Characterization and Empirical Evaluation of Bayesian and Credal Combination Operators

Bayesian theory [5] is one of the most commonly utilized theories for managing uncertainty in information fusion [20, 12]. The theory relies on two main assumptions: (1) a probability function should be used for representing belief and (2) Bayes’ theorem should be used for belief updating when a new observation has been made. The main criticism of Bayesian theory that can be found in the literature (e.g., [14, 25]) is that the first assumption is unrealistically strong since one is forced to quantify belief precisely even if one only possesses scarce information about the environment of interest. For this reason, a family of alternative theories has been introduced that usually goes under the name imprecise probability [26], where belief can be expressed imprecisely. One common theory that belongs to the family of imprecise probability is credal set theory [2, 3, 9, 10, 19], also known as “theory of credal sets” [11] and “quasi-Bayesian theory” [8], where one utilizes a closed convex set of probability functions (instead of a single function), denoted as a credal set [19], for representing belief. An attractive feature of credal set theory is that it reduces to Bayesian theory if singleton sets are adopted. Furthermore, credal set theory can be thought of as point-wise application of Bayes theorem on all probability (and likelihood) functions within operand sets (unlike, e.g., evidence theory [23], which is inconsistent with this point-wise Bayesian paradigm [2, 3]). Hence, credal set theory can be seen as the most straightforward generalization of Bayesian theory to imprecise probability. In this paper, we are interested in contrasting Bayesian theory with credal set theory when used for combining independent pieces of evidence, known as the combination problem [16]. Arnborg [2, 3] has previously characterized the relation between robust Bayesian theory, which can be seen as a sensitivity interpretation [4, 14] of credal set theory, and evidence theory [23] when used for the combination problem. We extend Arnborg’s work by characterizing the Bayesian and credal combination operators1 in terms of imprecision and conflict and by introducing methods for accounting for reliability of sources. In addition, we also empirically evaluate the use of the operators for decision making regarding some state space of interest. Since the credal combination operator is considerably more computational demanding than the Bayesian counterpart, such a evaluation can reveal whether or not the additional computational expense yields an increase in decision performance.