PhyslibAlpha.ProbabilisticTheory.Effect.Convex

The set of effects in a type $E$, denoted as `Effect E` and defined as the order interval $[0, 1] = \{A \in E \mid 0 \le A \le 1\}$, is a convex set over the real numbers $\mathbb{R}$.

For a type $E$ with a partial order, a zero element $0$, and a unit element $1$, let $e$ and $f$ be **effects** (elements in the order interval $[0, 1] \subseteq E$). Given a value $t$ in the unit interval $[0, 1]$, this function returns the **mixture** of $e$ and $f$, which is the effect defined by the convex combination $t e + (1 - t) f$. This represents randomizing between the two effects with probability $t$ of testing the first.

For any two effects $e, f$ in the interval $[0, 1] \subseteq E$ and a probability $t$ in the unit interval $[0, 1] \subseteq \mathbb{R}$, the mixture of $e$ and $f$ with parameter $t$ is equal to the convex combination: $$\text{mix}(e, f, t) = t \cdot e + (1 - t) \cdot f$$ where the equality holds for the corresponding elements in the underlying type $E$.