Theory Of Coherent Lower Previsions Research Articles

Conformity and independence with coherent lower previsions

We define the conformity of marginal and conditional models with a joint model within Walley's theory of coherent lower previsions. Loosely speaking, conformity means that the joint can reproduce the marginal and conditional models we started from. By studying conformity with and without additional assumptions of epistemic irrelevance and independence, we establish connections with a number of prominent models in Walley's theory: the marginal extension, the irrelevant natural extension, the independent natural extension and the strong product.

On the problem of computing the conglomerable natural extension

Embedding conglomerability as a rationality requirement in probability was among the aims of Walley's behavioural theory of coherent lower previsions. However, recent work has shown that this attempt has only been partly successful. If we focus in particular on the extension of given assessments to a rational and conglomerable model (in the least-committal way), we have that the procedure used in Walley's theory, the natural extension, provides only an approximation to the model that is actually sought for: the so-called conglomerable natural extension. In this paper we consider probabilistic assessments in the form of a coherent lower prevision P̲, which is another name for a lower expectation functional, and make an in-depth mathematical study of the problem of computing the conglomerable natural extension for this case: that is, where it is defined as the smallest coherent lower prevision F̲≥P̲ that is conglomerable, in case it exists. Past work has shown that F̲ can be approximated by an increasing sequence (E̲n)n∈N of coherent lower previsions. We solve an open problem by showing that this sequence can consist of infinitely many distinct elements. Moreover, we give sufficient conditions, of quite broad applicability, to make sure that the point-wise limit of the sequence is F̲ in case P̲ is the lower envelope of finitely many linear previsions. In addition, we study the question of the existence of F̲ and its relationship with the notion of marginal extension.

Conglomerable coherence

We contrast Williamsʼ and Walleyʼs theories of coherent lower previsions in the light of conglomerability. These are two of the most credited approaches to a behavioural theory of imprecise probability. Conglomerability is the notion that distinguishes them most: Williamsʼ theory does not consider it, while Walley aims at embedding it in his theory. This question is important, as conglomerability is a major point of disagreement at the foundations of probability, since it was first defined by de Finetti in 1930. We show that Walleyʼs notion of joint coherence (which is the single axiom of his theory) for conditional lower previsions does not take all the implications of conglomerability into account. Considering also some previous results in the literature, we deduce that Williamsʼ theory should be the one to use when conglomerability is not required; for the opposite case, we define the new theory of conglomerably coherent lower previsions, which is arguably the one to use, and of which Walleyʼs theory can be understood as an approximation. We show that this approximation is exact in two important cases: when all conditioning events have positive lower probability, and when conditioning partitions are nested.

Conglomerable natural extension

At the foundations of probability theory lies a question that has been open since de Finetti framed it in 1930: whether or not an uncertainty model should be required to be conglomerable. Conglomerability is related to accepting infinitely many conditional bets. Walley is one of the authors who have argued in favor of conglomerability, while de Finetti rejected the idea. In this paper we study the extension of the conglomerability condition to two types of uncertainty models that are more general than the ones envisaged by de Finetti: sets of desirable gambles and coherent lower previsions. We focus in particular on the weakest (i.e., the least-committal) of those extensions, which we call the conglomerable natural extension. The weakest extension that does not take conglomerability into account is simply called the natural extension. We show that taking the natural extension of assessments after imposing conglomerability—the procedure adopted in Walley’s theory—does not yield, in general, the conglomerable natural extension (but it does so in the case of the marginal extension). Iterating this process of imposing conglomerability and taking the natural extension produces a sequence of models that approach the conglomerable natural extension, although it is not known, at this point, whether this sequence converges to it. We give sufficient conditions for this to happen in some special cases, and study the differences between working with coherent sets of desirable gambles and coherent lower previsions. Our results indicate that it is necessary to rethink the foundations of Walley’s theory of coherent lower previsions for infinite partitions of conditioning events.

Conditional models: Coherence and inference through sequences of joint mass functions

We call a conditional model any set of statements made of conditional probabilities or expectations. We take conditional models as primitive compared to unconditional probability, in the sense that conditional statements do not need to be derived from an unconditional probability. We focus on two problems: ( coherence) giving conditions to guarantee that a conditional model is self-consistent; ( inference) delivering methods to derive new probabilistic statements from a self-consistent conditional model. We address these problems in the case where the probabilistic statements can be specified imprecisely through sets of probabilities, while restricting the attention to finite spaces of possibilities. Using Walley's theory of coherent lower previsions, we fully characterise the question of coherence, and specialise it for the case of precisely specified probabilities, which is the most common case addressed in the literature. This shows that coherent conditional models are equivalent to sequences of (possibly sets of) unconditional mass functions. In turn, this implies that the inferences from a conditional model are the limits of the conditional inferences obtained by applying Bayes’ rule, when possible, to the elements of the sequence. In doing so, we unveil the tight connection between conditional models and zero-probability events.

Conservative Inference Rule for Uncertain Reasoning under Incompleteness

In this paper we formulate the problem of inference under incomplete information in very general terms. This includes modelling the process responsible for the incompleteness, which we call the incompleteness process. We allow the process' behaviour to be partly unknown. Then we use Walley's theory of coherent lower previsions, a generalisation of the Bayesian theory to imprecision, to derive the rule to update beliefs under incompleteness that logically follows from our assumptions, and that we call conservative inference rule. This rule has some remarkable properties: it is an abstract rule to update beliefs that can be applied in any situation or domain; it gives us the opportunity to be neither too optimistic nor too pessimistic about the incompleteness process, which is a necessary condition to draw reliable while strong enough conclusions; and it is a coherent rule, in the sense that it cannot lead to inconsistencies. We give examples to show how the new rule can be applied in expert systems, in parametric statistical inference, and in pattern classification, and discuss more generally the view of incompleteness processes defended here as well as some of its consequences.

Forward irrelevance

We investigate how to combine marginal assessments about the values that random variables assume separately into a model for the values that they assume jointly, when (i) these marginal assessments are modelled by means of coherent lower previsions and (ii) we have the additional assumption that the random variables are forward epistemically irrelevant to each other. We consider and provide arguments for two possible combinations, namely the forward irrelevant natural extension and the forward irrelevant product, and we study the relationships between them. Our treatment also uncovers an interesting connection between the behavioural theory of coherent lower previsions, and Shafer and Vovk's game-theoretic approach to probability theory.

Finitely additive extensions of distribution functions and moment sequences: The coherent lower prevision approach

We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions.

Marginal extension in the theory of coherent lower previsions

We generalise Walley’s Marginal Extension Theorem to the case of any finite number of conditional lower previsions. Unlike the procedure of natural extension, our marginal extension always provides the smallest (most conservative) coherent extensions. We show that they can also be calculated as lower envelopes of marginal extensions of conditional linear (precise) previsions. Finally, we use our version of the theorem to study the so-called forward irrelevant product and forward irrelevant natural extension of a number of marginal lower previsions.

Belief models: An order-theoretic investigation

I show that there is a common order-theoretic structure underlying many of the models for representing beliefs in the literature. After identifying this structure, and studying it in some detail, I argue that it is useful. On the one hand, it can be used to study the relationships between several models for representing beliefs, and I show in particular that the model based on classical propositional logic can be embedded in that based on the theory of coherent lower previsions. On the other hand, it can be used to generalise the coherentist study of belief dynamics (belief expansion and revision) by using an abstract order-theoretic definition of the belief spaces where the dynamics of expansion and revision take place. Interestingly, many of the existing results for expansion and revision in the context of classical propositional logic can still be proven in this much more abstract setting, and therefore remain valid for many other belief models, such as those based on imprecise probabilities.

Measures of uncertainty in expert systems

This paper compares four measures that have been advocated as models for uncertainty in expert systems. The measures are additive probabilities (used in the Bayesian theory), coherent lower (or upper) previsions, belief functions (used in the Dempster-Shafer theory) and possibility measures (fuzzy logic). Special emphasis is given to the theory of coherent lower previsions, in which upper and lower probabilities, expectations and conditional probabilities are constructed from initial assessments through a technique of natural extension. Mathematically, all the measures can be regarded as types of coherent lower or upper previsions, and this perspective gives some insight into the properties of belief functions and possibility measures. The measures are evaluated according to six criteria: clarity of interpretation; ability to model partial information and imprecise assessments, especially judgements expressed in natural language; rules for combining and updating uncertainty, and their justification; consistency of models and inferences; feasibility of assessment; and feasibility of computations. Each of the four measures seems to be useful in special kinds of problems, but only lower and upper previsions appear to be sufficiently general to model the most common types of uncertainty.