[Paper Review] Independence with Lower and Upper Probabilities

Lonnie Chrisman|arXiv (Cornell University)|Feb 13, 2013

Logic, Reasoning, and Knowledge35 references15 citations

TL;DR

This paper demonstrates that interval probability representations cannot adequately capture epistemological independence except in degenerate cases—such as when bounds are point probabilities or vacuous. It argues that these limitations arise from misinterpreting probability intervals as epistemological uncertainty, and instead proposes an ontological interpretation where imprecision reflects structural approximation, not ignorance, resolving the apparent paradoxes.

ABSTRACT

It is shown that the ability of the interval probability representation to capture epistemological independence is severely limited. Two events are epistemologically independent if knowledge of the first event does not alter belief (i.e., probability bounds) about the second. However, independence in this form can only exist in a 2-monotone probability function in degenerate cases i.e., if the prior bounds are either point probabilities or entirely vacuous. Additional limitations are characterized for other classes of lower probabilities as well. It is argued that these phenomena are simply a matter of interpretation. They appear to be limitations when one interprets probability bounds as a measure of epistemological indeterminacy (i.e., uncertainty arising from a lack of knowledge), but are exactly as one would expect when probability intervals are interpreted as representations of ontological indeterminacy (indeterminacy introduced by structural approximations). The ontological interpretation is introduced and discussed.

Motivation & Objective

• To investigate whether lower and upper probabilities can represent epistemological independence—where knowledge of one event does not affect belief bounds about another.
• To identify the conditions under which epistemological independence is possible within interval probability models.
• To analyze the constraints imposed by 2-monotonicity and other classes of lower probability functions on independence.
• To argue that apparent limitations in modeling independence stem not from flaws in the framework, but from misinterpreting probability intervals as epistemic uncertainty rather than ontological indeterminacy.

Proposed method

• The paper analyzes the mathematical conditions under which two events can be epistemologically independent in a lower/upper probability framework.
• It examines 2-monotone lower probability functions and derives necessary conditions for independence to hold.
• It distinguishes between epistemological indeterminacy (lack of knowledge) and ontological indeterminacy (structural approximation in models).
• It introduces and defends an ontological interpretation of probability intervals, where imprecision arises from model simplification, not ignorance.
• It uses formal logic and probability bounds theory to show that independence only holds in degenerate cases under epistemic interpretation.
• It compares the behavior of interval probabilities under different interpretations to demonstrate consistency under the ontological reading.

Experimental results

Research questions

• RQ1 Under what conditions can two events be epistemologically independent in a lower and upper probability framework?
• RQ2 Why do interval probabilities fail to represent independence except in degenerate cases?
• RQ3 How does the interpretation of probability intervals—as epistemic or ontological—affect the validity of independence?
• RQ4 What mathematical constraints (e.g., 2-monotonicity) limit the existence of epistemological independence in imprecise probability models?
• RQ5 Can the apparent limitations of interval probabilities in modeling independence be resolved by reinterpreting the nature of imprecision?

Key findings

• Epistemological independence in interval probability models is only possible in degenerate cases—either when bounds are point probabilities or entirely vacuous.
• The failure to represent non-degenerate independence is not a flaw in the framework, but a consequence of interpreting imprecision as epistemic uncertainty.
• When probability intervals are interpreted ontologically—as reflecting structural approximation rather than ignorance—the limitations vanish and independence behaves as expected.
• The 2-monotonicity condition severely restricts the existence of epistemological independence, limiting it to trivial cases.
• The ontological interpretation provides a coherent foundation for imprecise probabilities, resolving paradoxes that arise under epistemic interpretation.
• The paper concludes that the observed limitations are artifacts of interpretation, not inherent flaws in lower and upper probability representations.