# Info-gap’s uncertainty model

Info-gap’s uncertainty model — designed to give expression to the uncertainty conditions that the theory deals with — is precisely the uncertainty model underlying radius of stability models, except that info-gap’s terminology is slightly different. In other words, info-gap’s uncertainty model consists of the following objects that are associated with a parameter of interest, call it $u$:

• An uncertainty space, $\mathscr{U}$, that is a set consisting of all the possible/plausible values of $u$.
• A point estimate of the true value of $u$, call it $\tilde{u}$.

As in the case of radius of stability models, info-gap decision theory imposes a neighborhood structure on $\mathscr{U}$. That is, a fundamental assumption of this theory is that there is a family of nested sets $N(\rho,\tilde{u}),\rho\ge 0,$ centered at $\tilde{u}$ where $N(\rho,\tilde{u})\subseteq \mathscr{U}$ denotes a neighborhood of size (radius) $\rho$ around $\tilde{u}$. These neighborhoods are assumed to have the following two basic properties:

• $N(0,\tilde{u}) = \{\tilde{u}\}$ (contraction)
• $N(\rho,\tilde{u}) \subseteq N(\rho + \varepsilon,\tilde{u}), \forall \rho,\varepsilon\ge 0$ (nesting)

The parameter $\rho$ representing the size (radius) of the neighborhoods is called the horizon of uncertainty.

The severity of the uncertainty under consideration is manifested in these three characteristics:

• The uncertainty space $\mathscr{U}$ can be vast (e.g. unbounded)
• The point estimate $\tilde{u}$ is poor and can be substantially wrong.
• The uncertainty is likelihood-free.

The last means, among other things, that there are no grounds to assume that the true value of $u$ is more/less likely to be in the neighborhood of any particular value of $u\in \mathscr{U}$. Specifically, there are no grounds to assume that the true value of $u$ is more/less likely to be in the neighborhood of the point estimate $\tilde{u}$ than in the neighborhood of any other $u\in \mathscr{U}$.

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