CONCENTRATION FUNCTION AND SENSITIVITY TO THE PRIOR. Consiglio Nazionale delle Ricerche. Institute of Statistics and Decision Sciences

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1 CONCENTRATION FUNCTION AND SENSITIVITY TO THE PRIOR Sandra Fortini 1 Fabrizio Ruggeri 1; 1 Consiglio Nazionale delle Ricerche Istituto per le Applicazioni della Matematica e dell'informatica Institute of Statistics and Decision Sciences Duke University AMS (1991) subject classication: Primary 6F15; Secondary 6F35. Key Words: Concentration function; Bayesian robustness; "{contaminations; Gini's concentration ratio; Pietra index. Running head: Concentration and sensitivity 1

2 Abstract. In robust Bayesian analysis, posterior ranges of some functions of the unknown parameters are usually considered when the prior probability measure varies in a class?. Such functions (probability of given sets, mean, variance, etc.) describe the variation of just one aspect of the posterior measure. The concentration function may describe more globally the changes in the posterior probability measure, because it gives simultaneously both the minimum and the maximum posterior probabilities, for any measure in?, achievable by all the sets with the same probability under a base posterior. In this paper, the concentration function is used to study the sensitivity of the posterior measures when some classes? of "{ contamination priors are considered. Besides, two well-known concentration indices are employed as a measure of the robustness and, nally, the results are applied to some examples. 1. Introduction. As described in Berger (1984, 1985, 199, 1994), the robust Bayesian approach usually deals with the uncertainty in specifying just one prior probability measure. Sometimes, a class? of probability measures seems to be the most plausible result of an elicitation process (e.g. it might consist of all the priors elicited by some experts). The class? should be easily specied, large enough to contain any plausible prior and small enough to exclude unnatural priors, and involving relatively simple computations. Posterior ranges of functions of the parameters (e.g. set probability, mean, etc.) are usually considered, as the prior measure varies in?. Small ranges mean that the inference (or decision) is not actually aected by a particular choice in?. As pointed out in Fortini and Ruggeri (1994a), a dierent robustness prob-

3 lem might be faced by means of the concentration function (c.f.), comparing the functional forms of any posterior probability measure, from a class? of priors, with a base posterior. The c.f., as dened by Cifarelli and Regazzini (1987), extends the notion of Lorenz curve and allows nding the range of the probabilities, under, of all the sets with equal probability x under. Robust analyses can be performed considering the largest range, for any x in [; 1], as the prior varies in?. The width of such intervals, expressed by means of the c.f., gives the distance between the posteriors and. Thus, it is worth computing the posterior c.f. for relevant classes of priors and the paper provides a simple, useful, expression for the "{contamination one; besides, the paper gives conditions under which the maximum range is easily found. Finally, another comparison is made possible by considering the concentration indices given in Cifarelli and Regazzini (1987). The paper complements the work in Fortini and Ruggeri (1994a) where the c.f. was mainly used to dene neighbourhoods of probability measures and computing bounds on posterior quantities over such classes. Here we develop their suggestion of considering a xed continuous, convex, monotone nondecreasing function which should lie below any posterior c.f. in order to ensure robustness. The results are then applied to some examples, including a model for the magnitudes of the major earthquakes in Lazio, an Italian region.. Concentration function. Cifarelli and Regazzini (1987) dened the c.f., as a generalisation of the Lorenz curve, described in Marshall and Olkin (1979, pag.5). The classical denition of concentration refers to the discrepancy between a discrete probability and a uniform one, say, and allows for the comparison 3

4 of the two probability measures, looking for subsets where is much more concentrated than (and vice versa). Cifarelli and Regazzini (1987) dened and studied the c.f. of with respect to (w.r.t.), where and are two probability measures on the same measurable space (; F). According to the Radon-Nikodym theorem, there is a unique partition fn; N C g F of and a nonnegative function h on N C such that, 8E F, (E) = s (N) = s (), where a () = Z Z E\N C h() (d) + s (E \ N), (N) =, \N C h() (d) and s denote the absolutely continuous and the singular part of w.r.t., respectively. Set h() = 1 all over N and dene H(y) = (f : h() yg) ; c x = inffy R : H(y) xg. Finally, let L x = f : h() c x g and L? x = f : h() < c x g. Denition 1 The function ' : [; 1]! [; 1] is said to be the concentration function of w.r.t. if '(x) = (L? x ) + c x fx? H(c? x )g for x (; 1), '() = and '(1) = a (). Observe that '(x) is a nondecreasing, continuous and convex function, such that '(x),?, '(x) = x 8x [; 1], = and '(x) = Z cx [x? H(t)]dt = Z x c t dt: (1) To better understand the behaviour of the c.f., two situations are described: '(1) = 1 denotes that is absolutely continuous w.r.t. while '(x) = ; x, means that gives no mass to a subset A F such that (A) =. To show how to practically draw the c.f. when and are absolutely continuous w.r.t. Lebesgue measure, consider, as an example, < +, the Borel {algebra F on it, G(; ) and E(1); then it follows that the Radon-Nikodym derivative is h() = 4 e?. 4

5 The c.f. '(x) is obtained by evaluating x = (L q ) and '(x) = (L q ), where L q = f : h() qg and q takes as many as possible values to draw the c.f. within the required accuracy. Probability measures can be compared, considering a partial ordering, induced by the c.f., in the space P of all probability measures. Denition Let ' 1, ' denote the c.f.'s of 1 and w.r.t. ; if ' (x) ' 1 (x) ; 8x [; 1]; we will say that is not less concentrated than 1 w.r.t.. Afterwards we will denote it by 1. A total ordering, consistent with the previous partial one, is achieved (Ci- R 1 farelli and Regazzini, 1987) when considering C () = fx? '(x)gdx and G () = sup x[;1] (x? '(x)) which are, respectively, the well-known Gini's area of concentration (Gini, 1914) and an index proposed by Pietra (1915), which equals half the total variation norm, as proved by Cifarelli and Regazzini (1987). Observe that C () = () G () = () = ; C () = 1 () G () = 1 ()? : The following Theorem, proved in Cifarelli and Regazzini (1987), states that '(x) substantially coincides with the minimum value of on the measurable subsets of with -measure not smaller than x. Theorem 1 If A F, (A) = x, then '(x) a (A). Moreover if x [; 1] is adherent to the range of H, then B x exists such that (B x ) = x and '(x) = a (B x ) = minf(a) : A F and (A) xg: () 5

6 If is nonatomic, then () holds for any x [; 1]. Such a Theorem is relevant in applying the c.f. to robust Bayesian analysis; in fact, given any x [; 1], the probability, under, of all the subsets A with {measure x, is such that '(x) (A) 1? '(1? x): (3) Two examples are presented to show how the c.f., far from substituting the other usual functions (e.g. the mean), furnishes dierent information about the probability measures. Comparisons among probability measures are sometimes made through their moments; it is well known that dierent measures could be found such that they share the rst n moments, while their dierence could be detected by the c.f.. As another example, consider two measures concentrated on disjoint but very close sets, say E 1 and E. In this case, the c.f. shows the dierence between them, which is large on the subsets which contain just one E i, i = 1;, although their means are very close. Finally, for completeness, it is worth mentioning all the other papers on the c.f., mainly applied to Bayesian robustness, which appeared after the works by Cifarelli and Regazzini: Fortini and Ruggeri (199, 1993, 1994b), Ruggeri and Wasserman (1991), Rotondi, Ruggeri and Vercesi (199), Carota and Ruggeri (1993) and Ruggeri (1993a). 3. Statement of the problem. In this paper, a class of posterior probability measures, corresponding to a class? of priors, is compared to a base posterior measure by means of their functional forms. Instead of looking at some 6

7 summarising quantity, like the mean, the concentration of the probability all over the parameter space is considered. The c.f. is the natural tool to perform such task, because it detects when a measure is more concentrated than another one. As mentioned in Fortini and Ruggeri (1994a), it is possible to specify a continuous, convex, nondecreasing function so that all the c.f.'s of the posterior measures w.r.t. the base one have to lie above it in order to have robustness. In this paper, following a suggestion in Regazzini (199), we consider the sensitivity to changes in the prior, looking for the lowest among all the c.f.'s of the posterior probability measures corresponding to some xed class of priors. If the lowest c.f. does not exist, then the inmum, pointwise, is considered. Besides, the robustness could be measured considering the distance of such function from the line connecting (; ) and (1; 1) by means of some indices, due to Gini and Pietra, and related to the c.f., as described in Cifarelli and Regazzini (1987). More formally, let X be a random variable on a dominated statistical space denoted by (X ; F X ; fp ; (; F)g); given a sample ~x from X, the experimental evidence about will be expressed by the likelihood function l ~x (), which we assume F X F-measurable. Let P denote the space of the probability measures on the parameter space (; F). Given a prior P, the posterior measure is dened by (A) = R A l ~x()(d) m, where m = R l ~x()(d) and A F. According to the robust Bayesian viewpoint, a class? of probability measures has to be considered, instead of eliciting just one prior. Suppose that there exists a base prior, like in the "{contamination class, and consider the class of the c.f.'s of,?, w.r.t.. Because of Theorem 1 and (3), it follows, for any? and A with (A) = x, 7

8 that b'(x) (A) 1? b'(1? x); where b'(x) = inf ' (x), for any x [; 1].? The interpretation of b', in terms of Bayesian robustness, is straightforward: the closest b'(x) and 1? b'(1? x) are for all x [; 1], the smallest the dierence of the functional forms of the posterior measures is. In particular, following a suggestion in Fortini and Ruggeri (1994), it should be checked that b'(x) g(x) for all x [; 1], where g(x) represents the maximum allowed discrepancy in order to ensure robustness. As an example, it could be g(x) = x, corresponding to the request sup j (A)? (A)j (A)(1? (A)). AF: (A)=x The computation of b' is quite simple when there exists b? such that b' 'b and this paper gives conditions ensuring its existence for some classes of priors; otherwise its computation becomes a harder task, which can be solved numerically. Besides, in this paper the comparison among probability measures is also possible through the concentration indices, considering C = sup C () and? G = sup G ():? The c.f. is well suited to satisfy the need, pointed out by Wasserman (199), of graphically summarising the results of a robust Bayesian analysis. In fact, the plot of b'(x) and 1? b'(1? x), like in Fig.1, gives a very intuitive description of the discrepancy between the base prior and the other measures. In Fig.1, '(x) denotes the c.f. of G(; ) with respect to E(1) and it is shown, as an example, that [:16; :559] is the range spanned by the probability, under, of the sets A with (A) = :4. 8

9 4. Analytic results. The c.f. and its derivatives are computed when the base prior is contaminated, by considering the class? " of the "{contaminated priors. Denition 3 Let be a xed prior probability measure and let " [; 1]. The class? " = f Q = (1?") +"Q; Q Qg, where Q P, is said to be an "{contamination class of priors. Some classes Q have been proposed and their properties are discussed in Berger (199). In this paper, we consider the class Q A of all the probability measures and, if is unimodal, the class Q U of all unimodal probability measures, with the same mode as. Proposition 1 Let and Q be probability measures on (; F); for any a [; 1], dene = (1? a) + aq. If ' and ' are the c.f.'s of and Q with respect to respectively, then '(x) = (1? a)x + a' (x): Let ' +(x) and '?(x) be, respectively, the right-hand and the left-hand derivatives of '(x), then it follows that ' +() = (1? a) + a inf ^h () and '?(1) = (1? a) + a sup ^h (); where ^h is the Radon-Nikodym derivative of Q with respect to. PROOF. The relation between ' and ' follows quite easily from the definition of c.f.. As pointed out by Cifarelli and Regazzini (1987), the right-hand derivative can be computed, because (1) implies that (' )? (1) = c 1. In the same way, (' ) + () is computed, observing that c t is right continuous at the origin. 9

10 Proposition Consider = (1? ") + "Q so that (1? ")D = (1? ")D + "D + "D Q Q ; with Q (1? ")D + "D Q Z D = l ~x () (d) and D Q = Z l ~x ()Q(d): (4) Let ' and ' denote the c.f.'s of and Q with respect to, respectively, so that '(x) = Moreover, it follows that ' +() = = '?(1) = = (1? ")D "D Q x + ' (x): (1? ")D + "D Q (1? ")D + "D Q (1? ")D (1? ")D + "D Q + "D Q inf (1? ")D + "D Q D (1? ") + " inf ^h () (1? ")D + "D Q (1? ")D (1? ")D + "D Q + ( D (1? ")D + "D Q ; h () "D Q sup h () (1? ")D + "D Q (1? ") + " sup ^h () where ^h and h are, respectively, the Radon-Nikodym derivatives of Q with respect to and of Q with respect to. PROOF. The expression about is well known (see Berger, 1985, p.6). The other results follow from Proposition 1 and the fact that ^h () D Q D because ^h () D Q l ~x ()(d) = l ~x ()Q(d) and h ()(d) = Q(d). D ) ; = h () The previous Propositions are now applied to the robustness analysis. The family of all posterior c.f.'s could be considered to get the inmum, pointwise, of the c.f.'s, as shown in Example 3. Besides, Example 4 will show how to assess robustness by means of a function lying under all the c.f.'s. We prefer now ending this Section by presenting some conditions ensuring the existence of the lowest c.f.. measures. After xing a nonatomic prior, let Q be a class of contaminating probability 1

11 Proposition 3 Suppose there exists b such that Q b Q, where Qb probability one to b. Given Q Q such that Z assigns l ~x ()Q(d) l ~x (b ); (5) then Q b Q : (1? ")D PROOF. Proposition and (1) imply that ' Q = b(x) (1? ")D + "l ~x ( ) x. (1? ")D Moreover, since l ~x ( ) D Q ; ' Q x ' b(x) Q (x) : (1? ")D + "D Q Condition (5) may be actually achieved, as shown in the examples of Section 4. The next Propositions follow immediately from Proposition 3. Proposition 4 Suppose that b is the maximum likelihood estimator of, given l ~x (). If Q b Q, then Q b Q for any Q Q. In particular, this result holds if Q = Q A. Proposition 5 Suppose that is unimodal with mode. If Q Q U and then Q Q for any Q Q. Z sup l ~x ()Q(d) l ~x ( ); QQ Another result is consequent upon Propositions 4 and 5. Proposition 6 Under the same hypotheses as Proposition 3, if (5) holds for any h i Q Q, then sup (A)? Q(A) (A) 1 + 1? " D?1, 8A F. Moreover C = G = 1 + 1? " D?1 QQ " l ~x ( ). " l ~x ( ) It follows from this Proposition that if (5) is fullled by any Q in Q, then there exists a contaminating measure Qb, which is the worst with regard to the 11

12 robustness, leading to the lowest c.f.. The same is not true if (5) does not hold, making the analysis more complex. Suppose that the space P is dominated by a measure and let denote the Radon-Nikodym derivative of with respect to ; assume that is the support of. Proposition 7 Let be unimodal with mode and let Q = Q U. Let h Q denote the Radon-Nikodym derivative of Q w.r.t.. Moreover let inf h Q () = hold for any Q Q. If there exists Q Q such that Z l ~x ()Q(d) > l ~x ( ); (6) then there is no b Q such that ' b Q (x) ' Q (x) for any x [; 1] and Q Q. PROOF. Let Q be the probability measure which gives probability one to. Since Q is a convex combination of absolutely continuous measures with respect to and Q, then ' Q (1) = ( Q ) a () < ( Q ) a() = ' Q (1). From (6) and Proposition, it follows (' Q ) () + < (' Q ) + (): Therefore, since the c.f. is continuous, there is a neighbourhood of the origin I() such that ' Q (x) < ' Q (x), for any x I(). The thesis follows immediately. If is unimodal and Q = Q U, then the behaviour of any contaminating measure, not covered by Propositions 4 and 7, depends on the value of ". Proposition 8 Let be unimodal with mode and let Q = Q U. Let h Q denote the Radon-Nikodym derivative of Q w.r.t.. Let Q be the probability measure which gives probability one to. If there exists Q Q such that inf h Q () > and R l ~x()q(d) > l ~x ( ); then Q Q if and only if inf q() () > D Q? l ~x ( ) D + " 1? " l ~x( ) : (7) 1

13 PROOF. The result follows from Proposition, by observing that (7) implies that (' Q ) +() > (' Q ) +(). As shown in the proofs of Propositions 7 and 8, the right-hand derivative of '(x) at x = is important in proving the existence of c.f.'s lying under the others just in a neighbourhood of the origin. Furthermore, the left-hand derivative at x = 1 plays a similar role. It is worth observing that such derivatives allow the approximation of the lower and upper bounds on the probability of the subsets A such that (A) is suciently close to. In fact, the minimisation of ' + (maximisation of '?) determines the contaminated probability measure whose c.f. lies under all the others in a neighbourhood of x = (x = 1). 5. Numerical examples. The c.f.'s are used in analysing four examples, in which the results shown in Section 3 are applied. Example 1. Assume that X N (; ), known, and N ( ; ), and known, with density function (). Let Q be the class of the probability measures which are either Q k U(? k; + k), k >, or Q 1, which assigns probability one to the point. Dene k = (1? ") + "Q k. Given a sample ~x from X, then the likelihood function is l ~x () = e?(?~x) =( ) p. The density of Q k, k >, is given by q k () = 1 k I [?k; +k]() where I A is the indicator function of the set A. (Note that Q k converges in distribution to Q 1 as k!.) According to the denitions as given in (4), it follows that D = e?(?~x) =(( + q )) and D ( + ) Qk = 1 Z +k l ~x ()d. k?k 13

14 Result 1 Given a sample ~x, then D Qk j ~x? j 1. PROOF. It holds that l ~x ( ) for any k > if and only if lim D k! + Q k = l ~x ( ) and lim D Q k!+1 k = +. Set h = k and v = j ~x? j. Then the derivative of D Qk w.r.t. h exists, D Qk (h) = 1 p (? R v+h v?h e?t = dt h + e?(v?h) = h + e?(v+h) = with lim f(h) = and f (h) = h[(v? h)e?(v?h) =? (v + h)e?(v+h) = ]. h! + h ) = 1 p f(h) h, For any h >, f (h) < if and only if g(h) = (v? h)e vh? (v + h)e?vh <. Such a condition is obviously satised if h v, so that D Qk l ~x ( ) for any k j ~x? j. Otherwise, suppose that v 1, then g(h) = ( 1 X v j+ h j+1? (j + 1)! 1X v j h j+1 (j)! ) ( 1 X v j h 1X j+1 (j + 1)!? ) v j h j+1. (j)! The convexity of D Qk implies that D Qk l ~x ( ) for any k > if v 1. Suppose now that v > 1, then lim D h! + Q k (h) = lim h! + f (h) that D Qk > l ~x ( ), for any k close enough to. 4h = e?v lim h(v? 1) = + so h! + Result Given a sample ~x, then the c.f. ' 1 of 1 w.r.t. is such that ' 1(x) ' k (x), for any x [; 1] and for any c.f. ' k of k w.r.t., if and only if j ~x? j 1. Given such a sample ~x, it follows that C = G = ( 1 + 1? " " s + exp? (~x? )!)?1 ( + ). Otherwise, there is no Qb k such that 'b k (x) ' k (x), for any x [; 1] and ' k. PROOF. Combining the previous Result and Proposition 5, it is immediate to prove that ' 1 lies under any other c.f. ' k if j ~x? j indices follows from Proposition The value of the 14

15 any k. Otherwise, dene h k () = q k( j ~x) ( j ~x) = q k() () D, then inf h k () =, for D Qk Given such a sample ~x, the previous Result implies that lim k! + D Qk = D + there exist innite values k > such that D Qk so that > l ~x ( ). From Proposition 7, the corresponding c.f.'s ' k lie under ' for some values x (; k ), with < k < 1. Since Q is the only measure in Q which is singular w.r.t., then it results ' k (1) > ' (1) so that no k, k >, is less concentrated than the other measures. Take =, = = 1, then it follows that C = G = ( 1 + 1? " p " exp? ~x 4!)?1. For any j~xj 1, it can be easily seen that the indices are small (large) when " is small (large), so that major contaminations lead to nonrobust situations, as intuitively expected. As an example, suppose ~x = :5, so that C = :15 if " = :1 and C = :69 if " = :5. Example. Assume that X E() and G(1 + ; 1), >, with density function (). Let Q be the class of the probability measures which are either Q G(1 + ; ), >, or Q 1, which assigns probability one to the point. Dene, for any >, = (1? ") + "Q. Given a sample ~x from X, then the likelihood function is l ~x () = e?~x. The measures Q are unimodal and their densities, for >, are given by q () = () e??( ). (Note that q 1 furthermore, Q converges in distribution to Q 1 as! +1.) and, According to the denitions as given in (4), it follows that 15

16 D = 1 + (1 + ~x) + and D Q = 1+ (1 + ). ( + ~x) + Result 3 Given a sample ~x, then D Q? p ~x + p. l ~x ( ) for any > if and only if PROOF. Set y = ~x and =, then D Q l ~x ( ) () 8 > f() = y + log(1 + 1 ) + ( + ) log( + y ). Note that lim f() =?1; lim f() = and f () =! +!+1 h() (1 + ) (y + ) where h() = 3 (y? 4y + ) + (1? 4y)? (y + y)? y : It holds lim h() =?y < and lim h() =! +!+1 8 >< >:?1? p y + p +1 o:w: The values y 6 [? p ; + p ] must be excluded because lim h() = +1!+1 implies that lim f() =!+1 + so that D Q > l ~x ( ) denitely as! +1. It is easy to prove that h () < for > so that f() is concave and negative for any >, for any value y [? p ; + p ]. : Result 4 Given a sample ~x, then the c.f. ' 1 of 1 w.r.t. is such that ' 1 (x) ' (x), for any x [; 1] and any c.f. ' of w.r.t., if and only if? p ~x + p. Given such a sample ~x, it follows that C = G = ( 1 + 1? " " 1 + (1 + ~x) + e ~x )?1. Otherwise, there is no b such that ' b (x) ' (x), for any x [; 1] and '. PROOF. Combining the previous Result and Proposition 5, it is immediate to prove that ' 1 lies under any other c.f. ' if? p ~x + p. The value of the indices follows from Proposition 6. 16

17 Otherwise, dene h () = q ( j ~x) ( j ~x) = q () () D D Q. Excluding the trivial case = 1 such that ' 1 (x) = x for any x [; 1], it may be proved that inf h () = 8 >< >: > 1?( + ) (?1) e? (?1) ( + ~x) +?( + )(1 + ~x) + > < < 1 : Given such a sample ~x, the previous Result implies that lim f() =!+1 + so that there exist innite values > 1 such that D Q > l ~x ( ). From Proposition 7, the corresponding c.f.'s ' lie under ' 1 for some values x (; ), with < < 1. Finally, ' (1) > ' 1 (1) implies that no, >, is less concentrated than the other probability measures. Take = 1, then it follows that C = G = ( 1 + 1? " " ) e ~x?1 (1 + ~x) 3 For any ~x such that? p ~x + p, it can be easily seen that the indices are small (large) when " is small (large), so that major contaminations lead to nonrobust situations, as intuitively expected. As an example, suppose ~x = 3, so that C = :16 if " = :1 and C = :6144 if " = :5.. Example 3. Assume that X b(1; ), i.e. a Bernoulli random variable, and has density function () = 8 >< >: 4 1= 4(1? ) 1= < 1: Consider the class of probability measures Q = fq 1 ; Q g, where Q 1 U(; 1), with density q 1 (), and Q U(; 1=), with density q (). 17

18 Dene k = (1? ") + "Q k, k = 1;. Given the sample ~x = 1, then the likelihood function is l ~x () =. According to their denitions as given in (4), it follows that D = 1=, D Q1 = 1= and D Q = 1=4. Dene h k () = q k( j ~x) ( j ~x) = q k() () D D Qk, k = 1;. It follows that h 1 () = 8 >< >: h () = (1? ) 8 >< >: 1 1= 1= < 1; 1= 1= < 1: Result 5 Let ' k be the c.f. of k w.r.t., k = 1;. Then it results ' 1 (x) = (1? ")x + "(1? p 1? x); x 1, and ' (x) = 8 >< >: (1? ")? " x x =3 (1? ") q? " x + "? " (1? 3 9(1? x) ) =3 < x 1: PROOF. Let ' k be the c.f. of Q w.r.t. k, k = 1;. Because of the denition of c.f. and h 1 () = h 1 (1? ), 1, it follows that ' 1 (x) = R 1?b b 1? b, where b = p 1? x=, b 1=, is the solution of x = R 1? b b x 1. q 1 (j~x)d = (j~x)d, Since h () = when 1= < 1, it follows that ' (x) = if x =3 = R 1= ([1=; 1]). When =3 < x 1, then ' (x) = q (j~x)d = 1? 4b, where b b = 3p 3(1? x)=, b 1=, is the solution of x = R 1= Applying Proposition, then ' 1 and ' are obtained. b (j~x)d + =3. Considering the convexity of the c.f.'s and evaluating the derivatives of ' 1 and ' at x = ; 1, it is proved that none of them lies under the other, so that 18

19 the robustness analysis has to be performed studying b'(x) = x [; 1], and the corresponding concentration indices. inf QQ ' Q(x), for any Result 6 Under the above conditions, it follows that b'(x) = 8 >< >: (1? ")? " x x =3 (1? ") q? " x + "? " (1? 3 9(1? x) ) =3 < x (1? ")x + "(1? p 1? x) < x 1; where is the unique solution of ' 1 (x) = ' (x), =3 < x 1. C = " 3? " and G = "? " " 11 15? 3p 9 5 (1? )5=3 # + " 3 (1? )3=. PROOF. The comparison of ' 1 and ' leads quite easily to nd b'. Furthermore The index C is the maximum between the Pietra's indices concerning ' 1 and '. The rst index is given by x? ' 1 (x) evaluated at x = =3 because it is the maximum for x =3 and ' 1 (=3) > 1 for any ". The second index is given by x? ' (x) evaluated at x = 3=4 but it is smaller than the rst one, for any ". The computation of the Gini's area of concentration G is indeed straightforward.. As an example, suppose " = :1 so that C = :3588 and G = :3973, while C = : and G = :4616 if " = :5. As shown by the indices, the base probability measure is just slightly aected by the contaminating priors, also under rather large values of ". Example 4. Consider the magnitude X of major earthquakes (i.e. with magnitude not less than 5) occurred in the Italian region Lazio since 186, extracted from a catalogue prepared by the Gruppo Catalogo dei Terremoti dell'istituto per la Geosica della Litosfera del Consiglio Nazionale delle Ricerche (CNR). Observe 19

20 that the real variable X is recorded at the values (5; 5:1; 5:; : : :), so that, it seems reasonable to consider, like in Ruggeri (1993b), the random variable Y = 1(X? 5) dened on N, distributed geometrically, with parameter. Such a simplication does not aect our analysis heavily, provided that we take ' 1, because the probability of Y being large is quite small since P 1 k=k (1? )k = (1? ) K ', even for a small K. A conjugate probability measure Be(8; ) and a noninformative one, U(; 1), are chosen as priors on the parameter. Their posteriors turn out to be, respectively, Be(11; :9) and Be(4; 1:9) and their posterior means are :7914 and :678, with respective standard deviations :153 and :1779. The two posterior means could be deemed enough close to ensure robustness, but a dierent picture is obtained when considering the c.f. of w.r.t. and the criterion for robustness is given by sup AF: (A)=x j (A)? (A)j (A)(1? (A)); i.e. '(x) g(x) = x ; 8x [; 1]. As shown in Fig., the c.f. '(x) lies below the curve g(x), showing that the functional forms of the posterior probability measures are quite dierent, at least according to our criterion. 6. Future researches. In this paper, the c.f. has been used to compare the functional forms of the posterior probability measures corresponding to a class of priors w.r.t. a base one. In principle, it is possible to dene a class of sampling models, so that a class of posterior probability measures is obtained as the likelihood changes. The same approach followed in this paper could then be applied, but we expect that computing the lowest c.f. will be an harder task.

21 Further developments are possible when considering either other classes of priors (possibly, nonparametric), like in Carota and Ruggeri (1993), or innitesimal variations of the base prior, like in Fortini and Ruggeri (1994b). Bibliography Berger, J. (1984). The robust Bayesian viewpoint (with discussion). Robustness of Bayesian Analysis, (ed. J.Kadane), Amsterdam: North Holland. Berger, J. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag. Berger, J. (199). Robust Bayesian Analysis : sensitivity to the prior. J. Statist. Plann. Inference, 5, Berger, J. (1994). An overview of Robust Bayesian Analysis. To appear in TEST. Carota, C. and Ruggeri, F. (1993). Robust Bayesian analysis given priors on partition sets. Quaderno IAMI 93.1, Milano: CNR-IAMI. Cifarelli, D.M. and Regazzini, E. (1987). On a general denition of concentration function. Sankhya B, 49, Fortini, S. and Ruggeri, F. (199). On dening neighbourhoods of measures through the concentration function. To appear in Sankhya A. Fortini, S. and Ruggeri, F. (1993). Concentration function and coecients of divergence for signed measures. To appear in J. Ital. Statist. Society. 1

22 Fortini, S. and Ruggeri, F. (1994a). Concentration functions and Bayesian robustness. To appear in J. Statist. Plann. Inference. Fortini, S. and Ruggeri, F. (1994b). Innitesimal properties of the concentration function. Unpublished Manuscript. Gini, C. (1914). Sulla misura della concentrazione della variabilita dei caratteri. Atti del Reale Istituto Veneto di S.L.A., A.A , 73, parte II, Marshall, A.W. and Olkin, I. (1979). Inequalities: theory of majorization and its applications. Academic Press, New York. Pietra, G. (1915). Delle relazioni tra gli indici di variabilita. Atti del Reale Istituto Veneto di S.L.A. A.A , 74, parte II, Regazzini, E. (199). Concentration comparisons between probability measures. Sankhya B 54, Rotondi, R., Ruggeri, F. and Vercesi, P. (199). A Bayesian approach to comparison of information sources. Quaderno IAMI 9.1, Milano: CNR-IAMI. Ruggeri, F. (1993a). Local and global sensitivity under some classes of priors. To appear in IMSIBAC IV. Amsterdam: North Holland. Ruggeri, F. (1993b). Bayesian comparison of Italian earthquakes. Quaderno IAMI 93.8, Milano: CNR-IAMI. Ruggeri, F. and Wasserman, L.A. (1991). Density based classes of priors: innites-

23 imal properties and approximations. Technical Report #58. Pittsburgh: Department of Statistics, Carnegie Mellon University. Wasserman, L.A. (199). Recent methodological advances in robust Bayesian inference. Bayesian Statistics IV. Oxford: Oxford Press. Sandra Fortini, Fabrizio Ruggeri Consiglio Nazionale delle Ricerche Istituto per le Applicazioni della Matematica e dell'informatica Via A.M. Ampere, 56 I-131 Milano, Italy. 3

24 Figure 1. Concentration function '(x) ( ) and 1?'(1?x) ( ) of G(; ) w.r.t. E(1). Figure. Concentration function ( ) between posteriors on and bound g(x) = x ( ). 4

25 f(1-x) f(x) x 5

26 f(x) g(x) x 6