On various ways of tackling incomplete information in statistics

This short paper discusses the contributions made to the featured section on Low Quality Data. We further refine the distinction between the ontic and epistemic views of imprecise data in statistics. We also question the extent to which likelihood functions can be viewed as belief functions. Finally we comment on the data disambiguation effect of learning methods, relating it to data reconciliation problems.     

Rejoinder on “Statistical reasoning with set-valued information: Ontic vs. epistemic views”

This note replies to comments made on our contribution to the Low Quality Data debate.     

A simple logic for reasoning about incomplete knowledge

The semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations.     

Comments on “Statistical reasoning with set-valued information: Ontic vs. epistemic view” by Inés Couso and Didier Dubois

I discuss some aspects of the distinction between ontic and epistemic views of sets as representation of imprecise or incomplete information. In particular, I consider its implications on imprecise probability representations: credal sets and sets of desirable gambles. It is emphasized that the interpretation of the same mathematical object can be different depending on the point of view from which this element is considered. In the case of a fuzzy information on a random variable, it is possible to define a possibility distribution on the simplex of probability distributions. I add some comments about the properties of this possibility distribution.     

Statistical aspects of fuzzy monotone set-valued stochastic processes. Application to birth-and-growth processes

The paper considers a particular family of fuzzy monotone set-valued stochastic processes. The proposed setting allows us to investigate suitable α-level sets of such processes, modeling birth-and-growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set-valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.     

Borderline vs. unknown: comparing three-valued representations of imperfect information

In this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and González-Rodríguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility–necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions.     

Quantity of information; Comparison between information systems: 1. Non-fuzzy states

In statistical theory, experiments or probabilistic information systems are supposed to be informative, since they reduce the amount of uncertainty associated with the states of nature. For the case that the available information systems are vague (fuzzy information systems), H. Tanaka, T. Okuda and K. Asai have proven, using the ‘measure of information’ as provided by ‘entropy’, that the fuzzy information systems are informative too.Now, we wish to state and to study a criterion in order to compare fuzzy information systems by the ‘quantity of information of a fuzzy information system’ (defined by Tanaka et al.).In this first paper we consider the situation where we require information about the original state space (non-fuzzy state space).The second paper will deal with the situation where we require only information on certain vague states (fuzzy states).     

Quantity of information; comparison between information systems: 2. Fuzzy states

In statistical theory, experiments or probabilistic information systems are supposed to be informative, since they reduce the amount of uncertainty associated with the states of nature. For the case that the available information systems are vague (fuzzy information systems), H. Tanaka, T. Okuda and K. Asai have proven, using the ‘measure of information’ as provided by ‘entropy’, that the fuzzy information systems are informative too.Now, we wish to state and to study a criterion in order to compare fuzzy information systems by the ‘quantity of information of a fuzzy information system’ (defined by Tanaka et al.).In the first paper we considered the situation where we require information about the original state space (non-fuzzy state space).This second paper deals with the situation where we require only information on certain vague states (fuzzy states).     

Allocation of information granularity in optimization and decision-making models: Towards building the foundations of Granular Computing

The highly diversified conceptual and algorithmic landscape of Granular Computing calls for the formation of sound fundamentals of the discipline, which cut across the diversity of formal frameworks (fuzzy sets, sets, rough sets) in which information granules are formed and processed. The study addresses this quest by introducing an idea of granular models – generalizations of numeric models that are formed as a result of an optimal allocation (distribution) of information granularity. Information granularity is regarded as a crucial design asset, which helps establish a better rapport of the resulting granular model with the system under modeling. A suite of modeling situations is elaborated on; they offer convincing examples behind the emergence of granular models. Pertinent problems showing how information granularity is distributed throughout the parameters of numeric functions (and resulting in granular mappings) are formulated as optimization tasks. A set of associated information granularity distribution protocols is discussed. We also provide a number of illustrative examples.     

Set-valued cooperative games with fuzzy payoffs. The fuzzy assignment game

In this paper we study cooperative games with fuzzy payoffs. The main advantage of the approach presented is the incorporation into the analysis of the problem of ambiguity inherent in many real-world collective decision situations. We propose extensions of core concepts which maintain the fuzzy nature of allocations, and lead to a more satisfactory study of the problem within the fuzzy context. Finally, we illustrate the extended core concepts and the approach to obtain the corresponding allocations through the analysis of assignment games with uncertain profits.     

L.P.L. a fuzzy programming language: 1. Syntactic aspects

The L.P.L. language (Linguistic oriented Programming Language) that we shall present in this paper and in a forthcoming one is a new language aiming at an implementation of the concepts of the fuzzy sets theory. In the first paper, after an introduction to the basic concepts. we shall describe the syntactic aspects of L.P.L., i.e., the data declarations structure, the statements structure, the general structure of L.P.L. models. The second paper will deal first with the basic semantic and the semantics of logical expressions, then the semantics of basic statements as well as the semantics of the control structure of programs.     

Utility maximization with partial information: Hamilton-Jacobi-Bellman equation approach

This paper deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modelled as an Ornstein-Uhlenbeck process. Here, only the stock price and interest rate can be observable for an investor. It is reduced to a partially observed stochastic control problem. Combining the filtering theory with the dynamic programming approach, explicit representations of the optimal value functions and corresponding optimal strategies are derived. Moreover, closed-form solutions are provided in two cases of exponential utility and logarithmic utility. In particular, logarithmic utility is considered under the restriction of short-selling and borrowing.      

The fuzzy decision problem: An approach to the point estimation problem with fuzzy information

This paper is devoted to the extension of the Bayesian method for the point estimation, when the available information is ‘vague’.In the nonfuzzy case, the parametric estimation can be approached as a particularization in the statistical decision problem. This motivates us to accomplish the mentioned extension by looking at the parametric estimation in the fuzzy case as a special situation in the fuzzy decision problem (defined by Tanaka, Okuda and Asia).In this way, concepts in the fuzzy decision problem are first ‘expressed’ in the estimation terminology. Then, on the basis of these concepts, we shall introduce some notions and state some interesting results. Finally, several illustrative examples will be exposed.     

Game-theoretic analyses of decentralized assembly supply chains: Non-cooperative equilibria vs. coordination with cost-sharing contracts

This paper considers a multiple-supplier, single manufacturer assembly supply chain where the suppliers produce components of a short life-cycle product which is assembled by the manufacturer. In this single-period problem the suppliers determine their production quantities and the manufacturer chooses the retail price. We assume that the manufacturer faces a random price-dependent demand in either additive or multiplicative form. For each case, we analyze both simultaneous-move and leader–follower games to respectively determine the Nash and Stackelberg equilibria, and find the globally-optimal solution that maximizes the system-wide expected profit. Then, we introduce appropriate buy-back and lost-sales cost-sharing contracts to coordinate this assembly supply chain, so that when all the suppliers and the manufacturer adopt their equilibrium solutions, the system-wide expected profit is maximized.     

“Bricks vs. Clicks”: The impact of manufacturer encroachment with a dealer leasing and selling of durable goods

In durable goods markets, many brand name manufacturers, including IBM, HP, Epson, and Lenovo, have adopted dual-channel supply chains to market their products. There is scant literature, however, addressing the product durability and its impact on players’ optimal strategies in a dual-channel supply chain. To fill this void, we consider a two-period dual-channel model in which a manufacturer sells a durable product directly through both a manufacturer-owned e-channel and an independent dealer who adopts a mix of selling and leasing to consumers. Our results show that the manufacturer begins encroaching into the market in Period 1, but the dealer starts withdrawing from the retail channel in Period 2. Moreover, as the direct selling cost decreases, the equilibrium quantities and wholesale prices become quite angular and often nonmonotonic. Among other results, we find that both the dealer and the supply chain may benefit from the manufacturer’s encroachment. Our results also indicate that both the market structure and the nature of competition have an important impact on the player’s (dealer’s) optimal choice of leasing and selling.