Epistemic Irrelevance Research Articles

Average behaviour in discrete-time imprecise Markov chains: A study of weak ergodicity

We study the limit behaviour of upper and lower bounds on expected time averages in imprecise Markov chains; a generalised type of Markov chain where the local dynamics, traditionally characterised by transition probabilities, are now represented by sets of ‘plausible’ transition probabilities. Our first main result is a necessary and sufficient condition under which these upper and lower bounds, called upper and lower expected time averages, will converge as time progresses towards infinity to limit values that do not depend on the process' initial state. Our condition is considerably weaker than that needed for ergodic behaviour; a similar notion which demands that marginal upper and lower expectations of functions at a single time instant converge to so-called limit—or steady state—upper and lower expectations. For this reason, we refer to our notion as ‘weak ergodicity’. Our second main result shows that, as far as this weakly ergodic behaviour is concerned, one should not worry about which type of independence assumption to adopt—epistemic irrelevance, complete independence or repetition independence. The characterisation of weak ergodicity as well as the limit values of upper and lower expected time averages do not depend on such a choice. Notably, this type of robustness is not exhibited by the notion of ergodicity and the related inferences of limit upper and lower expectations. Finally, though limit upper and lower expectations are often used to provide approximate information about the limit behaviour of time averages, we show that such an approximation is sub-optimal and that it can be significantly improved by directly using upper and lower expected time averages.

Modelling epistemic irrelevance with choice functions

We consider coherent choice functions under the recent axiomatisation proposed by De Bock and de Cooman that guarantees a representation in terms of binary preferences, and we discuss how to define conditioning in this framework. In a multivariate context, we propose a notion of marginalisation, and its inverse operation called weak (cylindrical) extension. We combine this with our definition of conditioning to define a notion of irrelevance, and we obtain the irrelevant natural extension in this framework: the least informative choice function that satisfies a given irrelevance assessment.

Credal networks under epistemic irrelevance

A credal network under epistemic irrelevance is a generalised type of Bayesian network that relaxes its two main building blocks. On the one hand, the local probabilities are allowed to be partially specified. On the other hand, the assessments of independence do not have to hold exactly. Conceptually, these two features turn credal networks under epistemic irrelevance into a powerful alternative to Bayesian networks, offering a more flexible approach to graph-based multivariate uncertainty modelling. However, in practice, they have long been perceived as very hard to work with, both theoretically and computationally.The aim of this paper is to demonstrate that this perception is no longer justified. We provide a general introduction to credal networks under epistemic irrelevance, give an overview of the state of the art, and present several new theoretical results. Most importantly, we explain how these results can be combined to allow for the design of recursive inference methods. We provide numerous concrete examples of how this can be achieved, and use these to demonstrate that computing with credal networks under epistemic irrelevance is most definitely feasible, and in some cases even highly efficient. We also discuss several philosophical aspects, including the lack of symmetry, how to deal with probability zero, the interpretation of lower expectations, the axiomatic status of graphoid properties, and the difference between updating and conditioning.

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.

Robust queueing theory: an initial study using imprecise probabilities

We study the robustness of performance predictions of discrete-time finite-capacity queues by applying the framework of imprecise probabilities. More concretely, we consider the Geo/Geo/1/L model with probabilities of arrival and departure that are no longer fixed, but are allowed to vary within given intervals. We distinguish between two concepts of independence in this framework, namely repetition independence and epistemic irrelevance. In the first approach, we assume the existence of time-homogeneous probabilities for arrival and departure, which leads us to consider a collection of stationary queues. In the second, the stationarity assumption is dropped and we allow the arrival and departure probabilities to vary from time point to time point; they may even depend on the complete history of queue lengths. We calculate bounds on the expected queue length, the probability of a particular queue length and the probability of turning on the server. For the expected queue length, both approaches coincide. For the other performance measures, we observe and discuss various differences between the bounds obtained for these two approaches. One of our observations is that ergodicity may break down due to imprecision: bounds on expected time averages of certain functions on the state space are not necessarily equal to the bounds on the expectation of that function at random instants in a steady-state queue.

Probabilistic Inference in Credal Networks: New Complexity Results

Credal networks are graph-based statistical models whose parameters take values in a set, instead of being sharply specified as in traditional statistical models (e.g., Bayesian networks). The computational complexity of inferences on such models depends on the irrelevance/independence concept adopted. In this paper, we study inferential complexity under the concepts of epistemic irrelevance and strong independence. We show that inferences under strong independence are NP-hard even in trees with binary variables except for a single ternary one. We prove that under epistemic irrelevance the polynomial-time complexity of inferences in credal trees is not likely to extend to more general models (e.g., singly connected topologies). These results clearly distinguish networks that admit efficient inferences and those where inferences are most likely hard, and settle several open questions regarding their computational complexity. We show that these results remain valid even if we disallow the use of zero probabilities. We also show that the computation of bounds on the probability of the future state in a hidden Markov model is the same whether we assume epistemic irrelevance or strong independence, and we prove a similar result for inference in naive Bayes structures. These inferential equivalences are important for practitioners, as hidden Markov models and naive Bayes structures are used in real applications of imprecise probability.

Credal networks under epistemic irrelevance: The sets of desirable gambles approach

We present a new approach to credal networks, which are graphical models that generalise Bayesian networks to deal with imprecise probabilities. Instead of applying the commonly used notion of strong independence, we replace it by the weaker, asymmetrical notion of epistemic irrelevance. We show how assessments of epistemic irrelevance allow us to construct a global model out of given local uncertainty models, leading to an intuitive expression for the so-called irrelevant natural extension of a credal network. In contrast with Cozman [4], who introduced this notion in terms of credal sets, our main results are presented using the language of sets of desirable gambles. This has allowed us to derive some remarkable properties of the irrelevant natural extension, including marginalisation properties and a tight connection with the notion of independent natural extension. Our perhaps most important result is that the irrelevant natural extension satisfies a collection of epistemic irrelevancies that is induced by AD-separation, an asymmetrical adaptation of d-separation. Both AD-separation and the induced collection of irrelevancies are shown to satisfy all graphoid properties except symmetry.

Irrelevant and independent natural extension for sets of desirable gambles

The results in this paper add useful tools to the theory of sets of desirable gambles, a growing toolbox for reasoning with partial probability assessments. We investigate how to combine a number of marginal coherent sets of desirable gambles into a joint set using the properties of epistemic irrelevance and independence. We provide formulas for the smallest such joint, called their independent natural extension, and study its main properties. The independent natural extension of maximal coherent sets of desirable gambles allows us to define the strong product of sets of desirable gambles. Finally, we explore an easy way to generalise these results to also apply for the conditional versions of epistemic irrelevance and independence. Having such a set of tools that are easily implemented in computer programs is clearly beneficial to fields, like AI, with a clear interest in coherent reasoning under uncertainty using general and robust uncertainty models that require no full specification.

Irrelevant and independent natural extension for sets of desirable gambles

The results in this paper add useful tools to the theory of sets of desirable gambles, a growing toolbox for reasoning with partial probability assessments. We investigate how to combine a number of marginal coherent sets of desirable gambles into a joint set using the properties of epistemic irrelevance and independence. We provide formulas for the smallest such joint, called their independent natural extension, and study its main properties. The independent natural extension of maximal coherent sets of desirable gambles allows us to define the strong product of sets of desirable gambles. Finally, we explore an easy way to generalise these results to also apply for the conditional versions of epistemic irrelevance and independence. Having such a set of tools that are easily implemented in computer programs is clearly beneficial to fields, like AI, with a clear interest in coherent reasoning under uncertainty using general and robust uncertainty models that require no full specification.

Modelling uncertainties in limit state functions

In this paper uncertainties in limit state functions g as arising in engineering problems are modelled by adding additional parameters and by introducing parameterized probability density functions which describe the uncertainties of these new additional parameters and of the basic variables of g. This will lead to a function p f ( a, b) for the probability of failure depending on parameters a and b corresponding to the two parameterized density functions. Further the parameters a and b are assumed to be uncertain. Using intervals, sets or random sets to model their uncertainty results in upper probabilities p ¯ f of failure. In this context we also discuss different notions of independence such as strong independence, epistemic irrelevance and random set independence and present a simple engineering example.

Epistemic irrelevance in credal nets: The case of imprecise Markov trees

We focus on credal nets, which are graphical models that generalise Bayesian nets to imprecise probability. We replace the notion of strong independence commonly used in credal nets with the weaker notion of epistemic irrelevance, which is arguably more suited for a behavioural theory of probability. Focusing on directed trees, we show how to combine the given local uncertainty models in the nodes of the graph into a global model, and we use this to construct and justify an exact message-passing algorithm that computes updated beliefs for a variable in the tree. The algorithm, which is linear in the number of nodes, is formulated entirely in terms of coherent lower previsions, and is shown to satisfy a number of rationality requirements. We supply examples of the algorithm’s operation, and report an application to on-line character recognition that illustrates the advantages of our approach for prediction. We comment on the perspectives, opened by the availability, for the first time, of a truly efficient algorithm based on epistemic irrelevance.

Epistemic irrelevance on sets of desirable gambles

This paper studies graphoid properties for epistemic irrelevance in sets of desirable gambles. For that aim, the basic operations of conditioning and marginalization are expressed in terms of variables. Then, it is shown that epistemic irrelevance is an asymmetric graphoid. The intersection property is verified in probability theory when the global probability distribution is positive in all the values. Here it is always verified due to the handling of zero probabilities in sets of gambles. An asymmetrical D-separation principle is also presented, by which this type of independence relationships can be represented in directed acyclic graphs.

Graphoid properties of epistemic irrelevance and independence

This paper investigates Walley's concepts of epistemic irrelevance and epistemic independence for imprecise probability models. We study the mathematical properties of irrelevance and independence, and their relation to the graphoid axioms. Examples are given to show that epistemic irrelevance can violate the symmetry, contraction and intersection axioms, that epistemic independence can violate contraction and intersection, and that this accords with informal notions of irrelevance and independence.

Metaphysics and the morality of abortion

Conclusions about the morality of abortion have been thought to receive some support from metaphysical doctrines about persons. The paper studies four instances in which philosophers have sought to draw such morals from metaphysics. It argues that in each instance the metaphysics makes no moral difference, and the manner of failure seems indicative of a general epistemic irrelevance of metaphysics to the moral issue.