Guided tour puzzle protocol

A credal set is a set of probability distributions or, equivalently, a set of probability measures. A credal set is often assumed or constructed to be a closed convex set. It is intended to express uncertainty or doubt about the probability model that should be used, or to convey the beliefs of a Bayesian agent about the possible states of the world.

Let $X$ denote a categorical variable, $P(X)$ a probability mass function over $X$, and $K(X)$ a credal set over $X$. If $K(X)$ is convex, the credal set can be equivalently described by its extreme points $\mathrm{e}\mathrm{x}\mathrm{t}[K(X)]$. The expectation for a function $f$ of $X$ with respect to the credal set $K(X)$ can be characterised only by its lower and upper bounds. For the lower bound,

$\underset{\_}{E}[f]={\min }_{P(X)\in K(X)}\sum _{x}f(x)P(x).$

Notably, such an inference problem can be equivalently obtained by considering only the extreme points of the credal set.Template:Cn

It is easy to see that a credal set over a Boolean variable cannot have more than two vertices, while no bounds can be provided for credal sets over variables with three or more values.Template:Cn

See also

• imprecise probability
• Dempster–Shafer theory
• probability box
• robust Bayes analysis
• upper and lower probabilities

Further reading

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