Probabilistic Inference with Interval Probabilities

: Probabilistic inference problems have very broad practical applications. To solve this kind of problems under conditions of certainty, an effective mathematical apparatus has been developed. In real situations, obtaining deterministic estimates of relevant probabilities is often difficult; therefore, problems with handling uncertain estimates of probabilities appear. This paper examines the problem of probabilistic inference with probability trees provided that the initial probabilities are given in the form of intervals of their possible values.


Introduction
In many practical problems, it is often necessary to determine the probabilities of events under consideration based on the probabilities of other events. Such problems are called probabilistic inference problems. Elementary tasks of probabilistic inference include [1,2]. More complicated are problems of probabilistic inference for probability trees and belief networks. More details about methods of solving this kind of tasks can be found in [1,2].
Effective techniques are developed for solving probabilistic inference problems for the cases when initial probability estimates are uncertain, namely, when the estimates are given in the interval or fuzzy form. This paper examines the problem of probabilistic inference under the condition that initial values of relevant probabilities are set as intervals of their possible values. In [3], a set of all possible such probabilities are formally defined as follows:

Basic Concepts and Definitions of Interval Probabilities
where   p A denotes a set of all possible probability estimates defined in the set of random events To avoid a situation when   P , boundary values of probability intervals have to satisfy these limiting conditions: Probability intervals satisfying conditions (2), in [3] are called proper intervals. It is evident that in tasks of interval probabilistic inference one should always operate with proper intervals only.
It means that deterministic probability values can be selected over the entire interval   , i i l u , including its boundaries. In [3], probability intervals meeting conditions (3) are called reachable intervals.In [3], it is proven that for reachable probability intervals these inequalities are valid: Calculations of relevant interval probabilities are made according to rules of interval arithmetic, as well as by some special expressions. The use of such special expressions is stipulated by the need to ensure reachable intervals of resulting probabilities. For illustration, the calculation of the posterior probabilities by Bayes' formula can be mentioned given that the initial probabilities are set in the interval form. Several methods for extending classical Bayes' formula to interval probabilities are known; one method was proposed in [4 -7]. The essence of this method is as constellation. Then the following information enables reconstruction of the initial field F :  -marginal F -field regarding the division C : "prior probabilities".

2.
    C -a set of fields of conditional F -probability according to the canonical concept.

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An F -field of complete probabilities can be calculated based on the assigned conditional For each B A  , intuitive concept of conditional probability creates an F -field Conditional probabilities   / ip A B are the desired posterior probabilities.
Although the application of the proposed method produces correct results, it will not be used in this paper due to the complexity and difficulties in interpretation of the results obtained. Instead, a method proposed in [8,9] will be used.
The method under consideration is based on the concept of generalized intervals. Classical interval is identified as a set of real numbers, whereas a generalized interval is identified with the help of predicates that are filled with real numbers; its boundaries are not ordered in common Operations on the generalized intervals are defined based on Kaucher arithmetic [10]. In the set of generalized intervals, these specific mathematical operations are defined.
Operation [7] results in a proper generalized interval.
The result of operation (8) is an improper generalised interval.
The operation that follows transforms a proper generalized interval into the improper In [8], author proposes this interval version of Bayes' formula: where i E , 1,..., It is necessary to understand that according to common expression (10), interval values of probabilities in both denominators in expression (11) are inverted values of initial interval values of When we have only two relevant events E and c E , boundary values of the posterior conditional probabilities can be calculated by these expressions [8]: It should be taken into consideration that in the denominators of expressions (12, a, b) not the initial boundary values of probabilities  

Case Study
Let us consider a "classical" task of assessing chances of the presence of oil on the site given that the prior evaluations of these chances and evaluations of conditional probabilities of the results of seismic exploration of the site are assigned. We have these initial data.
A set of random events ("states of nature") where event 1 a corresponds to the actual presence of oil on the site, but event 2 a corresponds to real absence of oil on the site. Let us call events 1 a and 2 a "geological events". Let us assume that based on the expert evaluation, the following interval values of probabilities of occurrence of these events are assigned: Assume that a manager of an oil mining company has made a decision to undertake seismic exploration of the site to re-evaluate the prior values of probabilities   The specifics of a seismic exploration is that it can both precisely confirm real presence or absence of oil on a site, and produce erroneous results, i.e., to show the presence of oil when it is missing in reality or to show the absence of oil when it is really present. Let us introduce this system of denotations: 1 1 / b a -seismic exploration has confirmed real presence of oil on the site; 2 1 / b a -seismic exploration has erroneously indicated the lack of oil on the site, though in reality oil is present;  To calculate the required posterior interval probabilities, it is necessary to divide the outcome probability values    It is easy to verify that the resulting intervals are valid probability intervals. The target state of information in the form of a decision tree is presented in Figure 2. (1) (2)  Note that on this probability tree, the numbering of outcomes corresponds to the numbering of outcomes on the probability tree in Figure 1. The probabilities of the respective outcomes in both figures are the same.

Algorithms for Finding Permissible Values of Probabilities on Sets of Their Interval Values
Let us introduce the concept of consistent probability intervals. Let us assume that a set of For clarity, three such conditional intervals are graphically presented in Figure 4. 2. For these intervals to be consistent, they must satisfy the following requirements: (1) 1 2 3 c + c + c = 1 ;(2) 1 1 Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.