r$ is a reduction of $m_f$, the posterior model odds $P(m_f \mid y)/P(m_r\mid y)$ equal the prior model odds times the Bayes factor $P(y \mid m_f)/P(y\mid m_r)$ [Friston & Penny 2011]. The related multiplicative structure — factorized evidence ratios multiply and sequential model-odds updates agree with one update by the product evidence (catalogue topic fep-120, theorem bayesFactor_multiplicative) — is available from the same definition layer as a further target.