Bayesian update of the parameters of probability distributions for risk assessment in a two-level hybrid probabilistic-possibilistic uncertainty framework

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Bayesian update of the parameters of probability

distributions for risk assessment in a two-level hybrid

probabilistic-possibilistic uncertainty framework

Nicola Pedroni, Enrico Zio, Alberto Pasanisi, Mathieu Couplet

To cite this version:

Nicola Pedroni, Enrico Zio, Alberto Pasanisi, Mathieu Couplet. Bayesian update of the parameters of probability distributions for risk assessment in a two-level hybrid probabilistic-possibilistic uncertainty framework. ESREL 2013, Sep 2013, Amsterdam, Netherlands. pp.1-8. �hal-00839962�

1 INTRODUCTION

We consider a framework of uncertainty representa- tion with two hierarchical levels (Limbourg & de Rocquigny 2010), in which risk analysis models of aleatory (i.e., random) events (e.g., failures) contain parameters (e.g., probabilities, failure rates, …) that are epistemically-uncertain, i.e., known with poor precision due to lack of knowledge and information.

Traditionally, both types of uncertainty are repre- sented by probability distributions (USNRC 2009) and Bayes’ rule is useful for updating the (probabil- istic) epistemic uncertainty representation as new in- formation (e.g., data) becomes available (Kelly &

Smith 2011).

However, in some situations, insufficient knowledge, information and data impairs a probabil- istic representation of epistemic uncertainty. A num- ber of alternative representation frameworks have been proposed for such cases, e.g., e.g., fuzzy set theory, evidence theory, possibility theory and inter- val analysis (Aven & Zio 2011).

In this paper, we adopt possibility distributions to describe epistemic uncertainty (Baudrit & Dubois 2006, Baudrit et al. 2008) and address the issue of updating, in a Bayesian framework, the possibilistic representation of the epistemically-uncertain pa- rameters of (aleatory) probability distributions. We take an approach of literature based on a purely possibilistic counterpart of the classical, well-

grounded probabilistic Bayes’ theorem: it requires the construction of a possibilistic likelihood function which is used to revise the prior possibility distribu- tions of the uncertain parameters (determined on the basis of a priori subjective knowledge and/or data) (Lapointe & Bobee 2000). To the best of the authors’

knowledge, this is the first time that the above men- tioned technique is applied to risk assessment prob- lems where hybrid uncertainty is separated into two hierarchical levels. To keep the analysis simple and retain a clear view of each step, the investigations are carried out with respect to a simple literature case study involving the risk-based design of a flood protection dike (Limbourg & de Rocquigny 2010).

Other methods have been proposed in the litera- ture to revise, in a Bayesian framework, non- probabilistic representations of epistemic uncertain- ty. In (Stein et al. 2013) a modification of the Bayes’

theorem is presented to account for the presence of fuzzy data and fuzzy prior PDFs. Finally, in (Smets 1993) a Generalized Bayes Theorem (GBT) is pro- posed within the framework of evidence theory: this approach is applied by (Le-Duy et al. 2011) to up- date the estimates of the failure rates of mechanical components in the context of nuclear Probabilistic Risk Assessment (PRA).

The remainder of the paper is organized as fol- lows. In Section 2, the representation of aleatory (probabilistic) and epistemic (possibilistic) uncer-

Bayesian update of the parameters of probability distributions for risk

assessment in a two-level hybrid probabilistic-possibilistic uncertainty

framework

N. Pedroni, E. Zio

Ecole Centrale Paris, Chatenay-Malabry, France & Supelec, Gif-Sur-Yvette, France

A. Pasanisi, M. Couplet

Electricitè de France, Chatou, France

ABSTRACT: Risk analysis models describing aleatory (i.e., random) events contain parameters (e.g., proba- bilities, failure rates, …) that are epistemically uncertain, i.e., known with poor precision. Whereas probability distributions are always used to describe aleatory uncertainty, alternative frameworks of representation may be considered for describing epistemic uncertainty, depending on the information and data available.

In this paper, we use possibility distributions to describe the epistemic uncertainty in the parameters of the (aleatory) probability distributions.

We address the issue of updating, in a Bayesian framework, the possibilistic representation of the epistemically-uncertain parameters of (aleatory) probability distributions as new information (e.g., data) be- comes available. A purely possibilistic counterpart of the classical, well-grounded probabilistic Bayes theorem is adopted.

The feasibility of the method is shown on a literature case study involving the risk-based design of a flood protection dike.

tainties in a “two-level” framework is provided; in Section 3, the method employed in this paper for the Bayesian update of the possibilistic parameters of aleatory probability distributions is described in de- tails; in Section 4, the case study concerning the risk-based design of a flood protection dike is pre- sented; in Section 5, the method of Section 3 is ap- plied to the case study of Section 4; finally, some conclusions are drawn in the last Section 6.

2 REPRESENTATION OF ALEATORY AND EPISTEMIC UNCERTAINTIES IN A TWO- LEVEL FRAMEWORK

In all generality, we consider an uncertain variable Y, whose uncertainty is described by the Probability Distribution Function (PDF) p^Y(y|1), where

} ..., , ..., , ,

{θ₁ θ₂ θ_m θ_P

=

1 is the vector of the corre-

sponding internal parameters. In a two-level frame- work, the parameters 1 are themselves affected by epistemic uncertainty (Limbourg & de Rocquigny 2010). In the present work, we describe these uncer- tainties by the (generally joint) possibility distribu- tion 1¹(1) (Baudrit et al. 2008).

For clarification by way of example, we may con- sider the generic uncertain variable Y described by a Gumbel PDF, i.e., Y ~ p^Y(y|1) = ^Gum

( )

^{1 =}

(

θ1 ,θ2

)

Gum = ^Gum

( )

γ ^,δ ^{= pY(y|}γ^,δ⁾. Parameter θ2

δ = is a fixed point-wise value (δ =θ₂ = 100), whereas parameter γ =θ₁ is epistemically-uncertain.

By hypothesis, the only information available on θ1

γ = is that it is defined on interval [a2, b2] = [900, 1300] and its most likely value is c₂ = 1100. This limited state of knowledge about γ =θ₁ can be de- scribed by a triangular possibility distribution π^γ(γ) with core c2 = 1100 and support [a2, b2] = [900, 1300] (Figure 1, top) (Baudrit & Dubois 2006).

Given the possibility distribution of γ =θ₁, we can define its 3-cut sets A_α^γ = {γ: π^γ(γ) ¹ α }, with 0 ≤ α 2 1. For example, A₀^γ_.5= [1000, 1200] is the set of γ values for which the possibility function is greater than or equal to 0.5 (dashed segment in Figure 1, top). Notice that the 3-cut set A_α^γ of pa- rameter 2 can be interpreted also as the (1 – 3)3100%

Confidence Interval (CI) for 2, i.e., the interval such that P[γ∈A_α^γ]≥1−α. For example, A₀^γ = [900, 1300] is the (1 – 0)3100% = 100% CI for 2, i.e., the interval that contains the “true” value of 2 with cer- tainty (solid segment in Figure 1, top); A₀^γ_.5 = [1000, 1200] (⊂ A₀^γ) is the (1 – 0.5)3100% = 50% CI

(dashed segment in Figure 1, top) and so on. In this view, the possibility distribution π^γ(γ) can be inter- preted as a set of nested CIs for parameter 2 (Baudrit

& Dubois 2006).

For each possibility (resp., confidence) level 3 (resp., 1 – 3) in [0, 1], a bundle of Cumulative Dis- tribution Functions (CDFs) for Y, namely

( )

{

^{FY y|γ,δ}

}

_α, can be generated by letting the epistemically-uncertain parameter γ range within the corresponding 3-cut set A_α^γ, i.e.,

{

^FY

(

^y|γ,δ

) }

_α ⁼

( )

{

F^Y y^{|γ,δ :γ∈}Aα^{γ,σ =100}

}

. This family of CDFs (of level 3) is bounded above and below by the up- per and lower CDFs, ^Fα^Y

( )

^y and ^FYα

( )

^y , defined as

( )

^y

F_α^Y = ^sup

{ (

^{| , =100}

) }

∈ γ δ

αγ

γ F^Y y

A

and ^Fα^Y

( )

^y =

( )

{

^{| , 100}

}

inf =

∈ _γ γ δ

γ α F^Y y

A , respectively. Since π^γ(γ) can be interpreted as a set of nested CIs for parame- ter 2 (see above), it can be argued that the 3-cuts of

) (γ

π^γ induce also a set of nested pairs of CDFs

( ) ( )

( )

{

Fα^Y y ^,Fα^Y y ^{:0≤α ≤1}

}

which bound the “true”

CDF ^FY

( )

^y of Y with confidence larger than or

equal to (1 – 3), i.e.,

( ) ( )

α

( )

α

α ≤ ≤ ]≥1−

[F y F y F y

P ^{Y Y Y} , with 0 ≤ α 2 1

(Baudrit et al. 2008). In passing, notice that the up- per and lower CDFs (of level 3), ^Fα^Y

( )

^y and ^FYα

( )

^y , can be referred to as the plausibility and belief func- tions (of level 3) of the set Z = (45, y], i.e.,

( )

^{y Pl}

( )

^Z

F_α^Y = _α^Y and ^Fα^Y

( )

^y =^Belα^Y

( )

^Z , respectively.

For illustration purposes, Figure 1, bottom, shows the bounding upper and lower CDFs of Y, ^Plα^Y

( )

^Z

and ^Belα^Y

( )

^Z , built in correspondence of the 3-cuts of level 3 = 0 (solid lines), 0.5 (dashed lines) and 1 (dot-dashed line) of the possibility distribution

) (γ

π^γ of parameter 2 (Figure 1, top).

Finally, the set of nested pairs of CDFs

( ) ( )

( )

{

Belα^Y Z ^,Plα^Y Z ^{:0≤α ≤1}

}

, Z = (45_, y], can be synthesized into a single pair of plausibility and be- lief functions as

( )

⁼

1

¹

( )

0

α Z dα Pl Z

Pl^{Y Y} and

( )

⁼

1

¹

( )

0

α Z dα Bel Z

Bel^{Y Y} , respectively (dotted lines in Figure 1, bottom). The plausibility and belief func- tions ^PlY

( )

^{Z and BelY}

( )

^{Z , Z = (45}, y], are shown to represent the “best bounds” for the “true” CDF

( )

^y

F^Y of the uncertain variable Y (Baudrit et al.

2008).

850 900 950 1000 1050 1100 1150 1200 1250 1300 1350

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ πγ ( γ )

A₀^γγγγ A₁^γγγγ

A_0.5^γγγγ

αααα = 0

αααα = 1

αααα = 0.5

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Y

Cumulative probability

α = 0 α = 0.5 α = 1 Pl(Z), Bel(Z)

Figure 1. Top: triangular possibility distribution π^γ(γ) ^{of the} uncertain parameter 2 of the Gumbel PDF of Y ~

(^y|γ^,δ ⁼¹⁰⁰)

p^Y ; in evidence the 3-cuts of level 3 = 0 (solid segment), 0.5 (dashed segment) and 1 (dot). Bottom: bounding upper and lower CDFs of Y, ^Plα^Y( )^Z and ^Belα^Y( )^Z , Z = (45, y], built in correspondence of the 3-cuts of level 3 = 0 (solid lines), 0.5 (dashed lines) and 1 (dot-dashed line) of π^γ(γ); the plau- sibility and belief functions ^PlY( )^{Z and BelY}( )^Z , Z = (45, y], are also shown (dotted lines)

3 BAYESIAN UPDATE OF THE POSSIBILISTIC PARAMETERS OF ALEATORY

PROBABILITY DISTRIBUTIONS

Let 1¹(1) be the (joint) prior possibility distribution for the parameters 1=[θ₁ ,θ₂ ,...,θ_m,...,θ_P] of the PDF p^Y(y|1) of variable Y(built on the basis of a priori subjective engineering knowledge and/or da- ta). For example, in the risk assessment context of this paper Y may represent the yearly maximal water flow of a river described by a Gumbel distribution:

thus, Y ~ p^Y(y|1) = Gum(1) = Gum(4₁, 4₂) = Gum(2, 5) = ^pY(^y|γ,δ) ^{and 11(1) = 1γ,δ(}γ^,δ^). Moreover, let y=[y₁,y₂,...,y_k,...,y_D] be a vector of D observed pieces of data representing the new in- formation/evidence available for the analysis: refer- ring to the example above, y may represent a vector

of D values collected over a long period time (e.g., many years) of the yearly maximal water flow of the river under analysis. The objective of the Bayesian analysis is to update the a priori representation

)

1(1

1 = ^1γ,δ(γ,δ) ^{of 1} = [2, 5] on the basis of the new evidence acquired, i.e., to calculate the posteri- or possibility distribution 1¹(1| y) (i.e.,

)

| ,

, ( δ y γ

δ

1γ ) of 1 after y is obtained.

The method considered in this paper is based on a purely possibilistic counterpart of the classical, probabilistic Bayes’ theorem (Lapointe & Bobée 2000):

{

^{( | ) ( )}

}

sup

) ( )

| ) (

|

( 1 y 1

1 y y 1

1 _{1 1}

1

1 1

1

1 1

1 1 1

L L

⋅

= ⋅ , (1)

where 1¹_L(1|y) is the possibilistic likelihood of the parameter vector 1 given the newly observed da- ta y, and quantities 1¹(1|y) and 1¹(1) are defined above. Notice that ^sup

{

^{1(1|y) 1(1)}

}

1

1

1_L ⋅ is a normal-

ization factor such that ^sup

{

^{1(1| y)}

}

1

1 = 1, as re- quired by possibility theory (Baudrit & Dubois 2006).

It is worth mentioning that forms of the possibilistic Bayes’ theorem alternative to (1) can be constructed as a result of other definitions of the op- eration of ‘conditioning’ with possibility distribu- tions: the reader is referred to (Lapointe & Bobée 2000) for technical details. In this paper, expression (1) has been chosen because “it satisfies desirable properties of the revision process and lead to contin- uous posterior distributions” (Lapointe & Bobée 2000).

The possibilistic likelihood 1¹_L(1|y) is here ob- tained by transforming the classical probabilistic likelihood function L¹(1|y) through normalization, i.e., 1_L¹(1|y) = ^{(1| y)/sup}

{

^{1(1| y)}

}

1

1 L

L . This choice

has been made for the following main reasons: (i) the transformation is simple and can be straightfor- wardly applied to any distribution; (ii) the resulting possibilistic likelihood is very closely related to the classical, purely probabilistic one (which is theoreti- cally well-grounded) by means of the simple and di- rect operation of normalization that preserves the

“original structure” of the experimental evidence;

(iii) it can be easily verified that the resulting possibilistic likelihood keeps the sequential nature of the updating procedure typical of the standard Bayes’ theorem. On the other hand, it has to be also admitted that the resulting possibility distributions

do not in general adhere to the probability-possibility consistency principle (Baudrit & Dubois 2006).

It is worth noting that other techniques of trans- formation of probability density functions into pos- sibility distributions exist, but the corresponding de- tails are not given here for brevity sake: the interested reader is referred to (Flage et al. 2013) for some proposed techniques, e.g., the principle of maximum specificity and the principle of minimal commitment. Also, it has to be noticed that other techniques are available to construct possibility dis- tributions (and, thus, possibilistic likelihood func- tions) directly from rough experimental data (i.e., without resorting to probability-possibility transfor- mations): see, e.g., (Serrurier & Prade 2011).

It is worth noting that the application of the ap- proach always produces a joint P-dimensional poste- rior possibility distribution 1¹(1|y) (whatever the state of dependence between the priors), character- ized by P-dimensional 3-cuts A_α^1|y, with 0 < 3 < 1:

as a consequence, there is an interactive dependence between the values that parameters {4_m: m = 1, 2, …, P} can take when ranging within a given 3-cut A_α^1|y. From 1¹(1|y) it is straightforward to obtain the marginal posterior possibility distribution

)

| ( _m y

4 4

1 ^m for each parameter 4m as 1^4m(4_m|y) = )}

| ( { max

, 1¹ 1 y

m

4_j∈ℜ j≠ , ∀θ_m∈ℜ, m = 1, 2, …, P:

)

| ( _m y

4 4

1 ^m is the projection of 1¹(1| y) onto the m- th axis. The (one-dimensional) 3-cut A_α^4m|y =

]

| ,

|

[θ_m,α y θ_m,α y of the marginal possibility distribu- tion 1^4m(4_m| y) is then related to the (P- dimensional) 3-cut A_α^1|y of the joint possibility dis- tribution 1¹(1|y) by the following straightforward relation, i.e., A_α^4m|y = [θ_m,α |y,θ_m,α|y] ⁼

}]

{ max }, { min

[ | | m

m A

A ^y θ ^y θ

α

α1 1 1

1∈ ∈ . In this view, notice that the use of the P-dimensional 3-cut A_α^1|_,Cart^y constructed by the Cartesian product of the (one-dimensional) 3- cuts A_α^4m|y of the marginal distributions, m = 1, 2, …, P (i.e., A_α^1|_,Cart^y = A_α^41|y x A_α^42|y x … x A_α^4m|y x … x

y

P|

A_α4 ) would (incorrectly) imply independence be- tween the posterior estimates of the parameters {4m: m = 1, 2, …, P}; however, since A_α^1|_,Cart^y completely contains A_α^1|y (i.e., by definition A_α^1|y ⊂ A_α^1|,Cart^y ), then conservatism would be still guaranteed (Stein et al. 2013).

4 CASE STUDY: FLOOD PROTECTION RISK- BASED DESIGN

The maximal water level of the river (i.e., the output variable of the model, Z ) is given as a func-_c tion of several (and some uncertain) parameters (i.e., the inputs to the model) (Limbourg & de Rocquigny 2010):

(

Q,Z ,Z ,K ,B,L

) (

f Y1,Y2,Y3,Y4,Y5,Y6

)

f

Z_c = _{m v s} = (2)

where: Y₁ = Q is the yearly maximal water dis- charge [m³/s]; Y2 = Z_mand Y3 = Z_vare the riverbed levels [m asl] at the upstream and downstream parts of the river under investigation, respectively; Y4 = Ks is the Strickler friction coefficient; Y₅ = B and Y6 = L are the width and length of the river part [m], respectively. Quantities Y5 = B (= 300m) and Y6 = L (= 5000m) are constant parameters, whereas quanti- ties Y1 = Q , Y2 = Z_m, Y3 = Z_v, Y4 = K_s are uncer- tain variables.

The n = 4 input variables Yi, i = 1, 2, 3, 4, are af- fected by aleatory and epistemic uncertainties. The aleatory part of the uncertainty is described by prob- ability distributions of defined shape. The parame- ters of the probability distributions describing the aleatory uncertainty are themselves affected by epis- temic uncertainty and represented in terms of possi- bility distributions.

The aleatory uncertainty in the yearly maximal water flow Y1 = Q is well described by a Gumbel probability distribution ^pQ

(

^qγ,δ

)

⁼ Gum

( )

γ,δ =

234 567 − 23

5 4 6

7 8

9 B A C

D −

− δ

γ δ

γ δ

q q exp exp

1exp

(Limbourg & de Rocquigny 2010). The extreme physical bounds on variable Q are Q_min =10m³/s (which is a typical drought flow level) and Q_max = 10000 m³/s (which is three times larger than the maximal flood ever oc- curred). The prior possibility distributions π^γ(γ) and π^δ(δ) for the epistemically-uncertain parame- ters γ ^and δ are subjectively chosen as triangular functions TR(a₂, c₂, b₂) and TR(a₅, c₅, b₅), respective- ly, with cores (i.e., preferred or most likely values) c2

= 955m³/s and c5 = 600m³/s, and supports [a2, b2] = [869, 1157] m³/s and [a5, b5] = [455, 660] m³/s, re- spectively. The Bayesian update of these uncertainty representations (based on prior subjective knowledge) is realized with the aid of a vector y1 = q

= [q₁, q₂, …, q_k, …, q₁₄₉] of D₁ = 149 (independent and identically distributed – iid) values of the annual maximal flow of the river. The point estimates γ^ˆMLE

and δ^ˆMLE for γ and δ obtained by the classical, purely probabilistic Maximum Likelihood Estima- tion (MLE) method are 1013.21 m³/s and 558.21 m³/s, respectively.

The aleatory part of the uncertainty in the up- stream riverbed level Y2 = Zm is represented by a normal distribution, i.e., Z ~ _m

(

m Zm Zm

)

Z z

p ^{m |}µ ^,σ ⁼

)

(

Zm Zm

N µ ^,σ (Limbourg & de Rocquigny 2010).

This distribution is truncated at the minimum and maximum physical bounds on Zm, i.e.,

5 .

min 53

, =

Zm m (given by plausible lower geomor- phologic limits to erosion) and Zm_,max =57m (given by plausible upper geomorphologic limits to sedi- mentation), respectively. The prior possibility distri- butionsfor µ_Zm ^and σZm are the triangular functions

) ( _Zm

Zm µ

π^{µ = TR(a}µ_Zm, c_µZm,

b_µZm) = TR(54.78, 54.93, 55.28) and π^σZm(σ_Zm) ^{= TR(a}σ_Zm,

c_σZm, b_σZm)

= TR(0.33, 0.51, 0.58), respectively. The Bayesian update of these uncertainty representations is carried out using a vector y2 = zm = [zm,1, zm,2, …, zm,k, …, zm,29] of D2 = 29 (iid) values of the upstream riv- erbed level. The MLE estimates of the parameters are µ^ˆ_Zm^MLE = 50.19 m and σ^ˆ_Zm^MLE = 0.38 m, respective- ly.

As for Y2 = Zm, the aleatory part of the uncertainty in the downstream riverbed level Y3 = Zv is repre- sented by a normal distribution, i.e., Z ~ _v

)

(

v Zv Zv Z z

p ^{v |}µ ^,σ ⁼ N

(

µZv^,σZv

)

, truncated at

min 48

, =

Zv m and Zv_,max =51m. As before, the prior possibility distributions are triangular functions, i.e.,

) ( _Zv

Zv µ

π^{µ = TR(a}µ_Zv , c_µZv,

b_µZv ) = TR(49.98, 50.11, 50.40) and π^σZv(σ_Zv) ^{= TR(a}σ_Zv,

c_σZv ,

b_σZv) = TR(0.23, 0.45, 0.54). These representations are up- dated by means of a vector y₃ = z_v = [z_v,1, z_v,2, …, z_v,k,

…, zv,29] of D3 = 29 (iid) values of the downstream riverbed level. The MLE estimates of the parameters are µ^ˆ_Zv^MLE = 55.03 m and σ^ˆ_Zv^MLE = 0.45 m, respective- ly.

The Strickler friction coefficient Y4 = K is the _s most critical source of uncertainty because it is usu- ally a simplification of a complex hydraulic model.

The absolute physical limits of K are 5 and 60, re-_s spectively (Limbourg & de Rocquigny 2010). The friction coefficient K is affected by random events _s modifying the river status (e.g., erosion, sedimenta- tion, …): the corresponding variability is typically described by a normal distribution, i.e., K ~ _s

)

(

s Ks Ks

K k

p ^{s |}µ ^,σ ⁼ N

(

µKs^,σKs

)

(Limbourg & de Rocquigny 2010). However, the parameters of this normal distribution are difficult to estimate because

data can only be obtained through “indirect calibra- tion characterized by significant uncertainty”

(Limbourg & de Rocquigny 2010): the uncertainty in these parameters is described by triangular possibil- ity distributions. The possibilistic functions

) ( s Ks

µK

π^{µ and (} Ks⁾

Ks σ

π^σ used to represent the a priori knowledge on µ_Ks^andσ_Ks ^{are TR(a}µ_Ks,

c_µKs, b_µKs) = TR(21.37, 25.23, 34.23) and TR(

a_σKs, c_σKs, b_σKs) = TR(1.16, 6.91, 9.37), respectively. The Bayesian revision of these a priori representations is performed by means of a vector y₄ = k_s = [k_s,1, k_s,2,

…, ks,k, …, ks,5] of D4 = 5 (iid) values of the Strickler friction coefficient. The MLE estimates of the pa- rameters are µ^ˆ_Ks^MLE = 27.8 and σ^ˆ_Ks^MLE = 5.26, respec- tively.

5 RESULTS

In order to simplify the notation, in what follows let 4 be one of the uncertain parameters of the PDFs of Y₁ = Q, Y₂ = Z_m, Y₃ = Z_v and Y₄ = K_s, i.e., 4 = γ ^, δ ^, µZm^, σZm^, µZv^, σZv^, µK_s ^or σKs. Figure 2 illustrates the possibility distributions of the epistemically- uncertain parameters of the aleatory PDFs ^pQ

(

^qγ,δ

)

(top) and

(

s Ks Ks

)

K k

p ^{s |}µ ^,σ (bottom) of the uncertain input variables Y1 = Q and Y4 = Ks, respectively; the parameters of the PDFs

(

m Zm Zm

)

Z z

p ^{m |}µ ^,σ ^and

)

(

v Zv Zv Z z

p ^{v |}µ ^,σ ^{of Y}2 = Zm andY3 = Zv, respectively, are not shown due to space limitations. In particular, the prior possibility distributions π^θ(θ) ⁽⁼ π^γ(γ^),

) (δ

π^{δ ,} π^µKs(µK_s^{) and (} Ks⁾

Ks σ

π^σ ) are shown as solid lines, whereas the marginal posterior possibil- ity distributions π^θ(θ |y) ⁽⁼ π^γ(γ ^|q^), π^δ(δ ^|q^),

)

|

( k_s

s Ks

µK

π^{µ and} π^σKs(σ_Ks^|ks⁾) obtained using D1

= 149 and D₄ = 5 pieces of data are shown in dashed lines, respectively; the point estimates θ^{ˆMLE (=}

γ^ˆMLE^, δ^ˆMLE, µ^ˆK^MLE_s ^and σ^ˆKs^MLE) produced by the clas- sical MLE method are also shown for comparison (dots) (see Section 4).

850 900 950 1000 1050 1100 1150 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

γ πγ (γ )

Prior Posterior MLE estimate

460 480 500 520 540 560 580 600 620 640 660

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δ πδ (δ )

20 25 30 35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

µ_K

s

πµKs (µKs )

1 2 3 4 5 6 7 8 9 10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

σK s

πσKs (σKs )

Figure 2. Possibility distributions of the epistemically-uncertain parameters of the aleatory PDFs ^pQ

(

^qγ,δ

)

(top) and

) ( s Ks Ks

K k

p ^{s |}µ ^,σ (bottom) of the uncertain input variables Y₁ = Q and Y₄ = K_s, respectively. Solid lines: priors; dashed lines: mar- ginal posteriors. The point estimates of the parameters obtained by the classical MLE method are also shown for comparison (dots)

From a mere visual and qualitative inspection of Figure 2 it can be seen that the approach is suitable for revising the prior possibility distributions (based on a priori purely subjective knowledge) by means of empirical data. In particular, it is evident that: (i) the most likely (i.e., preferred) values c₄ of the epistemically-uncertain parameters (i.e., those values in correspondence of which the possibility function equals 1) are moved towards the MLE estimates θ^ˆMLE in all the cases considered; (ii) the area S₄ un- derlying the corresponding possibility distributions is significantly reduced: noting that this area is relat- ed to the imprecision in the knowledge of the possibilistic parameter (i.e., the larger the area, the higher the imprecision), it can be concluded that the approach succeeds in reducing the epistemic uncer- tainty. With respect to that, Table 1 reports the most likely values c4 and the areas S4 underlying the (mar- ginal) possibility distributions of the uncertain pa- rameter 4 (= γ , δ , µ_Zm, σ_Zm^, µZv, σ_Zv,

Ks

µ ^and σKs) before and after the Bayesian update; the point estimates θ^ˆMLE obtained by the classical MLE meth- od are also reported for completeness. In addition, in order to provide a more quantitative assessment of the “updating power” of the method, two indicators are defined:

(i) the percentage relative difference R_θ^dist of the absolute distances, d_θ^Prior and d_θ^Posterior, between the prior and posterior most likely values, c_θ^Prior and

Posterior

c_θ , respectively, of parameter 4 and the corre- sponding MLE estimate θ^{ˆMLE, i.e.:}

(

⁻

)

^⋅100

= ^{Prior Posterior Prior}

dist d d d

R_{θ θ θ θ} , (3)

where d_θ^Prior = |cθ^Prior −θˆ^MLE |^/θ^{ˆMLEand d}θ^Posterior = ˆ |

|cθ^Posterior −θ^{MLE /}θ^ˆMLE. Obviously, the higher is R_θdist (i.e., the lower is d_θ^Posterior) the closer is the most likely value c_θ^Posterior to the MLE estimate θ^ˆMLE, i.e., the higher is the strength of the approach in updating the prior possibilistic distribution;

(ii) the percentage relative difference R₄ between the areas underlying the possibility distribution of parameter 4 before and after the Bayesian update, namely S_θ^Prior and S_θ^Posterior, respectively:

100 /

)

( − ⋅

= S^Prior S^Posterior S^Prior

R_{θ θ θ θ} . (4)

Again, the higher is R4, the higher is the reduction in the area (i.e., in the epistemic uncertainty) and, thus, the higher is the “updating strength” of the ap- proach.

Table 1. Most likely values c₄ of the parameters 4 = γ^, δ ^, µZm^, σZm^, µZv^, σZv^, µKs ^and σKs of the aleatory PDFs of Y₁ = Q, Y₂ = Z_m, Y₃ = Z_v and Y₄ = K_s and areas S₄ underlying the corresponding (marginal) possibility distributions before and after the Bayesian update. The point estimates θ^ˆMLE are also shown for comparison together with the values of R_θ^dist (3) and R₄ (4) (parentheses)

Possibility distributions update c₁ (R₁^dist) S₁ (R₁)

Y_j 1 1ˆ^MLE Prior Posterior Prior Posterior

Q 2 1013.21 955.55 1002.70 (81.77) 144.65 100.55 (30.49) 3 558.48 599.15 566.35 (80.65) 103.35 76.94 (25.56) Z_m 4_Zm 55.03 54.93 55.00 (70.00) 0.25 0.190 (24.00)

5Zm 0.45 0.51 0.47 (66.67) 0.12 0.110 (8.33)

Z_v 4_Zv 50.19 50.11 50.17 (75.00) 0.21 0.165 (21.43)

5_Zv 0.38 0.45 0.39 (85.71) 0.16 0.121 (24.38)

K_s 4_Ks 27.80 25.24 26.95 (66.80) 6.45 5.40 (16.28)

5_Ks 5.26 6.89 5.54 (82.82) 4.11 3.75 (8.76)

It is evident that the method succeeds in moving the most likely values c4 towards the corresponding MLE estimates θ^ˆMLE: actually, the values of R_θ^dist (3) range within 66.67% and 85.71%. From the analysis of quantitative indicator R4 (4) it can be seen that the method produces a consistent reduction in the area underlying the possibility distributions of the uncertain parameters: in particular, R4 (4) ranges between 8.76% and 30.49%. In addition, as ex- pected, the strength of the approach in reducing epis- temic uncertainty decreases with the size of the data set used to perform the Bayesian update. For exam- ple, the area S2 underlying the possibility distribution of 2 (Figure 2, top left) is reduced by 30.49% with the aid of a large data set of size D1 = 149; on the contrary, the area

S_σKs underlying the possibility dis- tribution of 6_Ks (Figure 2, bottom right) is reduced only by 8.76% by means of D4 = 5 pieces of data.

In order to show the effect that the reduction of the epistemic uncertainty in the distribution parame- ters has on the maximal water level of the river Z_c (i.e., the model output), Figure 3 shows the upper and lower CDFs

( )

c

Z z

F ^c and

( )

c Z z

F ^c (i.e., the plausibility and belief functions,

( (

c

] )

Z z

Pl ^c −∞, and

( ]

(

c

)

Z z

Bel ^c −∞, , respectively), of Zc obtained before (solid lines) and after the Bayesian update (dashed lines). Obviously, the gap between the plausibility and belief functions is larger before the Bayesian update: in particular, the ‘prior’ CDFs (solid lines) completely envelop the ‘posterior’ ones (dashed lines). This larger gap is explained by the larger area contained under the possibility distributions of the corresponding epistemically-uncertain parameters.

Then, in order to provide a fair and quantitative assessment of the approach adopted, proper indica- tors are computed. The final goal of the case study presented in Section 4 is to determine (i) the dike

level necessary to guarantee a given flood return pe- riod or (ii) the flood risk for a given dike level. With respect to issue (i) above, the quantity of interest that is most relevant to the decision maker is the 7·100%-th quantile of Zc (i.e., Z ): this corresponds _c^β to the yearly maximal water level with a 7·100-year return period. With respect to issue (ii) above, the quantity of interest is the probability that the maxi- mal water level of the river Z exceeds a given _c threshold *z , i.e., P[Z_c≥z_c*]: in the present paper,

c*

z = 55.5 m (Table 2). The intervals [

[ ]

^{FZc −1}

^{( )}

^{β ,}

[ ]

^{FZc −1}

^{( )}

^{β ] for Z , cβ 7} = 0.05, 0.50 and 0.95, are [50.70, 51.67], [52.16, 53.46] and [54.13, 56.44], re- spectively, before the Bayesian update; instead, they are [50.90, 51.56], [52.38, 53.23] and [54.21, 55.87], respectively, after the update. Also, the intervals [^1−FZc

( )

^{zc* ,1−FZc}

( )

^{zc* ] for P[Zc > zc}*] are [0.0054, 0.1092] and [0.0079, 0.0716], before and after the Bayesian update, respectively. Thus, the width of the intervals is reduced of 28.144_38.63%.

50 51 52 53 54 55 56 57 58 59

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Z_c

Cumulative probability

Prior Posterior

Figure 3. Plausibility and belief functions, (( c

]

)

Z z

Pl ^c −∞, and

(

]

( c )

Z z

Bel ^c −∞, , of the maximal water level of the river Z_c (i.e., the model output) before (solid lines) and after (dashed lines) the Bayesian update