$P(E)$ — the total probability of observing the evidence across all hypotheses — the denominator in Bayes theorem
$$P(E) = \int P(E \mid H) \cdot P(H)\, dH$$
the marginal likelihood: the probability of the data after integrating out (marginalizing over) all possible hypotheses, weighted by the prior.
as normalizing constant
the denominator ensures the posterior is a valid probability distribution:
$$P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)}$$
without $P(E)$, the right side is proportional to the posterior but not normalized — it doesn't sum to 1 over $H$. $P(E)$ is the unique constant that makes it a probability distribution. for computation, many algorithms (MCMC, variational inference) work with the unnormalized numerator $P(E \mid H) \cdot P(H)$ and avoid computing $P(E)$ directly.
as model evidence
for model comparison, $P(E \mid \mathcal{M})$ — the probability of the data under model $\mathcal{M}$ — measures how well the model fits. the Bayes factor compares two models directly:
$$\text{BF}_{12} = \frac{P(E \mid \mathcal{M}_1)}{P(E \mid \mathcal{M}_2)}$$
the Bayes factor is the update to the prior odds that the data provides. if $\text{BF}_{12} = 10$, the data is 10 times more probable under $\mathcal{M}_1$ than $\mathcal{M}_2$ — the posterior odds shift by a factor of 10 in favor of $\mathcal{M}_1$, regardless of the prior odds.
Occam's razor from the math
the marginal likelihood automatically penalizes model complexity. a complex model spreads its prior probability over many hypotheses — it can fit the data well under many of them, but the average (the marginal likelihood integral) is lower than a simpler model that concentrates its prior on the data-generating region.
formally: $P(E \mid \mathcal{M}) = \mathbb{E}_{H \sim P(H|\mathcal{M})}[P(E \mid H)]$. a model that only fits the data well for a small region of hypothesis space will have high likelihood in that region but the integral over the whole prior is penalized by the small support. parsimony emerges from marginalization without any explicit complexity penalty.
computational hardness
computing $P(E) = \int P(E \mid H) P(H)\, dH$ is analytically tractable only for conjugate prior-likelihood pairs. for everything else, approximation is necessary:
MCMC (Markov chain Monte Carlo). samples from the posterior $P(H \mid E) \propto P(E \mid H) P(H)$ without computing $P(E)$. the normalizing constant cancels in the acceptance ratio (Metropolis-Hastings). computationally expensive but asymptotically exact.
variational inference. approximates the posterior with a tractable family $q(H)$ by minimizing $D_{KL}(q \| P(H \mid E))$. this is equivalent to maximizing the ELBO (evidence lower bound): $\text{ELBO} = \mathbb{E}_q[\ln P(E \mid H)] - D_{KL}(q \| P(H))$. the ELBO is a lower bound on $\ln P(E)$.
importance sampling. estimates $P(E) = \mathbb{E}_{H \sim P(H)}[P(E \mid H)]$ by drawing samples from the prior and averaging their likelihoods. effective when the prior overlaps well with the likelihood. the same inverse-probability structure as proper scoring rules and ICBS settlement.
in cyber
in the cybergraph, the total evidence for a particle's relevance is the marginal over all neurons who linked to it:
$$P(\text{particle } q \text{ is relevant}) \propto \sum_\nu \sum_{\ell: \text{src}=p,\, \text{tgt}=q} P(\text{link} \mid \nu) \cdot P(\nu)$$
where $P(\nu)$ is the prior on that neuron (their karma) and $P(\text{link} \mid \nu)$ is the likelihood that their link is informative. cyberank approximates this marginal: it integrates out individual neuron contributions into a single relevance score for each particle.
the cyberlink market protocol's ICBS reserve ratio $q = r_{YES}/(r_{YES} + r_{NO})$ is the collective evidence for an edge: the market has integrated the likelihoods asserted by all positions into a single posterior probability. it is the practical analog of $P(E)$ computed not by integration but by market aggregation.
see Bayes theorem for the full formula. see likelihood for the numerator term. see prior and posterior for the other distributions. see KL divergence for the information-theoretic measure of how much evidence shifts belief.