a probability distribution over hypotheses held by an agent — quantified uncertainty about what is true
what a belief is
a belief is not a binary fact — it is a degree. to believe a hypothesis H is to assign it a probability $P(H) \in [0,1]$. this scalar encodes the agent's uncertainty: $P(H) = 0$ means certainty it is false, $P(H) = 1$ means certainty it is true, $P(H) = 0.7$ means 70% confident.
over multiple hypotheses, belief is a distribution: $\sum_i P(H_i) = 1$. the full distribution captures not just which hypothesis the agent favors but how spread the uncertainty is. a flat distribution expresses total ignorance. a peaked distribution near $H_k$ expresses near-certainty.
coherence
to be a valid belief, a probability assignment must satisfy Kolmogorov's axioms: non-negativity, normalization to 1, and additivity for disjoint events.
an incoherent belief system — one that violates these axioms — can be exploited by a Dutch book: a set of bets that the agent accepts as individually fair but that collectively guarantee a loss regardless of outcomes. coherence is the minimum rationality requirement for beliefs held under uncertainty.
the two interpretations of probability
frequentist. probability is a long-run frequency — the limit of the fraction of times an event occurs as the number of trials grows. $P(H)$ only makes sense for repeatable events. there is no frequentist $P(\text{"the Riemann hypothesis is true"})$ — it either is or it isn't.
Bayesian. probability is a degree of belief — a number encoding the agent's current epistemic state. $P(H)$ applies to any proposition, including unique events, unverifiable claims, and normative judgments. different agents can rationally hold different beliefs about the same proposition if they have different background knowledge.
the Bayesian interpretation is required for Bayesian Truth Serum, prediction markets, and the cyberlink market protocol — all of which involve beliefs about non-repeatable, non-resolvable, or subjective propositions.
belief update: Bayes theorem
beliefs are updated by Bayes theorem:
$$P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)}$$
the agent starts with a prior $P(H)$ and updates it to the posterior $P(H \mid E)$ upon observing evidence $E$. the update is optimal in the sense that it minimizes expected KL divergence between the agent's belief and the true distribution.
rational agents with the same prior who observe the same evidence reach the same posterior. agents with different priors converge over time as evidence accumulates — the Bernstein-von Mises theorem: posteriors from different priors merge when data is abundant.
belief and stake
in prediction markets and Bayesian Truth Serum, belief is made concrete by stake. an agent who claims $P(H) = 0.9$ but refuses to stake on H at 80:20 odds reveals their stated belief is not their actual belief.
stake is the mechanism that enforces honesty: expressing a belief at odds with your actual probability distribution costs expected money. the cyberlink in cyber is the unit of staked belief — creating a link with stake $(τ, a)$ is an economic assertion that the connection is meaningful. the valence $v$ is the meta-belief: the agent's prediction of how the collective will assess the link.
first-order vs second-order belief
first-order belief: $P(H)$ — what the agent thinks about the world.
second-order belief: $P_i(\text{crowd believes } H)$ — what the agent thinks about what others believe. this is the meta-prediction $m_i$ in Bayesian Truth Serum. the gap between first-order and second-order belief is where private knowledge lives: if you genuinely know something the crowd hasn't priced, your first-order belief exceeds your second-order belief (you think fewer others know than actually will).
Bayesian Truth Serum extracts private knowledge by rewarding agents whose first-order beliefs exceed their second-order predictions — beliefs that are more popular than they predicted they would be.
in cyber
every cyberlink is a staked belief. the cybergraph is the network of all beliefs ever asserted by all neurons, weighted by stake and validated by karma history.
prior: karma encodes the system's prior on how much to trust a neuron's new assertion. posterior: cyberank is the marginal posterior probability of a particle's relevance. syntropy: the aggregate information gain — how much collective beliefs sharpened from all assertions in an epoch.
the cyberlink market protocol converts beliefs into market positions. the inversely coupled bonding surface prices the collective belief about each edge. the Bayesian Truth Serum scores agents on how much their individual beliefs contributed to sharpening the collective.
see Bayes theorem for the update rule. see prior for the starting belief. see posterior for the updated belief. see Bayesian Truth Serum for honest belief elicitation. see prediction markets for belief markets.