Credal network

{{Short description|Probabilistic graphical models based on imprecise probability}}

{{confusing|date=November 2022}}

Credal networks are probabilistic graphical models based on imprecise probability.{{Cite journal |last1=Ding |first1=Song |last2=Pan |first2=Xing |last3=Zuo |first3=Dujun |last4=Zhang |first4=Wenjin |last5=Sun |first5=Liuwang |date=2022-02-01 |title=Uncertainty analysis of accident causality model using Credal Network with IDM method: A case study of hazardous material road transportation accidents |url=https://www.sciencedirect.com/science/article/pii/S0957582021006959 |journal=Process Safety and Environmental Protection |volume=158 |pages=461–473 |doi=10.1016/j.psep.2021.12.021 |bibcode=2022PSEP..158..461D |issn=0957-5820}} Credal networks can be regarded as an extension of Bayesian networks, where credal sets replace probability mass functions in the specification of the local models for the network variables given their parents. As a Bayesian network defines a joint probability mass function over its variables, a credal network defines a joint credal set. The way this credal set is defined depends on the particular notion of independence for imprecise probability adopted. Most of the research on credal networks focused on the case of strong independence. Given strong independence the joint credal set associated to a credal network is called its strong extension. Let $(X\_1,\backslash ldots,X\_n)$ denote a collection of categorical variables and $G$. If $K(X\_i\backslash mid\backslash pi\_i)$ is, for each $i=1,\backslash ldots,n$, a conditional credal set over $X\_i$, then the strong extension of a credal network is defined as follows:

: $K(X\_1,\backslash ldots,X\_n)=\backslash mathrm\{CH\}\backslash \{\; P(X\_1,\backslash ldots,X\_n):\; P(x\_1,\backslash ldots,x\_n)=\backslash prod\_\{i=1\}^n\; P(x\_i\backslash mid\backslash pi\_i)\; ,\; P(X\_i\backslash mid\backslash pi\_i)\; \backslash in\; K(X\_i\backslash mid\backslash pi\_i)\; \backslash \}$

where $\backslash mathrm\{CH\}$ denote the convex hull.