Houjun Liu

degrees of belief

degrees of belief help us quantify how much we believe some event \(A\) is more/less plausible than some event \(B\).

Let us take two statements:

  • \(A\) Taylor gets Nobel Prize in Literature
  • \(B\) Han shot first

For instance, if we want to express “I think its more likely that Taylor gets the prize than Han shot first”:

\begin{equation} A \succ B \end{equation}

axioms of degrees of belief

universal comparability

for two statements \(A, B\), only three states can exist:

  • \(A \succ B\) (A more likely)
  • \(A \prec B\) (B more likely)
  • \(A \sim B\) (equally likely)


if \(A \succeq B\) and \(B \succeq C\), then \(A \succeq C\)

language of probability

using this framework, we can then describe the events in terms of probability

  • \(P(A) > P(B) \Leftrightarrow A \succ B\)
  • \(P(A) = P(B) \Leftrightarrow A \sim B\)

See also axiom of probability