Maximum Likelihood Evidential Reasoning

Abstract

In this paper, we aim at generalising the evidential reasoning (ER) rule to establish a new maximum likelihood evidential reasoning (MAKER) framework for probabilistic inference from inputs to outputs in a system space, with their relationships characterised by imperfect data. The MAKER framework consists of three models: system state model (SSM), evidence acquisition model (EAM) and evidential reasoning model (ERM). SSM is introduced to describe system output in the form of ordinary probability distribution on singleton states of the system space to model randomness only, or more generally basic probability distribution on singleton states and their subsets, referred to as states for short, to depict both randomness and ambiguity explicitly. EAM is established to acquire evidence from a data source as system input in the form of basic probability distribution on the evidential elements of the data source, with each evidential element pointing to a state in the system space. ERM is created to combine pieces of acquired evidence, with each represented in the form of basic probability distribution on all the states and the powerset of the system space to facilitate an augmented probabilistic inference process where the trustworthiness of evidence is explicitly modelled alongside its randomness and ambiguity.

Within the MAKER framework, the trustworthiness of evidence is defined in terms of its reliability and expected weight to measure the total degree of its support for all states. Interdependence between pairs of evidence is also measured explicitly. A general conjunctive MAKER rule and algorithm are then established to infer system output from multiple inputs by combining multiple pieces of evidence that have weights and reliabilities and are dependent on each other in general. Several special MAKER rules and algorithms are deduced to facilitate inference in special situations where evidence is exclusive or independent of each other. Specific conditions are identified and proven where the MAKER rule reduces to the ER rule, Dempster's rule and Bayes’ rule. A bi-objective nonlinear pre-emptive minimax optimisation model is built to make use of observed data for optimal learning of evidence weights and reliabilities by maximising the predicted likelihood of the true state for each observation. Two numerical examples are analysed to demonstrate the three constituent models of the MAKER framework, the MAKER rules and algorithms, and the optimal learning model. A case study for human well-being analysis is provided where data from a panel survey are used to show the potential applications of the MAKER framework for probabilistic reasoning and decision making under different types of uncertainty.