$P(E \mid H)$ read as a function of $H$ with evidence $E$ fixed — how well each hypothesis explains the observed data
$$\mathcal{L}(H) = P(E \mid H)$$
same formula as Bayes theorem, different reading. when you fix the data and vary the hypothesis, it is no longer a probability distribution over $E$ — it is a scoring function over hypotheses. the likelihood does not integrate to 1 over $H$.
what it measures
the likelihood answers: if hypothesis $H$ were true, how probable is the data we actually observed? high likelihood = the hypothesis makes the observations unsurprising. low likelihood = the observations would be rare under this hypothesis.
the likelihood ratio $\mathcal{L}(H_1) / \mathcal{L}(H_2)$ compares two hypotheses head-to-head: how much more does the data support $H_1$ than $H_2$? this ratio is independent of the prior — it is the pure voice of the data.
log-likelihood
for independent observations $E = \{e_1, \ldots, e_n\}$:
$$\ell(H) = \ln \mathcal{L}(H) = \sum_{i=1}^n \ln P(e_i \mid H)$$
products become sums. numerically: probabilities multiply toward underflow; log-probabilities sum safely. theoretically: the log-likelihood is the natural bridge to entropy and KL divergence — the expected log-likelihood under the true distribution is $-H(P_\text{true}, P_H)$, the negative cross-entropy.
maximum likelihood estimation
MLE selects the hypothesis that maximizes the likelihood:
$$\hat{H}_{\text{MLE}} = \arg\max_H \mathcal{L}(H)$$
MLE is Bayesian inference with a flat prior: when all hypotheses are equally probable a priori, the posterior is proportional to the likelihood, so maximizing the posterior = maximizing the likelihood. MLE breaks from Bayesian inference only when the prior is non-uniform.
MLE is consistent (converges to the true $H$ as $n \to \infty$) and efficient (achieves the Cramér-Rao lower bound asymptotically). it is the standard for frequentist parameter estimation.
the likelihood principle
all evidence in the data about $H$ is contained in $\mathcal{L}(H)$. two datasets with proportional likelihood functions carry identical evidence about $H$, regardless of how they were collected, how the experiment was designed, or what data could have been observed but wasn't.
the stopping rule doesn't matter. an experiment stopped at $n=100$ because 100 flips were planned, and the same experiment stopped because the 100th heads appeared, give proportional likelihoods and therefore identical evidence — even though their sampling distributions differ. this is a fundamental departure from frequentist inference (where $p$-values depend on the stopping rule).
in Bayes theorem
in Bayes theorem, the likelihood is the bridge between prior and posterior:
$$\underbrace{P(H \mid E)}_{\text{posterior}} \propto \underbrace{P(E \mid H)}_{\text{likelihood}} \cdot \underbrace{P(H)}_{\text{prior}}$$
the likelihood re-weights the prior: hypotheses consistent with the data get upweighted, inconsistent ones get downweighted. the evidence (denominator) normalizes the result.
in cyber
every cyberlink created by a neuron is an implicit likelihood assertion: "if this connection is meaningful, I would expect the evidence I have seen." the stake $(τ, a)$ is the magnitude of the likelihood claim — how strongly the neuron asserts that the data (their knowledge, context, observation) supports this connection.
karma tracks the track record of a neuron's likelihood estimates over time: did their assertions prove correct? a neuron with high karma has a track record of high likelihoods for connections the market later validated.
the Bayesian Truth Serum scoring formula contains the likelihood ratio implicitly: the information gain term $D_{KL}(p_i \| \bar{m}_{-i}) - D_{KL}(p_i \| \bar{p}_{-i})$ measures how much the agent's belief departs from the crowd's prediction, weighted by the agent's own probability assessment — a log-likelihood ratio over the crowd's prior.
see Bayes theorem for the update rule. see evidence for the normalizing denominator. see prior and posterior for the other terms. see KL divergence for the information-theoretic connection.