C Baudrit

D Dubois

Practical Representations of Incomplete Probabilistic Knowledge

Keywords: Imprecise Probabilities, Possibility, Belief functions, Probability-Boxes

This article deals with the compact representation of incomplete probabilistic knowledge which can be encountered in risk evaluation problems, for instance in environmental studies. Various kinds of knowledge are considered such as expert opinions about characteristics of distributions or poor statistical information. Our approach is based on probability families encoded by possibility distributions and belief functions. In each case, a technique for representing the available imprecise probabilistic information faithfully is proposed, using different uncertainty frameworks (possibility theory, probability theory, belief functions...). Moreover the use of probability-possibility transformations enables confidence intervals to be encompassed by cuts of possibility distributions, thus making the representation stronger. The respective appropriateness of pairs of cumulative distributions, continuous possibility distributions or discrete random sets for representing information about the mean value, the mode, the median and other fractiles of ill-known probability distributions is discussed in detail.

Introduction

In risk analysis, uncertainties are often captured within a purely probabilistic framework. It suggests that all uncertainties whether of a random or an epistemic nature should be represented in the same way. Under this assumption, the uncertainty associated with each parameter of a mathematical model of some phenomenon can be described by a single probability distribution. According to the frequentist view, the occurrence of an event is a matter of chance. However, not all uncertainties Email addresses: baudrit@irit.fr (C. Baudrit), dubois@irit.fr (D. Dubois).

Preprint submitted to Elsevier Science 27 October 2005 are random nor can be objectively quantified, even if the choice of values for parameters is based as much as possible on on-site investigations. Due to time and financial constraints, information regarding model parameters is often incomplete. For example, it is quite common for a hydrogeologist to estimate the numerical values of acquifer parameters in the form of confidence intervals according to his/her experience and intuition (i.e expert judgment). We are then faced with a problem of processing incomplete knowledge.

Overall, uncertainty regarding model parameters may have essentially two origins. It may arise from randomness due to natural variability of observations resulting from heterogeneity (for instance, spatial heterogeneity) or the fluctuations of a quantity in time. Or it may be caused by imprecision due to a lack of information resulting, for example, from systematic measurement errors or expert opinions. As suggested by Ferson and Ginzburg [START_REF] Ferson | Different methods are needed to propagate ignorance and variability[END_REF] and more recently developed by Helton et al. [START_REF] Helton | ÒAn Exploration of Alternative Approaches to the Representation of Uncertainty in Model Predictions[END_REF], distinct representation methods are needed to adequately tell random variability (often referred to as "aleatory uncertainty") from imprecision (often referred to as "epistemic uncertainty").

A long philosophical tradition in probability theory dating back to Laplace demands that uniform distributions should be used by default in the absence of specific information about frequencies of possible values. For instance, when an expert gives his/her opinion on a parameter by claiming: "I only know that the value of x lies in an interval A", the uniform probability with support A is used. This view is also justified on the basis of the "maximum entropy" approach [START_REF] Gzyl | The Method of Maximum Entropy[END_REF]. However, this point of view can be challenged. Adopting a uniform probability distribution to express ignorance is questionable. This choice introduces information that in fact is not available and may seriously bias the outcome of risk analysis in a non-conservative manner [START_REF] Ferson | Different methods are needed to propagate ignorance and variability[END_REF]. A more faithful representation of this knowledge on parameter x is to use the characteristic function of the set A, say π such that π = 1, ∀ x ∈ A and 0 otherwise. This is because π is interpreted as a possibility distribution that encodes the family of all probability distribution functions with support in A (Dubois and Prade [START_REF] Dubois | When upper probabilities are possibility measures[END_REF]). Indeed, there exists an infinity of probability distributions with support in A, and the uniform distribution is just one among them.

In the context of risk evaluation, the knowledge really available on parameters is often vague or incomplete. This knowledge is not enough to isolate a single probability distribution in the domain of each parameter. When faced with this situation, representations of knowledge accepting such incompleteness look more in agreement with the available information. Of course, the Bayesian subjectivist approach maintains that only a standard probabilistic representation of uncertainty is rational, but this claim relies on a betting interpretation that enforces the use of a single probability distribution, in the scope of decision-making, not with a view to faithfully report the epistemic state of an agent (See Dubois, Prade and Smets [START_REF] Dubois | Representing partial ignorance[END_REF] for more discussions on this topic). In practice, while information regarding variability is best conveyed using probability distributions, information regarding imprecision is more faithfully conveyed using families of probability distributions [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF]. At the practical, level such families are most easily encoded either by probability boxes (upper and lower cumulative probability functions [START_REF] Ferson | Construction of Probability Boxes and Dempster-Shafer structures[END_REF] [START_REF] Ferson | Whereof one cannot speak: when input distributions are unknown[END_REF]) or by possibility distributions (also called fuzzy intervals) [START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF][START_REF] Dubois | Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps[END_REF] or yet by belief functions of Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. This article proposes practical representation methods for incomplete probabilistic information, based on formal links existing between possibility theory, imprecise probability and belief functions. These results can be applied for modelling inputs to uncertainty propagation algorithms.

In Section 2, we recall basics of probability-boxes ( upper & lower cumulative distribution functions ), possibility distributions and belief functions. We also recall the links between these representations. All of them can encode families of probability functions. In section 3, the expressive power of probability boxes and possibility distributions is compared. In section 4, some results on the relation between prediction intervals and possibility theory are recalled. It allows a stronger form of encoding of a probability family by a possibility distribution, whereby the cuts of the latter enclose the prediction intervals of the probability functions. In Sections 5 and 6, we consider a non exhaustive list of knowledge types that one may meet after an information collection step in problems like environmental risk evaluation. We especially focus on incomplete non-parametric models, for which only some characteristic values are known, such as the mode, the mean or the median and other fractiles of the distribution. For each case we propose an adapted representation in terms of p-boxes, belief functions (Section 5), and especially possibility distributions (Section 6).

Formal Frameworks for Representing Imprecise Probability

Consider a measurable space (Ω,A) where A is an algebra of measurable subsets of Ω. Let P be a set of probability measures on the referential (Ω,A). For all A ⊆ Ω measurable, we define: its upper probability P(A) = sup P∈P P(A) its lower probability

P(A) = inf P∈P P(A).
Such a family may be natural to consider if a probabilistic parametric model is used but the parameters such as the mean value or the variance are ill-known (for instance they lie in an interval). It can be also obtained if the probabilistic model relies on imprecise (e.g. set-valued) statistics (Jaffray, [START_REF] Jaffray | Bayesian updating and belief functions[END_REF]), or yet incomplete statistical information (only a set of conditional probabilities is available). In a subjectivist tradition, the lower probability P(A) can be interpreted as the maximal price one would be willing to pay for the gamble A, which pays 1 unit if event A occurs (and nothing otherwise) [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF]. Thus, P(A) is the maximal betting rate at which one would be disposed to bet on A. That means P(A) is a measure of evidence in favour of event A. The upper probability P(A) can be interpreted as the minimal selling price for the gamble A, or as one minus the maximal rate at which an agent would bet on A [START_REF] Walley | Statistical Reasoning with Imprecise Probabilities[END_REF]. That means P(A) measures the lack of evidence against A since we have:

P(A) = 1 -P(A c )
It is clear that representing and reasoning with a family of probabilities may be very complex. In the following we consider three frameworks for representing special sets of probability functions, which are more convenient for a practical handling.

In the following we review three modes of representation of uncertainty that can be cast in the imprecise probability model.

Probability boxes

Let X be a random variable on (Ω,A). Recall that a cumulative distribution function is a non decreasing function F : R → [0, 1] assigning to each x ∈ R the value P(X ∈ (-∞, x]). This function encodes all the information pertaining to a probability measure, and is often very useful in practice.

A natural model of an ill-known probability measure is thus obtained by considering a pair (F, F) of non-intersecting cumulative distribution functions, generalising an interval. The interval [F, F] is called a probability box (p-box) [21] [22]. A p-box encodes the class of probability measures whose cumulative distribution functions F are restricted by the bounding pair of cumulative distribution functions F and F such that

F(x) ≤ F(x) ≤ F(x) ∀x ∈ R
A p-box can be induced from a probability family P by:

∀x ∈ R F(x) = P((-∞, x]) ∀x ∈ R F(x) = P((-∞, x])
Let P(P < P) = {P, ∀A ⊆ Ω measurable, P(A) ≤ P(A) ≤ P(A)} be the probability family limited by upper P and lower P probabilities induced from P. Clearly P is a proper subset of P(P < P) generally. Let P(F ≤ F) be the probability family containing P and defined by:

P(F ≤ F) = {P, ∀x, F(x) ≤ F(x) ≤ F(x)}
Generally, P(F ≤ F) strictly contains P(P < P), hence also the set P it is built from. The probability box [F, F] provides a bracketing of some ill-known cumulative distribution function and the gap between F and F reflects the incomplete nature of the knowledge, thus picturing the extent of what is ignored. However, as we shall see, this representation method can be very imprecise.

Basics of Numerical Possibility Theory

Possibility theory [START_REF] Dubois | Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps[END_REF] is relevant to represent consonant imprecise knowledge. The basic notion is the possibility distribution, denoted π, an upper semi-continuous mapping from the real line to the unit interval. A possibility distribution describes the more or less plausible values of some uncertain variable X. Possibility theory provides two evaluations of the likelihood of an event, for instance whether the value of a real variable X does lie within a certain interval: the possibility Π and the necessity N. The normalized measure of possibility Π (respectively necessity N) is defined from the possibility distribution π : R → [0, 1] such that sup x∈R π(x) = 1 as follows:

Π(A) = sup x∈A π(x) (1)
N(A) = 1 -Π(A) = inf x A (1 -π(x)) (2)
• The possibility measure Π verifies :

∀A, B ⊆ R Π(A ∪ B) = max(Π(A), Π(B)) (3)
• The necessity measure N verifies :

∀A, B ⊆ R N(A ∩ B) = min(N(A), N(B)) (4)
A possibility distribution π 1 is more specific than another one π 2 in the wide sense as soon as π 1 ≤ π 2 , i.e. π 1 is more informative than π 2 .

A unimodal numerical possibility distribution may also be viewed as a nested set of confidence intervals, which are the α-cuts

[x α , x α ] = {x, π(x) ≥ α} of π. The degree of certainty that [x α , x α ] contains X is N([x α , x α ]) (= 1 -α if π is continuous).
Conversely, a nested set of intervals A i with degrees of certainty λ i that A i contains X is equivalent to the possibility distribution

π(x) = min i=1...n {1 -λ i , x ∈ A i }
provided that λ i is interpreted as a lower bound on N(A i ), and π is chosen as the least specific possibility distribution satisfying these inequalities [START_REF] Dubois | When upper probabilities are possibility measures[END_REF]. We can interpret any pair of dual functions necessity/possibility [N, Π] as upper and lower probabilities induced from specific probability families.

• Let π be a possibility distribution inducing a pair of functions [N, Π]. We define the probability family P(π) = {P, ∀A measurable, N(A) ≤ P(A)} = {P, ∀A measurable, P(A) ≤ Π(A)}. In this case, sup P∈P(π) P(A) = Π(A) and inf P∈P(π) P(A) = N(A) (see [START_REF] De Cooman | Supremum-preserving upper probabilities[END_REF][START_REF] Dubois | When upper probabilities are possibility measures[END_REF]) hold. In other words, the family P(π) is entirely determined by the probability intervals it generates. • Suppose pairs (interval A i , necessity weight λ i ) supplied by an expert are interpreted as stating that the probability P(A i ) is at least equal to λ i where A i is a measurable set. We define the probability family as follows: P(π) = {P, ∀A i , λ i ≤ P(A i )}. We thus know that P = Π and P = N (see [START_REF] Dubois | When upper probabilities are possibility measures[END_REF], and in the infinite case [START_REF] De Cooman | Supremum-preserving upper probabilities[END_REF]).

Imprecise probability induced by random intervals

The theory of imprecise probabilities introduced by Dempster [START_REF] Dempster | Upper and Lower Probabilities Induced by a Multivalued Mapping[END_REF] (and elaborated further by Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] and Smets [START_REF] Smets | The transferable belief model[END_REF] in a different context) allows imprecision and variability to be treated separately within a single framework. Indeed, it provides mathematical tools to process information which is at the same time of random and imprecise nature. Contrary to probability theory, which in the finite case assigns probability weights to atoms (elements of the referential), in this approach we may assign such weights to any subset, called focal set, with the understanding that portions of this weight may move freely from one element to another in a focal set. We typically find this kind of knowledge when some measurement device is tainted with limited perception capabilities and a random error (variability) due to the variability of a phenomenon. We may obtain a sample of random intervals of the form ([m iδ, m i + δ]) i=1...K supposedly containing the true value, where δ is a perception threshold, m i is a measured value and K is the number of interval observations. Each interval is attached a probability ν i of observing the measured value m i . That is, we obtain a mass distribution (ν i ) i=1...K on intervals, thus defining a random interval. The probability mass ν i can be freely re-allocated to points within interval [m iδ, m i + δ]. However, there is not enough information to do it. Like possibility theory, this theory provides two indicators, called plausibility Pl and belief Bel by Shafer [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. They qualify the validity of a proposition stating that the value of variable X should lie within a set A (a certain interval for example). Plausibility Pl and belief Bel measures are defined from the mass distribution assigning positive weights to a finite set F of measurable subsets of Ω:

ν : F → [0, 1] such that E∈F ν(E) = 1 (5)
as follows:

Bel(A) = E,E⊆A ν(E) (6)
Pl(A) = E,E∩A ∅ ν(E) = 1 -Bel( Ā) (7)
where E ∈ F is called a focal element.Bel(A) gathers the imprecise evidence that asserts A ; Pl(A) gathers the imprecise evidence that does not contradict A.

A mass distribution ν may encode the probability family P(ν) = {P, ∀A measurable, Bel(A) ≤ P(A)} = {P, ∀A measurable, P(A) ≤ Pl(A)} [START_REF] Dempster | Upper and Lower Probabilities Induced by a Multivalued Mapping[END_REF]. In this case we have: P = Pl and P = Bel, so that:

∀P ∈ P(ν), Bel ≤ P ≤ Pl (8)
This view of belief functions is at odds with the theory of evidence of Shafer and the transferable belief model of Smets, who never refer to an imprecisely located probability distribution. Originally, Dempster [START_REF] Dempster | Upper and Lower Probabilities Induced by a Multivalued Mapping[END_REF] considered imprecise probabilities induced from a probability space via a set-valued mapping. In this scope, Bel(A) is the minimal amount of probability that must be assigned to A by sharing the probability weights defined by the mass function among single values in the focal sets. Pl(A) is the maximal amount of probability that can be likewise assigned to A. We may define an upper F and a lower F cumulative distribution function (a particular p-box) such that ∀x ∈ R F(x) ≤ F(x) ≤ F(x) with :

F(x) = Pl(X ∈ (-∞, x]) (9) F(x) = Bel(X ∈ (-∞, x]) ( 10
)
But this p-box contains many more probability functions than P(ν).

The setting of belief and plausibility functions encompasses possibility and probability theories, at least in the finite case:

• When focal elements are nested, a belief measure Bel is a necessity measure, that is Bel = N. A plausibility measure Pl is a possibility measure, that is Pl = Π. • When focal elements are some disjoint intervals, plausibility Pl and belief Bel measures are both probability measures, that is, we have Bel = P = Pl, for unions of such intervals. Thus, all discrete probability distributions and possibility distributions may be interpreted by mass functions. However, continuous belief functions have not received much attention so far (except in the scope of random sets).

The above notions offer a common framework to treat the information of imprecise and random nature. However an obvious question is how to compare the expressivity of p-boxes, possibility distributions and belief functions. As we shall see, a p-box generally contains less information than a belief function and a possibility measure from which this p-box is derived. Possibility measures also offer the capability of approximating confidence intervals. A representation using belief functions is potentially more complex than the two other representation modes because a mass function must be specified for all subsets. However, using only a few focal subsets may be enough in practice. In the next section we focus on the respective expressive power of p-boxes and possibility measures.

Comparative expressivity of probability boxes and possibility distributions

Consider a unimodal continuous possibility distribution π with core {a} (i.e. Π({a}) = π(a) = 1 and ∀ x a, π(x) 1). We assume a unimodal π for simplicity. Results in this Section readily adapt to the case when the core of π is of the form [a, b]. The set of probability measures induced by π, that is, P(π), can be more conveniently described by a condition on the cumulative distribution functions of these probabilities (as first pointed out by Dubois and Prade [START_REF] Dubois | The mean value of a fuzzy number[END_REF]):

Theorem 1 Let π be a unimodal continuous possibility distribution with core {a}.

Then P(π) = {P, ∀x, y, x ≤ a ≤ y, F(x) + 1 -F(y) ≤ max(π(x), π(y))}

Proof. see appendix A

Note that we can choose x and y such that π(x) = π(y) in the expression of P(π), i.e. suppose that [x, y] is a cut of π. If I α is the α-cut of π, it holds that: P(π) = {P, P(I α ) ≥ N(I α ), ∀α ∈ (0, 1]}. Thus by putting ∀x ≤ a, f (x) = sup{y, π(x) ≥ π(y)}, we can prove that [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF]:

P(π) = {P, ∀x ≤ a, F(x) + 1 -F( f (x)) ≤ π(x)}.
Define a particular probability box [F,F] such that:

F(x) = Π(X ∈ (-∞, x]) (11) F(x) = N(X ∈ (-∞, x]). ( 12
)
It is clear that

F(x) = π(x) ∀x such that F(x) < 1 and F(x) = 1 -π(x) ∀x such that F(x) > 0. Define π + (x) =          π(x) f or x ≤ a 1 f or x ≥ a , π -(x) =          π(x) f or x ≥ a 1 f or x ≤ a
the functions π + (x) and 1-π -(x) can be equated to the cumulative distribution functions F and F. The probability box (F, F) = (π + , 1π -) has an important specific feature: there exists a real value a such that F(a) = 1 and F(a) = 0. It means that the p-box contains the deterministic value a, so that the two cumulative distributions are acting in disjoint areas of the real line separated by this value. We can retrieve a possibility distribution from such two cumulative distribution functions as π = min(F, 1 -F) and thus retrieve the possibility distribution that generated the p-box. However it is clear that this process applied to any p-box does not yield a normalized possibility distribution, when the cumulative distributions are too close.

A probability box can be a precise tool for approximating a probability distribution in the latter case, but it then forbids the case where the modelled unknown may be deterministic.

Moreover the two sets of probability functions P(π) and P(F < F) differ. The following results indicate that the former is more precise than the latter (Dubois and Prade [START_REF] Dubois | The mean value of a fuzzy number[END_REF]):

Theorem 2 The probability family encoded by the unimodal continuous possibility distribution π is included in the probability family encoded by the probability box

[F, F] induced from π:

P(π) ⊂ P(F < F) for F = 1 -π -and F = π +
Proof. Let be P ∈ P(π). As lim y→+∞ F(x)+1-F(y) = F(x) and lim y→+∞ max(π(x), π(y)) = π + (x), we obtain according to Theorem 1: F(x) ≤ π + (x). In the same way,

lim x→-∞ F(x) + 1 -F(y) = 1 -F(y) and lim x→-∞ max(π(x), π(y)) = π -(y), thus F(y) ≥ 1 -π -(y)
. Hence we have P ∈ P(F < F).

On the other hand, the other inclusion is false, indeed take for example the triangular possibility distribution π with support [0, 2] and core {1}.

Define P(F < F) = {P, ∀x, 1 -π -(x) ≤ F(x) ≤ π + (x)
} and P a probability measure such that P({0.5}) = 0.5 and P({1.5}) = 0.5. We do have

P ∈ P(F < F), however P P(π) because, for A = (-∞, 0.5] ∪ [1.5, +∞), it holds P(A) = 1 > Π(A) = 0.5.
We can systematize this counterexample and find probability families included in the probability box [F, F] induced by π, which are not present in P(π). The following result improves a previous one due to Dubois and Prade [START_REF] Dubois | The mean value of a fuzzy number[END_REF]:

Theorem 3 Let P be a probability measure in P(F < F) such that:

• There exists γ ≥ a satisfying P((-∞, γ]) = F(γ) (see Figure 1.a) (or P((-∞, γ]) = F(γ), (see Figure 1.b)). • There exists θ ∈ {x ≤ a/F(x) ≤ 1 -F(γ)} such that P((-∞, θ]) 0 (see Figure 1.a) (or θ ∈ {x ≥ a/1 -F(x) ≤ F(γ)} such that P((-∞, θ]) 1

), (see Figure 1.b)).

Then P P(π).

F F F F F F γ γ θ θ 0 0 1 a a Fig. 1. Probabilities in P(F < F) but not in P(π).
Proof. Let P ∈ P(F < F) with cumulative function F. Consider the case where γ ≥ a exists satisfying P((-∞, γ]) = F(γ) (see Figure 1.a).

Using Theorem 1 and the features of F on (-∞, θ],

we have P((-∞, θ] ∪ [γ, +∞)) = F(θ) + 1 -F(γ) > 1 -F(γ) = π(γ) = max(π(θ), π(γ)) Thus, P P(π). Similarly, if there exists γ ≤ a satisfying P((-∞, γ]) = F(γ), (see Figure 1.b); we have F(γ) + 1 -F(θ) > F(γ) = π(γ) = max(π(γ), π(θ)).
The probability box induced by π can thus contain multimodal distributions (if F(θ) = F(γ) for instance), and some unimodal distributions with mode different from a which are ruled out by the probability family encoded by the possibility distribution π.

Theorem 3 identifies a set of probability measures which are not in P(π). Consider π and F such that

F ≤ F = π + . If F is known on (-∞, a],
we can define a lower bound F * of F on [a, +∞) such that the corresponding probability measure P belongs to P(π) if and only if F ≥ F * . Consider the function g : y → min{x ≤ a|π(x) = π(y)}.

From theorem 1, we deduce that F ∈ P(π) if and only if

F(y) ≥ 1 -π(y) + F(g(y)), ∀y ≥ a
The function 1π(y) + F(g(y)) is not necessarily increasing (see Figure 2), we can thus define

F * (y) = max(F(a), 1 -π(y) + F(g(y)
)) for y ≥ a see Figure 2 Conversely if F is known on [a, +∞) an upper bound F * can be found on (-∞, a] such that P ∈ P(π) if and only if

F(x) ≤ F * (x) = min(F(a), π(a) -1 + F( f (x)))
It is clear that sets {P|F(x) ≤ F(x) ∀x ≤ a and F(x) = 1 ∀x ≥ a} and {P|F(x) ≥ 1-π(y)+F(g(y)) on [START_REF] Dou | Steady-state groundwater flow simulation with imprecise parameters[END_REF][START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF] ° F * on [START_REF] Bardossy | 1-, 2-and 3-dimensional modeling of groundwater movement in the unsaturated soil matrix using a fuzzy approach[END_REF][START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF] Fig. 2. Example of probability families included in P(π) induced by the triangular possibility distribution of core {10} and support [START_REF] Bardossy | 1-, 2-and 3-dimensional modeling of groundwater movement in the unsaturated soil matrix using a fuzzy approach[END_REF][START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF] F(x) ∀x ≥ a and F(x) = 0 ∀x ≤ a} are included in P(π). They correspond to probability densities with support [min, a] or limited by [a, max] and their mode is not the mode of π.

Conversely, suppose F < F is the available information, and there exists a real value a such that F(a) = 1 and F(a) = 0. The above results show that the pos-sibility distribution min(F, 1 -F) cannot encompass all probability distributions restricted by the p-box. An obvious example is as follows: consider a probability distribution P, and the probability box (F, F) such that F(x) = F(x) for x < a and 1 otherwise; F(x) = F(x) for x > a and 0 otherwise. The corresponding possibility distribution is π

(x) = F(x) if x < a, π(x) = 1 -F(x) if x > a and π(x) = 1 if x = a.
It can be checked that P P(π) while F < F < F.

Can we find a non-trivial possibility distribution π such that P(π ) contains the probability box [F, F] ? Note that

F(x) ≥ F(x) ≥ F(x) implies ∀x ≤ a ≤ y, F(x) + 1 -F(y) ≤ F(x) + 1 -F(y). Thus, with intervals [x, g(x)], where g(x) is such that F(x) = 1 -F(g(x)), ∀x ≤ a, F(x) + 1 -F(g(x)) ≤ 2F(x). Thus, by letting π + (x) = min(1, 2F(x)) and π -(y) = min(1, 2(1 -F(y)))
we do build a possibility distribution (often very imprecise) π such that P(F < F) ⊂ P(π ).

We may conclude: To represent knowledge using a possibility distribution is more precise than using the upper & lower cumulative distribution functions F, F it induces.

As a consequence, if we seek to estimate the probability P(X ∈ [x, y]) using the probability box [F, F] induced by π, for some x y, we may obtain a bracketing of this probability larger than that the one obtained from the possibility distribution. From the probability box, we can estimate a best envelope of probability P(X ∈ [x, y]) by: max(0,

F(y) -F(x)) ≤ P(X ∈ [x, y]) ≤ F(y) -F(x).
With a similar reasoning as in the proof of theorem 1, we can show ∃ (x, y), x y such that: max(0, F(y) -F(x)) < N([x, y])

Indeed, for x < a < y such that π + (x) > 0 and π -(y) > 0, we have:

max(0, F(y) -F(x)) = max(0, 1 -π -(y) -π + (x)) = 0 if π + (x) + π -(y) > 1; N([x, y]) = 1 -max(π + (x), π -(y)) > 0 It is clear that π + (x) + π -(y) > max(π + (x), π -(y)) which implies N([x, y]) > F(y) - F(x).
Note that the cumulative distributions describing any p-box can be generated by a belief function, contrary to the case of possibility distributions (see Ferson et al. [START_REF] Ferson | Construction of Probability Boxes and Dempster-Shafer structures[END_REF]). So an open interesting problem is to characterize the set of belief functions whose upper and lower cumulative distributions coincide with a (discrete) p-box, and among them the least informative one (for instance maximizing the expected size of the focal elements). This is left for further research.

Approximating probability families by possibility distributions

Let p be a unimodal probability density function. Denote by M the mode of p. Let P be the probability measure associated to p. So far we considered possibility distributions π which verify the following condition (dubbed Dominance condition): An approximate (covering) possibilistic representation of a given family P is any π such that P ⊆ P(π). Clearly it means that π dominates all probability functions in P. Ideally π should be such that P(π) is as small as possible. However such optimal covering approximations of probability families generally do not exist (see Dubois and Prade [START_REF] Dubois | Consonant approximations of belief functions[END_REF]). Nevertheless, in the remainder of the paper we shall lay bare various informative approximate covering possibilistic representations of probability families induced by incomplete probabilistic data.

P(A) ≤ Π(A),
However as previously seen, a possibility measure also encodes a set of nested confidence intervals provided by an expert. A possibility measure π such that P ∈ P(π) can be constructed as follows ( [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF], [START_REF] Dubois | On possibility/probability transformations[END_REF]):

Let J λ = [x(λ), y(λ)], for λ ∈ [0, 1] be a continuous nested interval family such that J λ ⊆ J β if λ ≥ β, J 0 = {x * } ⊂ supp(p)
and J 1 = supp(p) where supp(p) is the support of a unimodal probability density p. Then, the possibility distribution π given by :

π(x(λ)) = π(y(λ)) = 1 -P(J λ ) ∀λ (13)
dominates p. That is p ∈ P(π) (or P ≤ Π). If we choose J λ such that:

J λ = {x, p(x) ≥ λ}, ∀λ ∈ [0, sup(p)]. ( 14
)
J λ is of the form [x, f (x)] where f (x) = max{y, p(y) ≥ p(x)}. Then, J λ is the narrowest prediction interval of probability α = P(J λ ), x * is the mode M of p and J λ is also the most probable interval of length |y(λ)x(λ)| [START_REF] Dubois | Possibility theory, probability and fuzzy sets: misunderstandings, bridges and gaps[END_REF].

Hence, if we choose J λ as in ( 14) and π as in [START_REF] Dubois | The mean value of a fuzzy number[END_REF], we obtain a possibility dis-tribution π p such that Π ≥ P (dominance condition) and the α-cut of π p is indeed the narrowest prediction interval of p of confidence level 1α (prediction interval condition). Such π p is called the optimal transform of p,

π p (x) = 1 -P({y, p(y) ≥ p(x)}) = F(x) + 1 -F( f (x))
and π p (M) = 1. This transformation is optimal in the sense that it provides the most specific possibility distribution among those that dominate p, and preserve the order induced by p on the support interval.

It is clear that function π p is a kind of cumulative distribution. More precisely, given any total ordering of values on the real line, and any value x, let A (x) = {y, x y}, and assume A (x) is measurable for all x. The function F (x) = P(A (x))) is the cumulative function according to the order relation . If => the usual ordering on the real line, then F = F . Now, choosing the ordering induced by the density p, that is, x p y if and only if p(x) > p(y), then F p = π p .

Computing π p is not so obvious in general, but the case of symmetric densities has been considered in [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF]; it is shown that π p is convex on each side of the mode of p. This result no longer holds in the general case, but we can approximate any unimodal density p by means of a piecewise continuous function. Then we can easily show he following result:

Theorem 4 Let p be a unimodal continuous probability distribution function of mode M and of bounded support supp(p). If p is (piecewise) linear, then its optimal

transform is piecewise convex.

Proof. see appendix B

Using the idea of narrowest prediction intervals described above, it is also interesting to characterize approximate covering possibilistic representations of probability families P that account for such prediction intervals. A possibility distribution π is said to strongly dominate a probability measure P with density p if {x; p(x) ≥ λ} = J λ ⊆ {x, π(x) ≥ α} for α = 1 -P(J λ ) (dubbed prediction interval condition). Given a probability family P one may try to find the most specific possibility distribution π P thatstrongly dominates all P ∈ P such a possibility distribution is π P = sup p∈P π p . π P has the peculiarity that any of its α-cuts contain the (1α)-prediction interval of any p ∈ P. Note that this approach is enabled by the property P(π 1 ) ∪ P(π 2 )) ⊆ P(max(π 1 , π 2 )).

Simple Models of Incomplete Probabilistic Knowledge using p-boxes and belief functions

The preceding results can be applied to the definition of faithful representations of poor probabilistic knowledge by means of probability boxes, belief functions and especially possibility distributions. The extreme case is when an expert provides an interval containing the unknown value. Generally there is a little more information than a simple interval: an expert may have an idea on typical values in the interval: the median , the mean, the mode. Additional information on a distribution may be the knowledge of appropriate fractiles and confidence intervals. These pieces of information define constraints restricting a probability family. The problem is whether such family can be simply described or approximated by means of the simple tools described in the previous sections. These representation techniques suggest that simple non-parametric representations of available uncertain knowledge, where incompleteness and variability receive specific treatments, are feasible in the scope of further uncertainty propagation steps in risk analysis problems. This section recalls representation methods proposed by Ferson using p-boxes, when the mean-value of a density is known, and for the modelling of a small set of precise observations by means of imprecise probabilities. Moreover, belief functions can be directly used for exploiting the knowledge of fractiles.

Representations by Probability Boxes

As discussed earlier, probability boxes generalise the idea of interval from a pair of points to a pair of cumulative distribution functions. They are a very natural way of extending the notion of interval. They are especially informative when the two cumulative distributions are close to each other. They come up as a natural choice for imprecise parametric models with imprecise parameters. For instance, a Gaussian model where the mean-value and/or the variance is known to lie in a prescribed interval may naturally yield a narrow p-box (even if the latter contains non-Gaussian distributions). However, we do not deal with parametric models here.

The p-box model has been especially investigated by Ferson et al [START_REF] Ferson | Construction of Probability Boxes and Dempster-Shafer structures[END_REF]. We recall his proposals for representing distributions with fixed mean value as well as for using the Kolmogorov-Smirnov confidence limits in order to derive a p-box from small data samples.

Probability distributions with known mean and support

Suppose an expert supplies the mean µ and the support I = [b, c]. Let P mean I denote the set of probabilities with support I and prescribed mean, equal to µ. Ferson [START_REF] Ferson | Construction of Probability Boxes and Dempster-Shafer structures[END_REF] proposes to represent this knowledge by a probability box [F, F]. To ob-tain it, he separately solves two problems for each value x as follows: F(x) = sup F:E(X)=µ F(x) and F(x) = inf F:E(X)=µ F(x) (the unknown is F). Using the characteristic property of the mean:

F(x) =          x-µ x-b ∀x ∈ [µ, c] 0 ∀x ∈ [b, µ] , F(x) =          1 ∀x ∈ [µ, c] c-µ c-x ∀x ∈ [b, µ]
The probability box [F, F] (see Figure 3 for an example) defines a probability

(x) = 1 -x-µ x-b for x ∈ [µ, c].
But, as expected from the previous sections, the inclusion P mean I ⊂ P(π) does not hold. The probability P, defined by P(X = 2) = 3/5 and P(X = 7) = 2/5, is enough to show we do not have the inclusion. Indeed, we do have E(X) = 4 but P(X = 2 or X = 7) = 1 and Π(X = 2 or X = 7) = 0.6, which is contradictory with P ≤ Π. As pointed out earlier, the probability family P(π) such that π + (x) = min(1, 2F(x)) and π -(y) = min(1, 2(1 -F(y))) (see Figure 4), contains P mean I and P(F, F). However, it is clear that this p-box is poorly informative, and that the covering possibility is even more so. In fact, the mean value does not seem to bring much information on the distribution, and the problem of finding a better, tighter representation of this kind of information remains open. Moreover, while the average value is very easy and often natural to compute from statistical data, it is not clear that this value is cognitively plausible, that is, one may doubt that a single representative value of an ill-known quantity provided by an expert refers to the mean value. For instance, while some quantities like average income can be easily figured out, the average human size sounds like a very artificial notion, and would not be directly perceived by individuals.

Representing small data samples by a p-box

When the available knowledge is just a small data sample (x 1 , ..., x n ) coming from the unknown cumulative distribution function F, Ferson et al. [START_REF] Ferson | Construction of Probability Boxes and Dempster-Shafer structures[END_REF] define a probability box [F, F] by using Kolmogorov-Smirnov confidence limits (noted K.S.) [START_REF] Feller | On the Kolmogorov-Smirnov limit theorems for empirical distributions[END_REF][START_REF] Miller | Table of percentage points of Kolmogorov statistics[END_REF]. These confidence limits are distribution-free bounds about the sample empirical cumulative distribution function F n where n is the size of the sample. We can define F n as follows:

F n (x) =                                  0 for x < x (1) . . . i n for x (i) ≤ x < x (i+1)
. . .

1 for x ≥ x (n)
where x (i) are the order statistics of the sample.

F n and F n converge to the empirical cumulative distribution F n when the sample becomes very large, although convergence is rather slow. Kolmogorov-Smirnov limits require that the samples be independent and identically distributed. This is a very standard assumption, but it is sometimes hard to justify (if the values come from heterogeneous sources, for instance). To obtain these bounds, we use the maximum deviation D KS between F n and F defined as follows:

D KS = max i=1,...,n |F(x (i) ) - i n |, |F(x (i) ) - i -1 n |
D KS is a random variable whose exact distribution is not known but Kolmogorov found that √ nD KS has a limiting distribution given by:

∀ t ≥ 0 lim n→∞ P( √ nD KS ≤ t) = 1 -2 +∞ k=1 (-1) k+1 e -2k 2 t 2
This limit has been tabulated and allows for each confidence level α to find a value D n (α) such that P(D KS ≤ D n (α)) = 1α. To conclude, the K.S. bounds are computed with the expression min(1, max(0, F n (x) ± D n (α)) for a fixed confidence level α. For instance, at 95% confidence level, for a sample size of 10, the value of D n (α) is 0.40925 (see Figure 5). These limits are often used to express the reliability of results of a simulation or to test if the sampling from the simulation follows some probability laws. However, it is not common to use K.S. limits on input parameters to define a probability family respecting the available data. We must be aware that K.S. limits are not sure bounds but statistical ones. It means for instance that 95% of the time the true distribution will lie inside the bounds.

The obtained p-box cannot be obtained from a possibility distribution, as it gener- ally does not include the step-function corresponding to a deterministic value.

Discrete belief functions representations

Very naturally the representation of a family of probabilities by means of a belief function (a discrete random set) is appropriate if the probability of prescribed events is known. This is typically the case when only the median m of a distribution is known. The meaning of the median is: P(X ≤ m) = 0. This representation naturally extends to the case when some fractiles and the support I of the unknown probability distribution function are known. Suppose an expert supplies fractiles, say x 1 , x 2 and x 3 at 5%, 50% and 95%. Denote P x 1 ,x 2 ,x 3 I the set of probability distribution functions of support I = [b, c] and of fractiles x 1 , x 2 , x 3 . We can represent this knowledge in an exact way using a belief function by the following obvious mass function ν f ract : ν f ract ([b, x 1 ]) = 0.05, ν f ract (]x 1 , x 2 ]) = 0.45, ν f ract (]x 2 , x 3 ]) = 0.45 and ν f ract (]x 3 , c]) = 0.05. The belief function Bel f rac , deduced from ν f rac , is dominated by all probabilities with fractiles x 1 , x 2 and x 3 , i.e.,

P x 1 ,x 2 ,x 3 I = {P, ∀C, Bel f rac (C) ≤ P(C)}.
Note that the mass function induced by fractiles bears on a partition of the support. On the contrary if an expert, provides a confidence interval, x ∈ A ⊆ R with a certainty degree λ, the most cautious interpretation corresponds to an inequality P(A) ≥ λ. The corresponding mass function ν assigns λ to A and 1λ to the real line itself. This is called a simple support function by Shafer. Note that the two focal elements are nested. The knowledge of a confidence interval with confidence λ is less precise than a fractile: if A = [x 1 , x 3 ] with confidence at least λ, we cannot deduce the probability degrees associated to intervals (-∞, x 1 ] and [x 3 , +∞), except if we assume the symmetry of the underlying density.

A confidence interval can be represented by the possibility distribution [START_REF] Dubois | When upper probabilities are possibility measures[END_REF]:

∀x ∈ R π(x) =          1 i f x ∈ A 1 -λ i f x A
where π encodes the probability family P(π) = {P, λ ≤ P(A)}. When A is large enough, but, in practice, bounded, the level of confidence is 1. This representation extends when several nested confidence intervals {A 1 ⊂ A 2 ⊂ . . . A k } are obtained for several confidence levels {λ 1 < λ 2 < . . . λ k } as suggested previously. The corresponding mass assignment is ν(A i ) = λ i+1λ i , assuming λ 0 = 0 . It yields a discrete possibility distribution. The next section considers the case of continuous possibility distributions.

Representations by continuous possibility distributions

The use of continuous possibility distributions for representing probability families heavily relies on probabilistic inequalities. Such inequalities provide probability bounds for intervals forming a continuous nested family around a typical value. This nestedness property leads to interpreting the corresponding family as being induced by a possibility measure. While these bounds are usually used for proving convergence properties, we propose here to use them for representing knowledge. This is the case of the Chebyshev inequality for instance. The classical Chebyshev inequality [START_REF] Kendall | The Advanced Theory of Statistics[END_REF] defines a bracketing approximation on the confidence intervals around the known mean µ of a random variable X, knowing its standard deviation σ. The Chebyshev inequality can be written as follows:

P(|X -µ| ≤ kσ) ≥ 1 - 1 k 2 for k ≥ 1
By referring to the Section 2.2, Chebyshev inequality allows to define a possibility distribution π by considering intervals [µkσ, µ + kσ] as α-cuts of π and letting π(µ-kσ) = π(µ+kσ]) = 1/k 2 (see Figure 6). This possibility distribution (see [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF]) defines a probability family P(π) such that P µ,σ ⊆ P(π) containing all distributions with known mean and standard deviation, whether the unknown probability distribution function is symmetric or not, unimodal or not. If it is moreover assumed that the unknown probability distribution is unimodal and symmetric, we can improve the possibility distribution π by means of Camp-Meidel inequality [START_REF] Kendall | The Advanced Theory of Statistics[END_REF] (see Figure 6). Very often, and as seen above, the nested intervals share the same midpoints, thus yielding symmetric possibility distributions. In the following we do not make this restriction. On the contrary, we shall also rely on the most narrow intervals of fixed confidence levels as introduced earlier in this paper. It leads to exploit information on the mode of distributions rather than the mean. Moreover we make the additional assumption that the distributions have a bounded support. Some assumptions can be made on the shape of the density (without going to the point of choosing a particular mathematical model like a Gaussian): symmetry, convexity or concavity can be assumed, for instance.

P(|X -µ| ≤ kσ) ≥ 1 - 4 9k 2 for k ≥ 2 3 18

Distributions with known mode and support: simple dominance

Suppose the mode M and the support I of the unknown probability distribution function p is supplied by an expert. In this section unimodality of distributions is assumed. One might argue that the mode best corresponds to the notion of usual value, as being the most frequently observed value. Even if the mode is known to be difficult to extract from a sample of statistical data, one may consider that the most frequent value (or a most frequent small range of values) is the natural feature extracted from repeated observations by humans. So the problem of representing this kind of knowledge looks natural. We can take advantage of the fact that the cumulative distribution function F, associated to a unimodal (asymmetric) probability distribution function p with mode M and bounded support I, satisfies the following properties:

• F is convex on [inf(I), M] since p increases on [inf(I), M]. • F is concave on [M, sup(I)] since p decreases on [M, sup(I)].
Thus, the concavity of F changes at M. Let P M I be the set of probabilities with support I = [b, c] and with mode M. Ferson (in [START_REF] Ferson | Whereof one cannot speak: when input distributions are unknown[END_REF]) proposes to represent this knowledge by the probability box [F L , F L ] such that

• F L (x) = x-b M-b for x ∈ [b, M] and 1 otherwise. • F L (x) = x-M c-M for x ∈ [M, c] and 0 otherwise.
Indeed it is obvious that any probability distribution with mode M and support I is such that

F L > F > F L .
Theorem 5 The triangular possibility distribution π L = min(F L , 1 -F L ), with support [b, c] and core {M} dominates all probabilities lying in P M I .

Proof. consider the nested family of intervals [x, y] such that:

(x -b) M -b = (c -y) c -M .
They are cuts of the triangular possibility distribution π L . Define the cumulative distribution F L as follows:

F L (x) = F(M)(x-b) M-b
for x ≤ M, and

F L (x) = F(M) + (x-M)(1-F(M)) c-M
for x ≥ M. Due to the convexity of any F before the mode, and its concavity after the mode, it is clear that F L (x) ≥ F(x) for x ≤ M, and F L (x) ≤ F(x) for x ≥ M. Using [START_REF] Dubois | The mean value of a fuzzy number[END_REF] in Section 4, it is clear that:

∀(P, x) ∈ P M I × [b, M], P([x, y] c ) = F(x) + 1 -F(y) ≤ F L (x) + 1 -F L (y) = F(M)(x-b) M-b + 1 -(F(M) + (1-F(M))(y-M) c-M ) = x-b M-b = π L (x)
So it holds that Π L (A) ≥ P(A) ∀A, ∀P ∈ P M I .

Clearly this result corresponds to a Chebyshev-like probabilistic inequality built from the α-cuts of π L . The triangular possibility distribution π L of mode M is thus a more precise representation than the p-box [F L , F L ]. Namely, the probability family P(π L ) is a better approximation of P M I than the probability box [F, F] proposed by Ferson. Note that the assumption of bounded support is crucial in getting this piecewise linear representation. Moreover it is noticeable that this distribution does not depend on the value F(M).

Suppose now this value is known. Let P M,F(M) I be the set of probabilities with support I = [b, c], with mode M and value F(M) at M. The latter information can be modelled by a belief function (see Section 2.3) but we may wish to preserve a possibilistic representation and alter its shape so as to account for this fractile, still ensuring the Dominance condition. Assume F(M) ≤ 0.5. We choose nested intervals J x = [x, F -1 (1 -F(x))] around the median, and let

M = F -1 (1 -F(M)). We have seen F is concave on [M, c]. So, F > F L on [M, c]. Hence M ≤ M L = 1 1 -F(M) {(c -M)(1 -F(M)) -cF(M) + M}
Now we can use π p (x) = 1 -P(J x ) as a possibility distribution dominating p. Let

J x = [x, y].
The following possibility distribution π L,F(M) can then be used in place of π L :

• For x ∈ [b, M], π p (x) = 2 • F(x), and π p is convex. So π p (x) ≤ 2 • F L (x). So we let π L,F(M) (x) = 2 • F L (x). • For y ∈ [M, M L ], π p (y) = F(x) + 1 -F(y) ≤ F(M) + 1 -F(y), and the latter is convex. We let π L,F(M) (y) = F(M) + 1 -F L (y). • For y ∈ [M L , c], π ( y) = 2 • (1 -F(y)) is convex. π p (y) ≤ 2 • (1 -F L (y)). So we let π L,F(M) (y) = 2 • (1 -F L (y))
Thus, we have P M,F(M) I ⊆ P(π L,F(M) ). The obtained shape is more realistic than the triangular fuzzy interval especially when M is the center of I, because the lack of balance of the probability mass is reflected on the possibility distribution (see Figure 7). In the case where F(M) = 0.5, we obtain M = M and we thus retrieve the triangular π L with a support [b, c] and a core {M}. Note that it cannot be refined by exploiting the fact that both π L and the above derived π L,F(M) dominate P M,F(M) I to refine the result, considering min(π L,F(M) , π L ) as a tighter approximant. Indeed, as pointed out earlier min(π L,F(M) , π L ) will not dominate P M,F(M) I generally, as P(min(π L,F(M) , π L )) differs from P(π L,F(M) ) ∩ P(π L )

Accounting for fractiles in the continuous possibilistic representation

π m L (m) = π m L (m) = 0.5 + (1 -F(M)) m -b M -b where (m-b) M-b = (c-m) c-M .
Note that this possibility distribution π m L depends on F(M), and that if M > m, the inequality

F(M) ≥ M-b 2(m-b) holds, since π m L ≤ π L . When F(M) = M-b 2(m-b)
, the triangular possibility distribution π L is retrieved, for instance when the mode and the median coincide (F(M) = 0.5). If F(M) = 1 (the most asymmetric case) then π m L (m) = 0.5. Exploiting this representation needs an estimation of F(M). But this quantity is a good measure of the asymmetry of the distribution. This result is easily extended to any other fractiles, or any set of fractiles if they are known a priori. In particular, consider the case where an expert gives fractiles, say x 1 , x 2 and x 3 , at 5%, 50% and 95%, on top of the mode M. By definition x 2 is the median, and suppose that it coincides with the mode. Let P x 1 ,x 2 ,x 3 I be the probability family having these fractiles defined in the Section 5.2. With the same reasoning as above, we can represent this knowledge by the following (symmetric) possibility distribution : π(x 1 ) = π(x 3 ) = F(x 1 ) + 1 -F(x 3 ) = 0.1, π(x 2 ) = 1 and linear interpolations on [b, x 1 ], [x 1 , x 2 ], [x 2 , x 3 ] and [x 3 , c] for other values of π(x). Clearly P x 1 ,x 2 ,x 3 I ∪ P M,m ⊂ P(π) (respecting the Dominance condition defined in Section 4).

Distributions with known mode and support: bracketing prediction intervals

Suppose that I = [b, c] contains the support of the unknown probability distribution function p and the symmetry of p is assumed. Let P S I be the set of such probabilities. Their mode is b+c 2 due to symmetry (but it includes the uniform probability on I). If p is symmetric, the optimal transform π p around the mode is convex on each side of the mode [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF]. The symmetric triangular possibility distribution π S with support I and core b+c 2 is thus such that π S ≥ π p , ∀p, and is really equal to sup p∈P S I π p [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF]. So not only do P(π S ) contain P S I but also, the α-cuts of π S braket the narrowest prediction intervals of these probabilities. Nevertheless, P(π S ) also contains probability densities that are not symmetric and whose mode differ from b+c 2 (but they do not bracket their prediction intervals, necessarily). One may argue that the pbox [F , F ] defined by F (x) = x-b c-b if x ≤ b+c 2 , and 1 otherwise,

F (x) = x-b c-b if x ≥ b+c 2 ,
and 0 otherwise, is a more informative representation of symmetric densities with support in I. But note that in this case, the possibility distribution π(x) = min(F (x), 1 -F (x)) if x b+c 2 is also dominating all such symmetric distributions, and is even more precise than the p-box. But of course, it cannot bracket their prediction intervals. The specific merit of π S is precisely to bracked the prediction intervals in P S I , Interestingly, note that π S = 2 • min(F , 1 -F ) for x b+c 2 . If we know some fractiles, we can refine the representation as explained in the previous section. Such refinements would respect the prediction interval condition (see Figure 8) due to the symmetry assumption.

When p is asymmetric, the optimal transform π p , associated to p may fail to be convex on each side of the mode M. So the α-cuts of the triangular possibility distribution π L with core {M} do not always contain the optimal (1α)-prediction intervals of the probability measures of mode M, as clear from theorem 4 on the optimal transforms of piecewise linear densities. For instance consider the example on Figure 9 suggested in [START_REF] Dubois | Probability-possibility transformations, triangular fuzzy sets and probabilistic inequalities[END_REF], where: The interval [-1.4, 1.4] corresponding to the α-cut equal to 0.3 of the triangular possibility distribution does not contain the optimal 0.7-prediction interval of probability measure of mode 0, which is [-1.5, 1.5] : the optimal transform of p (in Section 4) is indeed not convex everywhere.

We can nevertheless find an upper bound of π p for a unimodal asymmetric continuous density p. Then, using the concavity of F and considering nested intervals J x = [x, max{y, p(y) ≥ p(x)} = f (x)] we have:

• For x ≤ M, π p (x) ≤ F(x)+1-F( f (x)) ≤ F L (x)+1-F(M) = F(M)(x-b) M-b +1-F(M). • For x ≥ M, π p (x) ≤ F( f -1 (x)) + 1 -F(x) ≤ F(M) + 1 -F L (x) = 1 -1-F(M) c-M (x -M)
Knowing the value F(M) is necessary to be able to define this approximation (see Figure 10 for instance). In general, it will be difficult to come up with a more informative possibility distribution which accounts for the prediction intervals of all probability measures on an interval I with fixed mode, due to the wide range of such distributions. Some additional assumptions must be made, for instance on the convexity-concavity of the unknown probability density function p. The assumption F(M) < 0.5 is consistent with the convexity of p on ]b, M[ and its concavity on ]M, c[. In this case, a possibility distribution linearly increasing from 0 to 1 on [b, M] covers all optimal transforms of such densities on this side. On the other side of the mode, using a linear shape is possible with π(c) = 1 -F(M) (see Figure 10). In summary, assuming F(M) is known and the assumption of theorem 5 on convexity & monotony of p holds, then a more informative possibility distribution whose cuts contain the confidence intervals of distributions of mode M having such characteristics can be computed.

Conclusion

The notion of imprecise probability offers a natural formal framework for representing imprecise knowledge on numerical quantities. Several types of information can be approximated by means of possibility distributions, others are directly and exactly representable by belief functions, yet other more naturally fit the probability box framework. In several cases, possibility distributions provide a concise approximate representation of a set of probability measures, sometimes interpretable in terms of confidence intervals of probabilities in such families. In fact each mode of representation seems to be adapted to the knowledge of specific characteristics of distributions. Only p-boxes seem to capture information about mean values in a reasonable way. Belief functions directly model fractile information, while possibility measures are particularly well-suited for representing families of distributions whose mode is known, and can integrate additional information on symmetry and concavity of densities, as well as known fractiles. The recent works of Neumaier [START_REF] Neumaier | Clouds, Fuzzy Sets and Probability Intervals[END_REF] focus on probability families P of the form P = P(π) ∩ P(1ρ) where π is a possibility distribution and ρ is a function I → [0, 1] acting as a lower bound of π, i.e. ρ ≤ π. The probability family P(π) = P is recovered when ρ = 0. The probability family P is more precise than P(π) and assessing its potential demands more future investigations.

Our representation tools using possibility theory are currently applied to risk management problems [START_REF] Baudrit | Joint Treatment of imprecision and Randomness in Uncertainty Propagation[END_REF][START_REF] Baudrit | Comparing Methods for Joint Objective and Subjective Uncertainty Propagation with an example in a risk assessment[END_REF]. In such problems, straightforward Monte-Carlo methods involve too rich assumptions of complete probabilistic knowledge and stochastic independence between parameters. Moreover, uncertainty due to variability and uncertainty due to incomplete knowledge are mixed up in the resulting distribution. In contrast, Bardossy et al. [START_REF] Bardossy | 1-, 2-and 3-dimensional modeling of groundwater movement in the unsaturated soil matrix using a fuzzy approach[END_REF], Bárdossy and Fodor [START_REF] Bárdossy | Evaluation of Uncertainties and Risks in Geology: New Mathematical Approaches for their Handling[END_REF], Dou et al. [START_REF] Dou | Steady-state groundwater flow simulation with imprecise parameters[END_REF][START_REF] Dou | Numerical solute transport simulation using fuzzy sets approach[END_REF] present applications of possibility theory to propagate imprecise information in environmental models. However a proper handling of real cases requires the propagation of heterogeneous uncertain information where imprecision and variability of parameters are separately accounted for and propagated through numerical models. Guyonnet et al. [START_REF] Guyonnet | Hybrid Approach for Addressing Uncertainty in Risk Assessments[END_REF] (see also Bárdossy and Fodor [START_REF] Bárdossy | Evaluation of Uncertainties and Risks in Geology: New Mathematical Approaches for their Handling[END_REF]) propose a method for the joint propagation of fuzzy intervals and probabilistic numbers. This method is further elaborated in [START_REF] Baudrit | Joint Propagation and Exploitation of Probabilistic and Possibilistic Information in Risk Assessment Models[END_REF][START_REF] Baudrit | Joint Treatment of imprecision and Randomness in Uncertainty Propagation[END_REF]. Various joint possibility-probability propagation techniques are compared in [START_REF] Baudrit | Comparing Methods for Joint Objective and Subjective Uncertainty Propagation with an example in a risk assessment[END_REF], some involving independence assumptions, other ones, more conservative, avoiding such assumptions. Comparison with p-box propagation is also made. For a stimulating discussion of various uncertainty propagation techniques, using random intervals imprecise probability and possibility theory, see [?].

The unified representation framework proposed here makes it easy to represent poor data of various types in a faithful and yet simple way. It facilitates the definition of a uniform mode of propagation in risk management, in spite of the heterogeneous character of the data collected and the computation of conservative estimates, something that is not allowed by traditional probabilistic methods.

P(A) ≥ N(A).

B Proof of theorem 4

First, we show that the optimal transform of a triangular density function p, is convex on each side of the mode M (see

π p (x) = F (x) -f (x)F ( f (x)) = p 1 (x) -f (x)p 2 ( f (x))
However p 1 (x) = p 2 ( f (x)), thus:

π p (x) = p 1 (x) 1 -f (x)
Hence differentiating again:

π p (x) = p 1 (x) 1 -f (x) -p 1 (x) f (x)
We know that p 1 (x) = p 2 ( f (x)); if we differentiate this equality, we obtain:

f (x) = p 1 (x) p 2 ( f (x))
The function p 1 increases on ]b, M[, then p 1 ≥ 0. The function p 2 strictly decreases on ]M, c[, then p 2 < 0. We thus deduce that f ≤ 0 ≤ 1. We conclude that:

( 1 -

 1 F(y))dyone obtains the following results

Fig. 3 .

 3 Fig. 3. Probability box built from x ∈ [2, 7] and E(X) = 4.

Fig. 4 .

 4 Fig. 4. Possibility distribution π containing the probability box [F, F].

10 FFFig. 5 .

 105 Fig. 5. Kolmogorov-Smirnov confidence limits (gray) about an empirical cumulative distributino function (black) assuming a sample size of 10.

5 .

 5 Let P med I be the set of probability functions with support I = [b, c] and with median m. This knowledge can be exactly represented by a mass function ν m such that ν m ([b, m]) = ν m (]m, c]) = 0.5. The belief function Bel m , deduced from ν m , encodes all probabilities with median m, i.e., P med I = {P, ∀C, Bel m (C) ≤ P(C)}.

Fig. 6 .

 6 Fig. 6. Optimal possibility distribution π p knowing µ = 2 and σ = 1 and using Bienaymé-Chebychev and Camp-Meidel inequality.

Fig. 7 .

 7 Fig. 7. M=4, F(M) = 0.4, min = 0 and max = 10.

  Suppose the expert provides the mode M and the median m of the probability distribution. Let P M,m I be the set of such unimodal probability functions bounded by I = [b, c] and assume m < M. Then we can refine the possibilistic approximation π L by accounting for the additional information on the median, namely that F(m) = 0.5. It means that F goes through the point of coordinates (m, 0.5). So, instead of F L , we can consider the piecewise cumulative distribution F m L made of the segments [(b, 0), (m, 0.5)], [(m, 0.5), (M, F(M))], [(M, F(M)), (c, 1)]. Clearly, F ≤ F m L < F L on [b, M]. Hence by choosing again the intervals [x, y] such that (x-b) M-b = (c-y) c-M , we obtain a more specific piecewise linear possibility distribution π m L ≤ π L which dominates all probability distributions with mode M and median m. That is, P M,m I ⊂ P(π m L ). In particular :

Fig. 8 .

 8 Fig. 8. Expert gives fractiles on 5%, 50% and 95% equal to 1, 5 and 9 on [0,10].

  p(x) = 0.6x + 1.2 on [-2, -1.5] p(x) = (0.2/3)x + 0.4 on [-1.5, 0] p(x) = -0.2x + 0.4 on [0, 2].

Fig. 9 .

 9 Fig. 9. Optimal transformation of p around the mode.

Theorem 6

 6 If the density function p is convex increasing on ]b, M[ and concave strictly decreasing on ]M, c[, then π p is also convex on ]b, M[. Proof. see appendix C.

Fig. 10 .

 10 Fig.10. the upper bound of π p and its improvement when the concavity-convexity of p is know for M=0 and F(M) = 0.4.

Fig. B. 1 .Fig. B. 3 .= 1 Fig. B. 4 .

 1314 Fig. B.1. Triangular probability density p on the left and the shape of its π p optimal transformation on the right. Now let p be piecewise linear and λ 1 ≤ λ 2 ≤ • • • ≤ λ n be the ordinates of the points where the slope changes. In particular p(M) = λ n and p(b) = p(c) = λ 1 = λ 2 = 0. For illustration, we picture the case where the density p is linear on 4 intervals [b = min(supp(p)), a 2 ], [a 2 , M], [M, a 4 ] ,[a 4 , c = max(supp(p))] (see Figure B.2). Consider index i such that λ i < λ i+1 , λ ∈ [λ i , λ i+1 ]. Denote [b i , b i ] and [b i+1 , b i+1 ] the intervals whose end-points have ordinates λ i and λ i+1 , and [x, y] such that p(x) =

p 1 (

 1 x)(1f (x)) ≥ 0, ∀x ∈]b, M[ By differentiating again f , we obtain f (x) = p 1 (x) -( f (x)) 2 p 2 ( f (x)) p 2 (x)We know that p is convex on ]b, M[ (resp. concave on ]M, c[), we have p 1 (x) ≥ 0 for all x ∈]b, M[ (resp. p 2 (x) ≤ 0 for all x ∈]M, c[). Hence, p 1 (x)-( f (x)) 2 p 2 ( f (x)) ≥ 0 for all x ∈]b, M[ and thus f (x) ≤ 0 for all x ∈]b, M[. We thus conclude that p 1 (x) f (x) ≤ 0, ∀x ∈]b, M[.

  for all measurable events A We say that the possibility measure dominates the probability measure and it means P ∈ P(π).More generally, π dominates a probability family P if and only if P ⊆ P(π). It holds that P(max(π 1 , π 2 )) is the convex closure of P(π 1 ) ∪ P(π 2 ), since the possibility distribution max(π 1 , π 2 ) generates the possibility measure max(Π 1 , Π

2 ). On the contrary, P(min(π 1 , π 2 )) P(π 1 ) ∩ P(π 2 ) in general, because min(Π 1 , Π 2 ) is not a possibility measure. So, if a probability family P is dominated by two possibility distributions π 1 , π 2 , one cannot deduce that P is dominated by min(π 1 , π 2 ), even if min(π 1 , π 2 ) is normalized.

Acknowledgements

This work is supported by the French Institutes B.R.G.M, I.R.S.N and I.N.E.R.I.S.

A Proof of theorem 1 ⊆: Let be P ∈ P(π) and an interval A such that A = [x, y] containing a.

By definition, N(A) ≤ P(A) is equivalent to

y] with y ≤ a, knowing that F is increasing and according to case (a), we have F(y) -F(x) ≤ F(y) ≤ π + (y). Hence P(A) ≤ Π(A). (d) For A = [x, y] with x ≥ a, knowing that F is limited by 1 and according to case (b), we have

. (e) For A, union of intervals such that Π(A) < 1. Suppose Π(A) is obtained for some y which lies on the right side of a. We may consider a set A = (-∞, x] ∪ [y, +∞) such that π(x) = π(y). Necessarily, A contains A, and we have Π(A) = Π(A ) = π(x) and P(A) ≤ P(A ). We have x ≤ a ≤ y ,thus

We then have P(A) ≤ Π(A). (f) For A, union of intervals such that Π(A) = 1, choose y on the boundary of A such that π(y) is maximal. Suppose that y is on the right of a, we can consider a set as A = [x, y] ⊂ A such that π(x) = π(y). We have Π(A) = Π(A ) = 1 and

y). we then have, N(A) = N(A ) ≤ P(A ) ≤ P(A), thus

To summarize, we have p 1 (x)(1f (x)) ≥ 0 and p 1 (x) f (x) ≤ 0, ∀x ∈]b, M[ We thus have proved that π p is positive on ]b, M[, and hence the convexity of π p on ]b, M[.


Maria Patrizia Carrieri

Camelia Protopopescu
email: camelia.protopopescu@inserm.fr

Zobair Younossi

Antoine Vilotitch

Hélène Fontaine

Ventzislava Petrov-Sanchez

Marc Marcellin

Fabrice Carrat

Christophe Hézode

Marc Bourlière

Protopopescu C Younossi Z Et

Fabienne Marcellin

Younossi Z Et Al

Health-Related Quality of Life in Chronic HCV-Infected Patients Switching to Pegylated-Interferon-Free Regimens (ANRS CO20 CUPIC Cohort Study and SIRIUS Trial)

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Key Points for Decision Makers

This study examines health-related quality of life (HRQL) in HCV-infected patients switching, after treatment failure, from a pegylated-interferon (PEG-IFN)-based HCV treatment (ANRS CO20 CUPIC cohort) to a PEG-IFN-free treatment (SIRIUS trial).

The results highlight a relative improvement in both physical and mental HRQL during HCV treatment, but no major improvement in physical HRQL after treatment end, when comparing PEG-IFN-free regimens with PEG-IFN-containing regimens. This suggests that in the PEG-IFNfree regimens era, screening and comprehensive care of comorbidities and residual somatic symptoms during treatment, and especially after HCV clearance, are still needed to improve patient outcomes.

Introduction

For several decades HCV chronic infection was regarded as an incurable disease, leading ineluctably to the death of a great many people from liver cirrhosis or liver cancer sequelae. Pegylated-interferon (PEG-IFN) plus ribavirin (RBV) was the first regimen to result in promising HCV clearance rates, although its effectiveness was strongly linked to HCV genotype and continuity of treatment (persistence). In 1999, the European Consensus Conference on Hepatitis C established that the gold standard for treatment of chronic hepatitis C was a combination of PEG-IFN and RBV [START_REF] Palumbo | PEG-interferon in acute and chronic hepatitis C: a review[END_REF]. However, increasing access to treatment and ensuring adherence to the treatment regimen were a challenge with this combination. The burden of anemia and/or influenza-like symptoms, as well as psychiatric side-effects such as depression, fatigue and anger [START_REF] Marcellin | Adding HCV treatment to HIV treatment in HIV-HCV coinfected patients: the impact on the different dimensions of fatigue and self-reported side effects[END_REF][START_REF] Marcellin | Using patientreported outcomes to improve the management of co-infection with HIV and HCV: the ANRS CO13 HEPAVIH cohort[END_REF], represented a major barrier to initiation and continuity of treatment [START_REF] Bonner | Self-efficacy and adherence to antiviral treatment for chronic hepatitis C[END_REF][START_REF] Hauber | Patient preferences and assessment of likely adherence to hepatitis C virus treatment[END_REF][START_REF] Broers | Barriers to interferon-alpha therapy are higher in intravenous drug users than in other patients with acute hepatitis C[END_REF]. HCV-related fatigue and treatment-related side effects negatively affected health-related quality of life (HRQL) and were identified as predictive factors of treatment discontinuation [START_REF] Guadagnino | Does depression symptomatology affect medication compliance during the first weeks of anti-HCV therapy in intravenous drug users? Digestive and liver disease : official journal of the Italian Society of Gastroenterology and the Italian Association for the Study of the[END_REF].

The arrival of the first protease inhibitors -boceprevir and telaprevir -in 2012, which were administered in combination with PEG-IFN and RBV, improved rates of sustained virological response (SVR) in genotype 1 subjects [START_REF] Hezode | Triple therapy in treatment-experienced patients with HCV-cirrhosis in a multicentre cohort of the French Early Access Programme (ANRS CO20-CUPIC) -NCT01514890[END_REF], but increased toxicity (cytopenia, anemia, asthenia, myalgia, skin rash) [START_REF] Chopra | How to optimize HCV therapy in genotype 1 patients: management of side-effects[END_REF] resulting in treatment discontinuation. This highlighted the need for appropriate somatic and psychiatric clinical management. These regimens were also associated with significant impairment of HRQL [10,[START_REF] Vera-Llonch | Health-related quality of life in genotype 1 treatment-naive chronic hepatitis C patients receiving telaprevir combination treatment in the ADVANCE study[END_REF].

One of the new Direct-Acting Antiviral Agents (DAA) against HCV is sofosbuvir (SOF), a nucleotide analog inhibitor of the NS5B polymerase enzyme, approved in June 2013 by the US Food & Drug Administration (FDA) [START_REF]Food and Drug Administration Approves Gilead's Sovaldi™ (Sofosbuvir) for the Treatment of Chronic Hepatitis C[END_REF] and in January 2014 by the European MEdicines Agency (EMEA) [START_REF]European Commission Grants Marketing Authorization for Gilead's Sovaldi® (Sofosbuvir) for the Treatment of Chronic Hepatitis C Infection[END_REF] for different treatment combinations. It may be used with PEG-IFN and RBV for 12 weeks or with RBV only for 12 or 24 weeks, according to the HCV genotype. SOF + RBV was the first interferon-free HCV treatment regimen available, and is also recommended for patients with severe fibrosis. Carrieri MP., Protopopescu C., Younossi Z. et al SOF is characterized by high efficacy and tolerance, which result in high response rates (depending on HCV genotype, extent of liver damage and previous treatment history) and few side-effects, especially when not combined with PEG-IFN. HRQL data for SOF regimens with and without PEG-IFN and/or RBV have recently been published. These data show that PEG-IFNcontaining SOF regimens significantly impaired patients' HRQL (as captured by the Medical Outcomes Study 36-item Short-Form questionnaire (MOS SF-36)) [START_REF] Younossi | Minimal impact of sofosbuvir and ribavirin on health related quality of life in chronic hepatitis C (CH-C)[END_REF]. Moreover, the addition of SOF to the treatment regimen did not add to the HRQL burden of PEG-IFN + RBV. In contrast, PEG-IFN-free regimens with SOF + RBV resulted in a slight impairment of HRQL, which was not related to treatment duration [START_REF] Younossi | Minimal impact of sofosbuvir and ribavirin on health related quality of life in chronic hepatitis C (CH-C)[END_REF]. It has also been shown that other patient-reported outcomes, including work productivity, were more negatively affected in patients receiving SOF with PEG-IFN than in those on SOF regimens without PEG-IFN [START_REF] Younossi | Effects of sofosbuvir-based treatment, with and without interferon, on outcome and productivity of patients with chronic hepatitis C. Clinical gastroenterology and hepatology : the official clinical practice[END_REF].

The objective of this study was to compare, for the first time, HRQL in patients receiving boceprevir or telaprevir triple combination PEG-IFN-containing therapy (ANRS CO20 CUPIC cohort study [START_REF] Hezode | Triple therapy in treatment-experienced patients with HCV-cirrhosis in a multicentre cohort of the French Early Access Programme (ANRS CO20-CUPIC) -NCT01514890[END_REF]) who subsequently switched to SOF combined with a NS5A inhibitor (ledipasvir (LDV)) PEG-IFN-free therapy with or without RBV (SIRIUS trial [START_REF] Bourliere | Ledipasvir-sofosbuvir with or without ribavirin to treat patients with HCV genotype 1 infection and cirrhosis non-responsive to previous protease-inhibitor therapy: a randomised, doubleblind, phase 2 trial (SIRIUS)[END_REF]). The comparison focuses specifically on patient HRQL during HCV treatment in a first analysis, and after the end of treatment in a second analysis.

Methods

Study Design

Analyses are based on data collected from HCV-chronic patients who consecutively participated for a new drug. The study prospectively followed 674 patients who received triple therapy with PEG-IFN, RBV and either boceprevir or telaprevir for 48 weeks, with approval from the French Health authorities [START_REF] Hezode | Triple therapy in treatment-experienced patients with HCV-cirrhosis in a multicentre cohort of the French Early Access Programme (ANRS CO20-CUPIC) -NCT01514890[END_REF]. Drug safety profiles and virological response to combined treatment were assessed [START_REF] Hezode | Triple therapy in treatment-experienced patients with HCV-cirrhosis in a multicentre cohort of the French Early Access Programme (ANRS CO20-CUPIC) -NCT01514890[END_REF]. SIRIUS (ClinicalTrials.gov identifier: GS-US-337-0121) is a phase II multicenter, randomized, double-blind trial [START_REF] Bourliere | Ledipasvir-sofosbuvir with or without ribavirin to treat patients with HCV genotype 1 infection and cirrhosis non-responsive to previous protease-inhibitor therapy: a randomised, doubleblind, phase 2 trial (SIRIUS)[END_REF], which evaluated the efficacy and safety of interferon-free SOF/LDV treatment with or without RBV, in treatment-experienced patients with compensated cirrhosis.

Eligible patients were genotype 1 individuals with prior virological failure after both PEG-IFN + RBV regimen and triple combination therapy (containing either telaprevir or boceprevir). The 154 patients enrolled in SIRIUS were randomized into the following two arms: 24-week treatment with SOF/LDV as a fixed-dose combination (FDC) with placebo RBV (arm 1: "FDC + Placebo RBV", n=78 patients), and 12-week treatment with placebo, followed by 12-week treatment with FDC and RBV (arm 2: "Placebo -> FDC + RBV", n=76 patients).

Among the 154 patients enrolled in SIRIUS between 21st October and 28th November 2013, 47 were identified as having previously participated in the CUPIC cohort (CUPIC enrolment date between March 2011 and April 2012). According to SIRIUS eligibility criteria, none of these 47 patients had achieved sustained virological response (SVR) after treatment from CUPIC, and none had either hepatocellular carcinoma or decompensated hepatic disease. All these patients achieved SVR by the end of the treatment in SIRIUS trial. The analyses of this study will use data from these 47 patients.

Assessment of HRQL

In the CUPIC study, patient HRQL was assessed using the Medical Outcomes Study 12-item Short-Form questionnaire (MOS SF-12), a short version of the MOS SF-36 questionnaire [START_REF] Ware | Interpreting SF-36 summary health measures: a response[END_REF], which has been widely used for studies on HCV-infected patients in France and other European countries [START_REF] Dacosta Dibonaventura | The burden of viral hepatitis C in Europe: a propensity analysis of patient outcomes[END_REF]19]. Patients answered the SF-12 at several time points during follow-up, as Carrieri MP., Protopopescu C., Younossi Z. et al follows: baseline (Week 0), during treatment (Weeks 4, 8, 12, 24, 36, and 48), and after treatment (Week 72, i.e., 24 weeks after treatment end).

A similar self-administered questionnaire including the MOS SF-36 questionnaire [START_REF] Ware | Interpreting SF-36 summary health measures: a response[END_REF] was proposed to participants in the SIRIUS trial during and after the treatment period: at the baseline visit (Week 0), during treatment (Weeks 4, 12, 16, 24), and after the end of treatment (Weeks 28, 36, 48, i.e. respectively 4, 12 and 24 weeks after treatment end) [START_REF] Younossi | The impact of ledipasvir/sofosbuvir on patient-reported outcomes in cirrhotic patients with chronic hepatitis C: the SIRIUS study[END_REF].

Two HRQL aggregate scores -the physical component summary (PCS) for physical HRQL and the mental component summary (MCS) for mental HRQL -were calculated in both studies using the 12 items common to the SF-12 and the SF-36 questionnaires. These summary scores were calculated using the U.S. population norms to the mean of 50 and a standard deviation of 10, over a range of 0 to 100 points, with higher values denoting better HRQL.

Statistical Analyses

Given the repeated-measure design of the study and the presence of two arms in the SIRIUS trial with different treatment durations and follow-up schedules, for the purpose of standardization, analyses were restricted to the data collected at the following selected visits during CUPIC and SIRIUS follow-ups. Accordingly, we performed the following two analyses:

During-treatment analysis: This analysis aimed at comparing PCS and MCS during treatment in SIRIUS (using data at Weeks 4, 12, 16, 24) versus CUPIC (using data at Weeks 4, 8, 12, 24, 36, and 48), after adjustment for gender, age at each visit and pre-treatment HRQL values (baseline data). Patients with available HRQL scores at least once during treatment and at baseline, in at least one of the two studies, were included in this analysis. The baseline time point was set at Week 0. However for SIRIUS-arm 2 patients (who had taken the Placebo during the first 12 weeks), the baseline was set at Week 12 and only subsequent visits (at Weeks 16 and 24) were included in the "during-treatment" analysis. In addition, in order to maximize the sample size, we performed a last observation carried forward imputation for SIRIUS-arm 2 patients, using the score values at Week 0 as baseline values when the values at Week 12 were missing (this was the Carrieri MP., Protopopescu C., Younossi Z. et al case for 2 patients). A linear mixed-effects regression model -adjusted for baseline PCS and MCS values in CUPIC and SIRIUS -was used to verify whether PCS and MCS changed significantly during the PEG-IFN-free regimens (SIRIUS) with respect to the boceprevir/telaprevir PEG-IFNcontaining regimens (CUPIC).

Post-treatment analysis: This second analysis aimed at comparing PCS and MCS after treatment end in SIRIUS (data at Weeks 36 and 48) and CUPIC (data at Week 72), after adjustment for gender, age at each visit and pre-treatment HRQL values (baseline data). To this end, we employed the same method used for the "during-treatment" comparison. Patients with available HRQL scores at least once after end of treatment and at baseline, in at least one of the two studies (CUPIC and SIRIUS), were included in this analysis. Nevertheless, the visit at 28 weeks in SIRIUS (i.e. 4 weeks after treatment end) was not included in this analysis, to keep a reasonable minimum delay of at least 12 weeks after the end of treatment.

In all analyses the computation of p-values and confidence intervals was based on the robust sandwich method.

The descriptive statistics (mean ± standard deviation, median and interquartile range (IQR)) of PCS and MCS scores for each CUPIC and SIRIUS visit are reported in Table 1. The scores' distributions for both studies and for each visit are also described using boxplots in Fig. 1.

Stata software release 12.1 was used for all statistical analyses [START_REF]Stata Statistical Software[END_REF].

Results

Among the 47 patients enrolled in both studies (SIRIUS trial and CUPIC cohort), 42 were present in the CUPIC HRQL database (the remaining 5 patients did not respond to the HRQL questionnaire during the CUPIC follow-up). Overall, 43 patients were eligible for the "duringtreatment" analysis, having available HRQL scores at least once during treatment and at baseline, in at least one of the two studies, for a total of 212 HRQL assessments (109 visits from CUPIC and 103 visits from SIRIUS). Forty-three patients were eligible for "post-treatment" analysis, having HRQL scores available at least once after end of treatment and at baseline, in at least one of the two studies, for a total of 82 HRQL assessments (7 visits from CUPIC and 75 visits from SIRIUS, see flow chart in Fig. 2).

The "during-treatment" study group of 43 patients mainly included men (n=35 (81.4%)), median

[IQR] age at baseline in CUPIC was 56 [50-63] years. Among these 43 patients, 33 had HRQL data during treatment and at baseline in CUPIC, and 41 had HRQL data during treatment and at baseline in Sirius (18 in "FDC + Placebo RBV" arm and 23 in "Placebo -> FDC + RBV" arm) (see Fig. 2).

The "post-treatment" study group of 43 patients had similar characteristics (in fact, 41 patients were common to both the "during-treatment" and "post-treatment" study groups). It included 36 men (83.7%), with a median [IQR] age at baseline in CUPIC of 56 [51-64] years.

In the "during-treatment" analysis (Table 2, In the "post-treatment" analysis (Table 2, model B), significantly higher MCS values were found after PEG-IFN-free treatment (SIRIUS), with an average increase of 3.95 points (95% CI: [0.38; This study has two main results. First, compared with PEG-IFN-containing regimens, we observed a significant improvement in both the physical and mental dimensions of HRQL during PEG-IFN-free treatment. Second, in the post-treatment analysis, a significant improvement was observed in the mental dimension of HRQL after completing PEG-IFN-free treatments compared with PEG-IFN-containing treatments. By contrast, no significant improvement was found for post-treatment values of physical HRQL.

The improvement observed during treatment in both physical and mental HRQL when moving from telaprevir/boceprevir-based triple combination therapy to the PEG-IFN-free SOF-based treatment is consistent with previous results [START_REF] Younossi | The impact of ledipasvir/sofosbuvir on patient-reported outcomes in cirrhotic patients with chronic hepatitis C: the SIRIUS study[END_REF] and may be due to the benefits of PEG-IFN-free treatment in terms of increased efficacy and fewer treatment-related side-effects. Moreover, the former can be considered the "worst case scenario" in terms of toxicity burden, while the latter has already proven itself to be quite well tolerated for most patients.

Results from the "post-treatment" analysis showed there was no significant difference between physical HRQL values after PEG-IFN-containing and PEG-IFN-free treatment completion. This may be attributable to fatigue, which is known to persist after completing PEG-IFN-containing regimens [START_REF] Younossi | The effects of HCV infection and management on healthrelated quality of life[END_REF], and which may also continue for a substantial period (in our case 12 to 24 weeks) after completing PEG-IFN-free regimens, despite HCV clearance. In fact, recent data show that fatigue may decrease over time after post HCV clearance [START_REF] Gerber | Effects of Viral Eradication With Ledipasvir and Sofosbuvir, With or Without Ribavirin, on Measures of Fatigue in Patients With Chronic Hepatitis C Virus Infection[END_REF], which would suggest that even Carrieri MP., Protopopescu C., Younossi Z. et al with the advent of novel powerful and less toxic regimens, the management of physical health disturbances cannot be neglected, and needs to be incorporated into standard HCV care.

The significant improvement found in the mental HRQL dimension after the end of PEG-IFNfree regimens suggests that HCV clearance may have important benefits for patients including decreased anxiety and a reduced impact of HCV-related psychiatric symptoms on daily life. This improvement found here is consistent with previous research in HIV-HCV co-infected patients which highlighted that the low rates of suicide ideation exhibited by HCV-clearers were similar to those in people who had never been HCV co-infected [START_REF] Fressard | Suicide risk in people living with HIV: the possible role of HCV co-infection and experience of discrimination[END_REF]. This is an additional argument for promoting access to better tolerated treatment regimens for HCV-infected patients.

In addition, given that the minimal clinically important difference for the two summary scores (PCS and MCS) is generally in the range of 3 to 5 points [START_REF] Younossi | Systematic review: patient-reported outcomes in chronic hepatitis C-the impact of liver disease and new treatment regimens[END_REF][START_REF] Hays | The RAND-36 measure of health-related quality of life[END_REF][START_REF] Spiegel | Impact of hepatitis C on health related quality of life: a systematic review and quantitative assessment[END_REF], we can conclude that the statistically significant improvements observed in this study can also be interpreted as clinically important, except for the mental HRQL dimension during treatment. For the latter, the slight improvement of 2.49 points, albeit statistically significant, was not clinically significant. We suppose that depressive and anxiety symptoms (not measured in this study but known to be highly correlated with mental HRQL) may persist during any HCV treatment because of patient expectations about treatment outcomes, and also because of the burden of HCV on mental health [START_REF] Fletcher | Hepatitis C virus and the brain[END_REF].

Limitations

The main limitation of this study is the use of a generic scale for the assessment of HRQL, which is perhaps unable to capture specific dimensions of improvement (or absence of improvement) in chronically-infected HCV patients' quality of life. Although the sample size is rather limited and this may hide potential differences, we used robust estimates to compensate for this reduced power. Carrieri MP., Protopopescu C., Younossi Z. et al Moreover, the lack of comparable scales of self-reported symptoms and fatigue in both studies did not enable us to document the residual burden of some symptoms during and after PEG-IFNcontaining and PEG-IFN-free regimens.

Finally, in this study we were not able to compare the specific dimensions (subdomains) of the SF-12 scale which could have captured specific aspects like fatigue or emotional problems. This was due to the fact that the HRQL questionnaire in CUPIC was based on SF-12 version 1, while the HRQL questionnaire in SIRIUS was based on SF-36 version 2. Consequently, although the PCS and MCS summary scores were comparable between versions 1 and 2 of the reduced SF-12 scale, this was not the case for the SF-12 scale's specific dimensions, which had different scoring algorithms in the two versions.

Conclusion

To conclude, our results highlight an improvement in both physical and mental HRQL during HCV treatment, but no major improvement in physical HRQL after treatment end, when comparing PEG-IFN-free regimens with PEG-IFN-containing regimens. Accordingly, these results may have important implications for clinical management of HCV chronically-infected patients. They suggest that in the PEG-IFN-free regimen era, screening and comprehensive care of comorbidities and residual somatic symptoms during and especially after HCV clearance, are still needed in order to improve patient outcomes.

Insert

  Carrieri MP., Protopopescu C., Younossi Z. et al -Yannicq Bacq, MD, Service d'Hépatologie et Gastroentérologie, Centre Hospitalier Universitaire Trousseau, Tours, France -Paul Calès, MD, Service d'Hépatologie et Gastroentérologie, Centre Hospitalier Universitaire, Angers, France -Patrick Delasalle, MD, Service d'Hépatologie et Gastroentérologie, Clinique du Palais, Grasse, France -Denis Ouzan, MD, Service d'Hépatologie et Gastroentérologie, Institut Arnaud Tzanck, Saint-Laurent du Var, France -Jean-Baptiste Nousbaum, MD, Service d'Hépatologie et Gastroentérologie, Centre Hospitalier Universitaire de la Cavale Blanche, Brest, France -Christine Sylvain, MD, Service d'Hépatologie et Gastroentérologie, Centre Hospitalier Universitaire, Poitiers, France -Didier Ribard, MD, Service d'Hépatologie et Gastroentérologie, Centre Hospitalier Universitaire Caremeau, Nîmes, France -Gilles Gatineau-Sailliant, MD, Centre Hospitalier, Meaux, France -Stéphanie de Montigny-Lenhardt, MD, Centre Hospitalier Edmond Garcin, Aubagne, France -Philippe Renard, MD, Centre Hospitalier Victor Dupouy, Argenteuil, France -Christophe Pilette, MD, Centre Hospitalier, Le Mans, France -Jacques Denis, MD, Centre Hospitalier Sud Francilien, Corbeil-Essonnes, France -Caroline Lascoux-Combe, MD, Service de Maladies Infectieuses, Hôpital Saint-Louis, AP-HP, Paris, France -Laurent Abel, MD, Unité de génétique, INSERM U550, Paris, France -Matthew Albert, PhD, Unité d'Immunologie, Institut Pasteur, Paris, France -Olivier Chazouillères, MD, Service d'Hépatologie, Hôpital Saint-Antoine, AP-HP, Paris, France -Jean Dubuisson, PhD, Unité de Virologie, INSERM U1019, CNRS UMR8204, Lille, France -Francesco Negro, MD, Hôpital Cantonal, Genève, Suisse -Georges-Philippe Pageaux, MD, Hôpital Saint Eloi, Montpellier, France Carrieri MP., Protopopescu C., Younossi Z. et al -Valérie Paradis, MD, Service d'anatomopathologie, Hôpital Beaujon, AP-HP, Clichy, France -Bruno Spire, PhD, Santé Publique, Sciences Humaines et Sociales, INSERM, Marseille, France -Anne-Marie Taburet, MD, Pharmacologie, Hôpital Bicêtre, AP-HP, Le Kremlin-Bicêtre, France -Jean-Claude Trinchet, MD, Service d'Hépatologie, Hôpital Jean Verdier, AP-HP, Bondy, France -Yazdan Yazdanpanah, MD, Economie de la Santé, Hôpital Bichat, AP-HP, Paris, France -Cécilie Dufour, INSERM UMR-S 707, Université Pierre et Marie Curie Paris 6, Paris, France -Céline Fréhaut, INSERM UMR-S 707, Université Pierre et Marie Curie Paris 6, Paris, France -Marion Pirot, INSERM UMR-S 707, Université Pierre et Marie Curie Paris 6, Paris, France -Aurélie Lesel, INSERM UMR-S 707, Université Pierre et Marie Curie Paris 6, Paris, France -Nathalie Zahraa, INSERM UMR-S 707, Université Pierre et Marie Curie Paris 6, Paris, France -Frédéric Chau, INSERM UMR-S 707, Université Pierre et Marie Curie Paris 6, Paris, France Running Heading Health-Related Quality of Life in HCV Patients Switching to Pegylated-Interferon-Free Regimens Abstract word count: 242 (max 250) Text word count: 3105 (max 4000) Running heading character count: 94 (max 100)

  in the following two studies conducted in France: the ANRS CO20 CUPIC cohort (in 2011-2012) and the SIRIUS trial (in 2013-2014). ANRS CO20 CUPIC (ClinicalTrials.gov number NCT01514890) was a multicenter cohort study which started in February 2011. It included HCV genotype 1 patients with compensated cirrhosis who had failed to eradicate HCV after completing a previous standard PEG-IFN/RBV therapy, and who were subsequently provided with protease inhibitor-based HCV therapy in the context Carrieri MP., Protopopescu C., Younossi Z. et al of the French Early Access Program and the cohort's Temporary Authorisation for Use (ATU)

  model A), results showed significantly higher values for both PCS and MCS scores during PEG-IFN-free treatment (SIRIUS) than with PEG-IFNcontaining treatment combined with telaprevir or boceprevir (CUPIC), after adjustment for baseline score values (before treatment), gender and age at each visit. The average increase in HRQL values during the follow-up of SIRIUS versus CUPIC -estimated by the coefficient in the multivariate linear regression mixed model -was 4.57 points (95% confidence interval (CI): [2.44; 6.71], p-value<0.0001) for the PCS score and 2.49 points (95% CI: [0.28; 4.69], p=0.027) for the MCS score.

7 .

 7 52], p-value=0.030) compared with after PEG-IFN-containing treatment (CUPIC), after adjustment for baseline PCS values, gender and age at each visit. However, for PCS score, there was a trend towards better HRQL when comparing PCS values after PEG-IFN-free treatment versus after PEG-IFN-containing treatment, but this increase of 2.07 points (95% CI: [-4.69; 8.84], p-value=0.548) was not statistically significant. 4 Discussion Carrieri MP., Protopopescu C., Younossi Z. et al To our knowledge, this is the first study to compare changes in HRQL in the same group of chronic HCV-infected patients switching from a PEG-IFN-containing regimen (including telaprevir or boceprevir) to a PEG-IFN-free regimen, while taking into account within-subject variability. The repeated-measure design of the study allowed us to control for pre-treatment values of HRQL while comparing HRQL values assessed during treatment and in post-treatment comparison.

Fig. 1 Fig. 1 Fig. 2

 112 Figures Legend

4 )Fig. 1

 41 Fig. 1 Boxplots of PCS and MCS scores in CUPIC and SIRIUS, during and post treatment comparisons

Table 1 .

 1 Descriptive statistics of PCS and MCS scores in CUPIC and SIRIUS, for the during-and post-treatment analyses Carrieri MP., Protopopescu C., Younossi Z. et al

			CUPIC		FDC + Placebo RBV		Placebo->FDC + RBV
			(PEG-IFN regimens)		(SIRIUS arm 1)		(SIRIUS arm 2)
		n	mean ± SD	n	mean ± SD	n	mean ± SD
			median (IQR)		median (IQR)		median (IQR)
	PCS score
	Baseline §	33	43.4 ± 6.6	19	46.7 ± 6.3	24	47.1 ± 7.7
			43.2 (38.7-49.5)		46.2 (42.6-51.0)		48.2 (40.1-54.0)
	During-treatment
	Week 4	30	41.0 ± 5.8	14	45.5 ± 4.6
			40.8 (38.0-45.9)		46.0 (41.6-49.4)
	Week 8	24	43.2 ± 6.0
			43.0 (39.8-47.5)
	Week 12	24	40.2 ± 5.5	17	46.6 ± 7.7
			41.2 (36.9-43.5)		44.6 (39.5-54.0)
	Week 16			16	46.8 ± 7.3	21	47.2 ± 7.6
					45.8 (41.5-53.8)		48.3 (44.6-52.1)
	Week 24	14	37.8 ± 6.6	17	45.8 ± 7.7	18	47.0 ± 6.9
			39.1 (34.5-43.8)		46.4 (42.0-50.2)		47.9 (43.3-52.7)
	Week 36	9	41.4 ± 6.2
			42.0 (36.2-44.9)
	Week 48	8	41.6 ± 3.6
			42.1 (39.9-44.3)
	Post-treatment
	Week 36			18	45.7 ± 8.4	19	46.0 ± 7.7
					44.0 (38.6-53.4)		46.1 (39.3-53.4)
	Week 48			18	47.9 ± 7.5	20	51.2 ± 5.2
					49.2 (43.9-54.8)		51.9 (49.0-55.0)
	Week 72	7	42.9 ± 7.3
			42.8 (36.7-50.4)
	MCS score
	Baseline §	33	42.3 ± 5.8	19	42.4 ± 10.1	24	40.8 ± 12.2
			41.2 (38.6-45.4)		44.0 (34.1-49.3)		39.7 (29.8-51.9)
	During-treatment

Acknowledgements

We thank all the members of the CUPIC Study Group*. We especially thank all the patients who agreed to participate in the two studies and who completed the self-administered questionnaires.

We thank Bérengère Davin (PhD, Health Economist, ORS PACA, Marseilles, France) for suggestions about data analyses and interpretation of the results. Finally, we thank Jude Sweeney (Milan, Italy) for the English revision and editing of our manuscript.

Funding

The ANRS CO20 CUPIC cohort study was sponsored and funded by The National Agency for Research on AIDS and Viral Hepatitis (ANRS). The ANRS CO20 CUPIC cohort study was conducted with the support and participation of the Association Française pour l'Etude du Foie (AFEF). This work was supported by the French ANRS, with the participation of Gilead Sciences.

Author's contributions

Conflict of Interest

Zobair Younossi: research support and consultant for Gilead Sciences.

Hélène Fontaine: speaker fees from MSD, Janssen, Roche, BMS.

Christophe Hézode: consultancy fees from Abbott, MSD, Roche, Janssen, BMS, Gilead, and speaker fees from MSD, Janssen, Roche, BMS.

The other authors (Maria Patrizia Carrieri, Camelia Protopopescu, Antoine Vilotitch, Ventzislava Petrov-Sanchez, Fabienne Marcellin, Fabrice Carrat, and Marc Bourlière) declare that they have no conflict of interest.

Informed Consent

Written informed consent was obtained from all patients included in the study.

Ethical Approval

All procedures performed in this study were in accordance with the ethical standards of the