Hyperthesis LogoHyperthesis

Belief Distribution

A belief distribution is a set of subjective probabilities, assigned by the user, covering all possible outcomes for an event.

Examples

Binary Outcome
Two possible outcomes

"I believe Team A wins = 70%, Team B wins = 30%"

P(A)+P(B)=70%+30%=100%P(A) + P(B) = 70\% + 30\% = 100\%
Multi-Outcome
Three or more possible outcomes

Soccer match: "Team A wins = 50%, Team B wins = 30%, Draw = 20%"

P(A)+P(B)+P(Draw)=50%+30%+20%=100%P(A) + P(B) + P(\text{Draw}) = 50\% + 30\% + 20\% = 100\%
Zero Probability
When you're certain an outcome won't occur

"Team A = 0%, Team B = 80%, Draw = 20%"

Mathematical Properties

For nn possible outcomes, a belief distribution is a vector:

B⃗=[p1,p2,...,pn]\vec{B} = [p_1, p_2, ..., p_n]

Where:

  • pi≥0p_i \geq 0 for all ii (non-negative)
  • ∑i=1npi=1\sum_{i=1}^{n} p_i = 1 (sums to 100%)

Each pip_i represents your subjective probability that outcome ii will occur.

Why Belief Distributions Matter

Express Uncertainty

Real-world beliefs are rarely binary. You might think Team A is likely to win (70%) but still acknowledge Team B has a chance (30%). Belief distributions capture this nuance.

Risk Management

Your exposure scales with your confidence. If you're only 30% confident in an outcome, you'll only risk a proportional amount of your bet on that outcome.

Value Discovery

By expressing beliefs across all outcomes, the system can find value wherever market prices diverge from your beliefs, not just on your "favorite" outcome.

One-Sided Conditional Bets

You can place a 100% belief on a single outcome, creating a one-sided conditional bet:

A 100% bet on Team A:

B⃗=[100%,0%]\vec{B} = [100\%, 0\%]

Think of it as a hybrid between a traditional market order (immediate execution) and a limit order (price protection), but with dynamic optimization.