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10
RISK ANALYSIS ON THE BASIS OF PARTIAL INFORMATION ABOUT
QUANTILES
Lev V. Utkin
Department of Computer Science, St. Petersburg Forest Technical Academy,
St. Petersburg, Russia e-mail: utkin@stat.uni-muenchen.de
Thomas Augustin •
Department of Statistics, University of Munich, Germany e-mail: thomas@stat.uni-muenchen.de
Abstract. Risk analysis under partial information about probability distributions of states of nature is studied. An efficient method is proposed for a case when initial information is elicited from experts in the form of interval quantiles of an unknown probability distribution. This method reduces a difficult to handle non-linear optimisation problem for computing the optimal action to a simple linear one. A numerical example illustrates the proposed approach.
Keywords: risk analysis, imprecise probability theory, optimisation problem, quantile, decision making, randomized strategy, expert opinions
INTRODUCTION
One of the main objectives of performing risk analyses is to support decision-making processes. Risk analysis provides a basis for comparing alternative concepts, actions or system configurations under uncertainty. A variety of methods has been developed for estimating losses and risks. When events occur frequently and when they are not very severe, it is relatively simple to estimate the risk exposure of an organization, as well as a reasonable premium when, for instance, an insurance transaction is made. Commonly used methods rely on variations of the principle of maximizing expected utility, tacitly assuming that all underlying uncertainty can adequately be described by a precise and completely known probability measure. However, when the uncertainty is complex and the quality of the estimates is poor, e.g., when evaluating low-probability, catastrophic events, the customary use of such rules together with overprecise data could be harmful as well as misleading. Therefore, it is necessary to extend the principle of maximizing expected utility to deal with complex uncertainty. This allows powerful evaluation under vague and numerically imprecise information. An efficient way for realizing such methods is the framework provided by imprecise probability theory [3,5,6].
Very often the initial data about unwanted events are elicited from experts, who are typically asked about quantiles of a random quantity (states of nature). Based on this information, and on the choice of a parameterized family of distribution functions, a fitted distribution function is chosen that represents the available information in some best way to some extent. However, as pointed out, for instance, in [2], experts better supply intervals of quantiles rather than point-values because their knowledge is not only of limited reliability, but also imprecise. Moreover, as discussed above, the choice of one particular distribution function fitted to the quantiles would lead to substantial errors in risk analysis. Therefore, new procedures for computing optimal actions under conditions of partial information about states of nature in the form of imprecise quantiles are proposed in the paper. Efficient methods for computing optimal unrandomized and randomized actions based on solving the linear optimisation problems are investigated. A numerical example illustrates the methods.
A GENERAL APPROACH TO RISK ANALYSIS AND IMPRECISE QUANTILES
Consider the basic model of decision theory: One has to choose an action from a non-empty, finite set A={a1v..,an} of possible actions. The consequences of every action depend on the true, but unknown state of nature t eQ = tm}. The corresponding outcome is evaluated by the utility function
u : (A x Q) a R (a, t) a u( a, t)
and by the associated random variable u(a) on (Q, Po(Q)) taking the values u(a,t). Alternatively a loss function l(a,t) is assigned, which can be embedded into the framework proposed by setting u(a, t) = -l(a, t). Often it makes sense to study randomized actions, which can be understood as a
probability measure A = (\,...,An) on (A,Po(A)). Then u( ) and u() are extended to randomized actions
by defining u(A, t) := Vn= u(as, t)As.
If the states of nature are produced by a perfect random mechanism (e.g. an ideal lottery), and the corresponding probability measure p on (Q, Po(Q)) is completely known, the Bernoulli principle is nearly
unanimously favored. One chooses that action A* which maximizes the expected utility
E p u(A) := V mm=j u(A, tj) p(tj) among all A. Here E p is the expectation operator with respect to the
distribution p.
Suppose that information about states of nature is represented as a set of r judgements bi - Epf - bi, i=1,...,r, on the expectations of some random quantities fj,..., fr. This set restricts all distributions p on (Q, Po(Q)) by a set M such that every distribution p from M satisfies all the inequalities. An action A* is optimal iff for all A, EM u(A ) > EM (u(A)). Here EM is the lower prevision (expectation) taken over all probability distributions p from M. Then the optimal action A* can be obtained by maximizing EM (u(A)) subject to A +... + An = 1. This leads to the non-linear optimisation problem:
min V m Vn u (as, t.) -As ■ p(t.) a max (1)
peM^J=J^s=j ^ s' ]J s yyjJ As>0 V '
subject to b - Epf - bi, i=1,...,r, \ +... + An = 1.
Similar expressions can be written in a case of the continuous set of states of nature Q = [ A, B]. In this
CB
case, the expected utility is E u(A) := I u(A, t)p(t)dt. Here p(t) is a density function which is consistent
p J A
with the set of initial judgements about states of nature. In the paper, we will consider the continuous set of states of nature.
In the probabilistic approach, experts are typically asked about quantiles of a random variable X defined on a continuous sample space Q. The smallest number t e Q, such that Pr {X - t} = k /100, is called the k% quantile and denoted qk%. In this approach, the experts are often asked to supply the 5%, 50% and 95% quantiles. In other words, an expert supplies t1, t2, t3 such that Pr{X - t1} = 0.05, Pr{X - t2} = 0.5,
Pr{X -13} = 0.95, respectively. Generally, if r experts provide their judgements about qi quantiles, i=1,...,r, of an unknown cumulative probability distribution of the continuous random variable X, this information can be represented as Pr{X - tt} = qt, i=1,...,r. In terms of the imprecise probability theory, qi
can be viewed as identical lower and upper previsions (expectations) of the gamble I[0t ](X), i.e.,
EI[0t ](X) = EI[0t ](X). Here I[0t ](X) is the indicator function taking the value 1 if X e[0, tt ] and 0 if
X £ [0, tt ]. However, judgements elicited from experts are usually imprecise and unreliable due to the
limited precision of human assessments. In other words, experts provide some intervals of quantiles in the form Xi = [itj, ti ]. This can be formally written as
Pr{X < [ti,U]} = qt, i = 1,...,r (2)
Every interval X i produces a set of probability distributions such that the lower distribution contains the point qi (ti) and the upper one contains the point qt (ti) .
Decision making with imprecise quantiles
Let us define what EM u(A) means in the case when initial information about p is given in the form of quantile intervals. Suppose that we knew precise values of qi quantiles ti, i=1,...,r. Denote T = (t1,..., tr) and the set of possible vectors T by {T}. Let EM (u(A) | T) be the lower expectation of the function u (A) under condition of precise values T of quantiles. Since at least one of the points tk belonging to the interval Xi = [t_i, ti ] is a true value of the corresponding quantile, then there holds
EM u(A) = w min EM (u(A) | T).
VteXi ,1 =1,...,r
By using the natural extension [3,4,5] for computing the lower prevision EM (u(A) | T), we get the following linear programming problem:
em (u(A) 1 T) = niax (c + ^ r=1 wtqt) (3)
subject to wi, c e R, i = 1,..., r, and c + ^=1 wiI[0^t ](t) < u(A, t), Vt eQ.
UNRANDOMIZED STRATEGY
The unrandomized strategy supposes that A = (0,..., 0, As ,0,..., 0), As = 1. Let us consider how to find the value s corresponding to the optimal action.
Proposition 1. Suppose that q1 < q2 < ... < qr and A = (0,...,0, As,0,...,0), As = 1. Denote q0=0, qr+i=1, to=A, tr+i=B. Then the solution to problem (3) exists if (i) t1 < t2 < ... < tr, (ii) tt < ti+1 for qt < qi+1, 7=1,...,r, and this solution is
em (u(A) IT ) = X r=0 (q+1- q) t ît?1" u(as,t ).
7=0 t<=-iti ,ti+1]
Let us consider an approximate solution of the decision making problem in the case of interval quantiles. Let us divide the sample space Q into N intervals by points A = t0,t1,..., tn-1,tn = B . Then the
set {T} becomes finite and contains vectors of the form (rl(0),T(1),...,T(r)) such that Tl^) e [t7,ti], i.e., l(i) is an index of a point belonging to X7 .
Proposition 2. Suppose tt e [t^, ti ], i=1,...,r. If there exist such i and j that î7 > tj and qt < qj, then judgements are conflicting, otherwise the optimal action is
as = ai-g max min^M u = arg ( mi]tl )Xi=0 (qi+1 - q>) [ min ]u(as, t) .
s s=1,...,nTe{T} s s=1,...,n (Tl(0),TlC1),...,Tl(r)) '=0 te[Tl(iTl(i+1)]
In particular, if all utility functions u(as,t), s=1,...,n, are decreasing as t is increasing, then aopt = argmax (■=0(^+1 -q)u(as,ti+1)), if u(as,t) , s=1,...,n, are increasing, then
aopt = arg mi« r=0(q+1- q)u (as, ti)) • RANDOMIZED STRATEGY
The technique proposed in the previous sections leads to a series of non-linear optimisation problems in the case of the randomized strategy. Therefore, it is necessary to consider a different method for computing A • Here the modification of an approach proposed by Augustin [1] based on using sets of extreme points is applied. The optimisation problem for computing the optimal randomised action is
mn ¡A (Zn=1u (a,t)A)))dt ^ max, (4)
subject to A+ ••• + An = 1 and Pr{X < [^, ti ]} = qi, i = 1,..., r . Let us introduce the variable
g=mn ¡A (z n=1u (as,t )A) p(t)dt
and consider the sense of (2). If to call the expectation EM (u(A)| T) and the set of constraints
Pr{X < tt} = qt, i=1,...,r, for every fixed Tby an imprecise model, then (2) corresponds to the union of a set
of imprecise models taken over all possible vectors T, i.e., the set M of distributions p restricted by constraints (1) is the union of sets MT . According to [3], a set of extreme points of the united model is the union of extreme points of the imprecise models corresponding to vectors T, i.e., extr(M) = uTe{T} extr(MT). This implies that a set of problems (4) can be reduced to the problem:
max G
A eR+ ,GeR
subject to
G <£(zn=1 u(as,t)A)p(t)dt, p e Ut£{T}extr(MT), Z1= = 1. (5)
Now we have to find the extreme points for each T e {T}. Let us rewrite the available information about quantiles corresponding to T in the following form:
¡A p(t)dt = q, ¡A p(t)dt = q2- q^^.^ ¡t p(t)dt =1 - qr .
All equalities can be considered independently in the sense that they do not have common variables. If we approximately represent the integrals as sums, then the i-th hyperplane produced by the i-th equality has the following extreme points:
(q -q_1,0,...,0), (0,qt -q,._1,...,0),...,(0,0,...,qt -q^) .
Hence the set MT has the extreme points of the form:
p(t)=Z r=0(q+1 - q> )s(t-t ), t e [^, ti+l],
where 5(t - zt) is the Dirac function which has unit area concentrated in the immediate vicinity of the point
t; t0 = A, tr+1 = B, q0 =0, qr+1 =1-
After substituting these extreme points into constraints (5), we get
G < Z n=1 As Z r=0 f+1 u (as, t)p(t)dt, Vp e extr (MT ) .
ti
If we take one set of extreme points by fixed T, then there holds
G <Zn=1As Zr=0 u (as t )(q+1 - qi X VT e [ti, t+1]. (6)
Let us consider an approximate solution of the decision making problem in the case of interval quantiles. By dividing the sample space Q into N intervals by points A = t0,t1,...,tn-1, TN = B (see the section "Unrandomized strategy"), we get a finite set of constraints
G < Z n=1 As ZL u(as t №+1- q), vt e [T «T o+1)], Vl (i) (7)
Proposition 3. Suppose ti e [^, ti ], i=1,...,r. If there exist such i and j that ti > tj and q < qj, then
judgements are conflicting, otherwise the optimal randomized action is approximately defined by solving the following linear programming problem:
max G,
AseR+ ,GeR
subject to (7) and Z = As = 1.
Since the right side of (7) has to be as small as possible and qi+1 - qi > 0, then the set of constraints is reduced to one constraint in the case of increasing or decreasing utility functions. By considering the set {T}, we can say that constraints (7) have to be satisfied for arbitrary values tt and ti+1 such that tt e Xi and
ti+1 e Xi+1, i=0,...,r. It is obvious that min u(as,Ti) by tt = ti for increasing utility functions (by ti = ti+1
T e[ti ,ti+1] "
for decreasing utility functions) is less than by any tt > ti (ti < ti). This implies that we remain one constraint
G < s=1 As Z ' r=0 (qi+1
qt )u (as, ti) (8)
in the case of increasing utility functions or one constraint
G < Z h-A Z r=o(q+1 - q > (a, _'+O (9)
in the case of decreasing utility functions.
Proposition 4. The linear optimisation problems with constraints (8) or (9) have the following solution:
G=ma* Z r=0 (q+1- q>)u (as, T ),
11, k = arg max Z r=0 (q+1- q)u (as, t )
Ak =< s s=i, ,n ,
0, otherwise
where Ti = ti for increasing utility functions, Ti = ti+1 for decreasing utility functions.
Proposition 4 implies that the randomised optimal action for the considered decision problem is equivalent to the unrandomized one (see Proposition 2).
NUMERICAL EXAMPLE
Suppose experts provide 5%, 50%, 95% quantiles of the probability distribution of a random variable defined on the sample space Q = [0,120]. This implies r=3 and q1=0.05, q2=0.5, q3=0.95. Expert
judgements are given in Table 1. Suppose we have to choose one of two actions {a1, a2} in accordance with utility functions
u(a1, t) = exp(-0.1t), u(a2, t) = 0.5 - 0.012x .
Table 1. Interval quantiles provided by experts
5% 50% 95%
Lower Upper Lower Upper Lower Upper
2 4 12 15 19 19
Since the utility functions are decreasing, it follows from Proposition 4 or Proposition 2 that
k = arg m ax Z ;=0 (q+i- q )u (a, ~ti+i ) •
If s=1, then we get the lower expected utility
(q - qo )u («1, ti) + (q2 - q )u (a, 12) + (q - q2 )" (a, t3) + (q4 - )u (ax, t4 ) = (0.05 - 0) exp(-0.ix 4) + (0.5 - 0.05) exp(-0.i x i5) +(0.95 - 0.5) exp(-0.ixi9) + (i - 0.95) exp(-0.i x 20) = 0.208.
If s=2, then the lower expected utility is
(qi- q0)u (a2, ti)+(q2- qi)u (a2,12)+( q3- q2 )u (a2,13)+q - q3)u (a, r 4)
= (0.05 - 0)(0.5 - 0.0i2 x 4) + (0.5 - 0.05)(0.5 - 0.0i2 x i5)
+(0.95 - 0.5)(0.5 - 0.0i2 x i9) + (i - 0.95)(0.5 - 0.0i2 x 20) = 0.302.
The above numerical results imply that the optimal action is a2.
SOME REMARKS ABOUT DISCRETE STATES OF NATURE
If the set of states of nature is discrete, Q = {t1,..., tm }, then information about interval quantiles can be represented as
q < Pr{X < [ti, 1 ]} < q+1,1 -qt < Pr{X > [t_t, 1 ]} < 1 -q-1,i = 1,...,r . In this case, the linear programming problem for computing EM (u(A) | T) is of the form:
em (u(A) 1 T) = m,ax (c+Zr=1 (ca - diq+1 + wi(1 - q) - v(1 - q-1))
cc,diwi v \ —1-1 /
subject to c, diwi, v e R +, c e R, i = 1,..., r, and
c + Z r=1 ((Ci - dt) Im ] (t) + (Wi - Vi) I[ti ,m] (t) ) < u (A, t), Vt eQ .
Generally, it is difficult to find any solution to the above problem in the explicit form. However, this problem can be numerically solved for every T e {T}, and the optimal action is computed by maximizing
min EM (u(A) | T) over all possible actions.
T e{T }
CONCLUSION
Computationally simple algorithms have been obtained for calculating optimal actions under partial information about quantiles. It is worth noticing that we have focused in this paper on the basic decision problem. However, the ideas of this paper should be also applicable to more complex decision problems, for example, multi-criteria decision making, or the case where additional sample information is available.
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