2. plausibility function pl : 2 → [0, 1] is defined as: Shafer : bel is ‘normalized’ => closed world assumption=> bel(Ω)=1, pl(Ω)=1,m(∅)
Observations in belief functions
Belief functions
Ω the frame of discernment(elements of the set Ω are called ‘worlds’) One “actual world” ω0. But which? An agent can only express the strength of his/her opinion (called belief) degree of belief that the actual world belongs to this or that subset of Ω. [0, Shafer belief function bel : 2 → [0 1] bel(A) denotes the strength of Agent’s belief that ω0∈A. bel satisfies the following inequalities:
Not supporting any strictly more specific propositions A basic belief mass given to a set A supports also that the actual world is in every subsets that contains A. The degree of belief bel(A) for A quantifies the total amount of justified specific support given to A. We say justified because we include in bel(A) only the basic belief masses given to subsets of A. m({x,y}) given to {x,y} could support x if further information indicates this.However given the available information the basic belief mass can only be given to {x,y}. We say specific because the basic belief mass m(Ø) is not included in bel(A) as it is given to the subset Ø.
Dempster–Shafer approach
Answering the question ‘‘what is the belief in A, as expressed by the probability that the proposition A is provable given the evidence?’’ An alternative to traditional probabilistic theory for the mathematical representation of uncertainty. Whereas a Bayesian approach assesses probabilities directly for the answer, the Dempster–Shafer approach assesses evidence for related questions. -combination of evidence obtained from multiple sources and the modeling of conflict between them. -Allocation of a probability mass to sets or intervals Pros -Ability to model various types of partial ignorance, limited or conflicting evidence -more flexible model than Bayes’ theorem. -computationally simpler than Bayes’ theorem. -No assumption regarding the probability of the individual constituents of the set or interval. -evaluation of risk and reliability in engineering applications when it is not possible to obtain a precise measurement from experiments, or when knowledge is obtained from expert elicitation. Cons -Can produce conclusions that are counter-intuitive. Dempster–Shafer is most suited to situations where beliefs are numerically expressed and where there is some degree of ignorance, i.e. there is an incomplete model.
Entertained beliefs and beliefs in a decision context
Uncertainty induces beliefs=“graded dispositions that guide our behavior” ‘rational’agent behavior described within decision contexts “It has been argued that decisions are ‘rational’ only if we use a probability
Introduction to Evidential Reasoning
Belief Functions
Ignorance Types
Incompleteness: Combinatory Combinatory Imprecision: Combinatory Combinatory Interval theory Fuzzy sets Possibility Theory (physical form) Uncertainty: Probability Theory Upper-Lower Probabilities the chance of it being "heads" when tossing a coin. John's wife is Jill or Joan. Jill is not John's wife. Paul's height is between 170 and 180. Paul is tall. the possibility for Paul's height to be about 175 cm. John is married, but his wife's name is not given All computer scientists like pizza, but their names are not available.
Other useful functions (‘1-1’ with bel) 1.basic belief assignment (bba) m : 2 → [0, 1] defined as:
m(A) for A ⊆ Ω is called the basic belief mass (bbm) given to A. It may happen that m(∅) > 0. The relation from m to bel is given by:
Answering ‘‘what is the belief in A? as expressed by the unconditional probability that A is true given evidence, e ?’’ Assumption: precise probabilities can be assessed for all events. Too rare… Why believe one hypothesis other than that provided by the evidence?? The evidence have to be re-organised so that probabilities sum to unity. Pros -rules of probability calculus : uncontroversial, constant conclusions with the probability assessments. -Bayesian theory is easy to understand. Cons It is least suited to problems where there is -partial or complete ignorance -limited or conflicting information due to assumptions made(e.g. equi-probability) Cannot deal with imprecise, qualitative or natural language judgements such as ‘‘if A then probably B’’.