Introduction

Abstract

The first think to do is to explain what is our “new approach” and how it fits into the area of uncertain probabilities. We first consider a very simple example using interval probabilities. Let X = {x ₁, x ₂, x ₃} be a finite set and let P be a probability function defined on all subsets of X with P({x _i }) = a _i , 1 ≤ i ≤ 3, 0 < a _i < 1 all i and $\sum\nolimits_{{i = 1}}^{3} {{a_{i}} = 1}$. X together with P is a discrete (finite) probability distribution. In practice all the a _i values must be known exactly. Many times these values are estimated, or they are provided by experts. We now assume that some of these a _i values are uncertain and we will model this uncertainty using intervals. Suppose we estimate a ₁ as 0.2 ± 0.1, a ₂ = 0.5 ± 0.2 and a ₃ = 0.3 ± 0.1. Then we would have these probabilities in intervals a ₁ ∈ [0.1, 0.3], a ₂ ∈ [0.3, 0.7] and a ₃ ∈ [0.2, 0.4]. What if we now want the probability of the event A = {x ₁ , x ₂}, it would also be an interval, say [A ₁, A ₂], and we would compute it as follows

$$ \left[ {{A_{1}},{A_{2}}} \right] = \left\{ {\left. {{a_{1}},{a_{2}}} \right|{a_{1}} \in \left[ {0.1,0.3} \right],{a_{2}} \in \left[ {0.3,0.7} \right],{a_{3}} \in \left[ {0.2,0.4} \right],{a_{1}} + {a_{2}} + {a_{3}} = 1} \right\} $$

((1.1))

.