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FAQs of Imprecise Probability

This page answers some questions on our imprecise probability for aleatory and epistemic uncertainty quantifications.

What is imprecise probability?

Imprecise probability can be looked as a generalization of the traditional probability. Instead of a precise value of the probabilistic measure Pr(A)=p associated with an event A, a pair of lower and upper probabilities Pr(A)=[p1,p2] are used to include a set of probabilities and quantify the aleatory and epistemic uncertainties simultaneously. The range of the interval [p1,p2] captures the epistemic uncertainty component.

Several theories and representations of imprecise probability have been proposed. The Dempster-Shafer evidence theory (Dempster, 1967; Shafer, 1976) characterizes evidence with discrete probability masses, where Belief-Plausibility pairs are used to measure incertitude. The coherent lower prevision theory (Walley, 1991) models epistemic uncertainty with the lower prevision (supremum acceptable buying price) and the upper prevision (infimum acceptable selling price) with behavioral interpretations, following the notations of de Finetti's subjective probability theory. Probability bound analysis (Ferson et al., 2003) captures uncertain information with pairs of lower and upper distribution functions or called p-boxes. F-probability (Weichselberger, 2000) incorporates intervals into probability values which maintain the Kolmogorov properties. Others include the possibility theory (Dubois & Prade, 1988) that represents incertitude with Necessity-Possibility pairs. The fuzzy probability (Moller & Beer, 2004) considers probability distributions with fuzzy parameters. A random set (Malchanov, 2005) is a multi-valued mapping from the probability space to the value space. A cloud (Neumaier, 2004) combines fuzzy sets, intervals, and probability distributions, although there is as yet no way to compute with these structures. More recently, a new form of imprecise probability based on the generalized interval (Wang, 2008; 2010) was proposed.

Why don't we just use the traditional precise probability to quantify both variability and incertitude?

Probability theory does provide the common ground to quantify both aleatory and epistemic uncertainties and so far is the most popular approach. Traditionally probabilistic properties are quantified by precise values of probability measures and their parameters (e.g. means and higher-order moments).

However, the traditional probability theory has limitations to represent epistemic uncertainty. The most significant one is that it does not differentiate the total ignorance from other probability distributions. The total ignorance means that there is absolutely no information about the system or subject under study. Based on the principle of maximum entropy, uniform distributions are usually assumed when the precise probability theory is applied in this case. A problem arises because introducing any form of distribution itself has introduced extra knowledge in. This leads to the Bertrand-style paradoxes such as the van Fraasen's cube factory (van Fraasen, 1989). "Knowing the unknown" is not the total ignorance. In contrast, imprecise probability just uses a probability P=[0,1] which perfectly represents the absolute ignorance.

Another limitation of precise probability to quantify epistemic uncertainty is to represent personal indeterminacy and group inconsistency in the context of subjective probability. It lacks mechanisms in capturing the true subjective beliefs of those who have hesitation and indeterminacy without assertive confidence under certain situations. There is no mechanism to represent the situation that subjective probabilities from different people are inconsistent either, since subjective probability is personal. As a result, rational group decision making based on probability theory is not possible based on Arrow' impossibility theorem (Arrow 1951). Precise probability does not capture a range of opinions or estimations adequately without assuming some consensus of precise values on the distributions of opinions. "Agreeing the disagreed" is not the best way to capture inconsistency.

Why choose interval to represent epistemic uncertainty?

Interval is as simple as a pair of numbers, i.e. the lower and upper bounds. The reason to choose interval is two-fold.

First, interval is natural to human users and simple to use. It has been widely used to represent a range of possible values, an estimate of lower and upper bounds for numerical errors, and the measurement error because of the available precision as the result of instrument calibration.

Second, interval can be regarded as the most suitable way to represent the lack of knowledge. Compared to other forms, interval has the least assumption. It only needs two values for the bounds. In contrast, statistical distributions need assumptions of distribution types, distribution parameters, and the functional mapping from events to real values between 0 and 1. Fuzzy sets need assumptions of not only lower and upper bounds, but also membership functions. Given that the lack of knowledge is the nature of epistemic uncertainty, a representation with the least assumption is the most desirable. Notice that an interval [L,U] only specifies its lower bound L and upper bound U. It does not assume a uniform distribution of values between L and U.

Probability theory is a rigorous theory. Are imprecise probability theories also rigorous ones?

A theory is rigorous if and only if its definitions, properties and relationships are self-consistent and derived with logical reasoning. Similarly, its applications in either engineering decision making or any others are rigorous if and only if its users rigorously follow the true meanings of properties and relationships and do not misuse or abuse. For instance, the values of lower and upper probabilities are not arbituarily assigned. Rather they should satisfy some restrictions, such as the coherence and avoid sure loss constraints in Walley's coherent lower prevision theory (Walley, 1991) and the logic coherence constraint in our generalizer interval probability.

Imprecise probability is an extension and generalization of precise probability. Many concepts of the precise probability theory can appropriately be generalized to imprecise probabilities. Similar to other forms of imprecise probabilities, the proposed imprecise probability theory is an extension, generalization, and enhancement of the traditional precise probability. All principles of precise probability are applicable in the proposed imprecise probability, such as the Kolmogorov axioms, independence, and Bayes' rule. Those principles and properties are defined in the precise probability theory, and they form the fundamental framework of the precise probability theory. Similarly, they are rigorously defined in imprecise probability theories. Our proposed one has the most similar form of the precise probability theory, compared to other imprecise probability theories.

Probabilistic approach is the only rigorous approach that is consistent with the fundamentals of decision theory. Specifically, when implementing the imprecise probabilities, is it possible to define a rigorous decision making criterion?

Decision theory includes theories of preference, utility and value, subjective probability and ambiguity, decision under risk or uncertainty, Bayesian decision analysis, probabilistic choice, social choice, and elections (Fishburn, 1991).

Advances in decision theory have often been driven by paradoxes. One of the recent ones is Ellsberg's paradox (Ellsberg, 1961), which is about subjective expected utility theory and ambiguity. Ambiguity has been well recognized in both modeling and experimental studies. Ambiguity is the quality depending on the amount, type, reliability, and unanimity of information, whereas Frisch and Baron (1988) defined ambiguity as uncertainty about probability, created by missing information that is relevant and could be known.

In the original expected utility theory (von Neumann and Morgenstern, 1947), it is assumed that all probabilities of outcomes are objectively known, which is hardly the case in decision processes. Therefore, the subjective expected utility theory (Savage, 1954) is applied more widely with probabilities are subjective or personal. Yet, various studies suggest that how much people know about a state's probability does influence their willingness to bet on the state. A simple example is the bet on two coins. The first one has been flipped thousands of times and people concluded it is fair. The second one has been thrown twice and the result is one head and one tail. People believe both coins are fair but still prefer to bet on the first one. Ellsberg's paradox (Ellsberg, 1961) is the most studied problem. As a simple example, for two urns, the first one contains 50 red and 50 black balls whereas the second one contains an unknown combination of red and black balls. Many people prefer to bet on red from urn 1 rather than betting on red from urn 2. That is, ambiguity aversion is common.

Consequently, to further model ambiguity, researchers of decision theory have proposed various methods, including second-order probability, probability set, non-additive probability, ambiguity-dependent utility function, etc. Camerer and Weber (1992) provided a comprehensive review.

Additionally, one misconception of the expected utility theory is that decision making has to be based on the predetermined precise values of probabilities. That is, given a sigma algebra with all possible events, the subjective probability of every event should be given a unique value. Otherwise, it is impossible to make decision. In other words, unique probability values is both sufficient and necessary conditions of rational decision making. However, Savage's expected utility theory (Savage, 1954) only poses restrictions on the coherence of personal preferences and nothing more. Savage did not forbid preferences on actions, where he used a 'grand-world small-world' illustration in his theory. Therefore, without precise probability, coherent decision is still possible.

If an interval is used to represent uncertainty, what information is used and how do you define the lower and upper bounds of the interval?

There is no fundamental difference between the interval used in interval uncertainty and the confidence interval in aleatory probability to characterize uncertainty. Therefore, the simplest way to determine the lower and upper bounds is just to use confidence intervals as regularly used in P(lowerBound ≤ x ≤ upperBound)=1-α. Intervals then are the "vehicles" used to estimate uncertainty accumulation and propagation. The interval propagation can be easily computed with the confidence of completeness of estimation. Furthermore, if the new generalized interval is adopted, the soundness of range estimation can be ensured as well. Completeness and soundness cannot be verified easily with probabilistic models. Therefore, any information/data available to estimate the parameters of probability distributions for a probabilistic model of uncertainty can at the same time be used to determine the lower and upper bounds of intervals.

There are also other approaches to determine the lower and upper bounds for specific purposes. The width of an interval is determined the robustness measurement or confidence you would like to have. If no statistical data are available, engineers (domain experts) usually tend to estimate ranges such as 65+/-5% based on experiences, etc. These are interval bounds in nature.

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