Société de Calcul Mathématique, S. A. Algorithmes et Optimisation

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1 Société de Calcul Mathématique, S. A. Algorithmes et Otimisatio The iformatio associated with a samle Berard Beauzamy Société de Calcul Mathématique SA May, 009 Abstract We collect a samle : distict values x,..., x, reeated resectively,..., times. Let be the size of the samle. Let,..., be the (uow) robability of each value ; oe usually uses the estimate which is correct oly asymtotically. Here, we itroduce a robability law o the the ule,..., (ot just a estimate, but a whole robability law). We study the margial laws for each ad we show that the global variace of this robability law is a good idicator whether the samle is sufficiet or ot. Acowledgemets : We tha Igor Carro for his hel i the resetatio of the aer. Siège social et bureaux :, Faubourg Sait Hooré, Paris. Tel. : Fax : Société Aoyme au caital de Euros. RCS : Paris B SIRET : APE : 79Z

2 Table of cotets The iformatio associated with a samle... I. Itroductio...3 II. Two differet situatios : static ad dyamical...3 III. Some termiology...3 IV. The static situatio...4. Comutig the margial law for...6. Comutig the exectatios ad the variaces Deedece uo the umber of classes Cofidece iterval for each Asymtotic estimates for the cofidece iterval for each A first examle : ollutio i rivers A secod examle : Trais beig late Proerties of the recisio width w...8 V. The dyamical situatio...0 Refereces...4 Iformatio coected with a samle. Berard Beauzamy, May 009

3 I. Itroductio We cosider the followig questio: we collect a samle (from a hysical exerimet, or a comutatioal code, or from olls), ad we wat to ow if this samle is "sufficiet" or ot. The word "sufficiet" requires clarificatio. Roughly seaig, it meas: does the samle give a correct idea of the "real" robability law behid the exerimet? Or, more recisely, if we collected a samle which would be 0 times bigger, or 00 times bigger, ad so o, would it give sigificatly differet iformatio? I ractice, this questio is of cosiderable imortace, for two reasos: If the samle is recogized as "isufficiet", ot to tae wrog decisios ; Reduce, if ossible, the time eeded to acquire a "sufficiet" samle, which is usually costly. As a examle of the first tye, we may metio all questios coected to the warraty o idustrial roducts. For istace, the warraty o a car (say 3 years) may be too log, or too short, if the samle uo which it is based is isufficiet. As a examle of the secod tye, we may metio a cotract we had with the Frech "Istitut de Radiorotectio et de Sûreté ucléaire" : it dealt with the imrovemet of uclear measuremets. But these measuremets require eutro coutig, which taes time ; so the questio is whe to sto? (see [IRS-SCM] for a ublished descritio of this wor). To decide whether a samle is sufficiet or ot is ot the same as recostructig missig data, as we did i [BBOZ]. I the recostructio of missig data, we have some owledge of the uderlyig robability law, ad, whe a samle is isufficiet, recisely we do ot have this owledge. II. Two differet situatios : static ad dyamical There are obviously two differet situatios : either the samle is give to you at oce, as a big database, with o iformatio o how it was collected, or you see it growig (or, which is the same, you have a iformatio about the date whe each data was collected). We call the first situatio "static" ad the secod oe "dyamical". ote that there may be itermediate situatios : oe samle, made for examle of a static database ad of a dyamical database. But i this case we cosider the whole samle as static. III. Some termiology Before we eter the discussio, let us give some termiology. We call a "class" a ossible value for the results. For istace, if you measure the heights of eole betwee.5 ad meters, with recisio 0. m, you have 5 classes, amely.5 -.6,.6 -.7,.7 -.8,.8 -.9,.9 -. So, the umber of classes deeds : Iformatio coected with a samle. Berard Beauzamy, May 009 3

4 O the rage of the measuremet (here the rage is.5 - ), which usually deeds o what you wat to measure ad for what reaso (this is "exert owledge") ; O the recisio of the measuremet. This recisio deeds itself o the recisio of the measuremet device, but also o the objectives of your measuremet. For istace, if you simly wat to establish correlatios betwee food ad height, a recisio of 0. m might be eough, but if you are a tailor, you might wat a recisio of cm or eve below. Mathematically, ad i terms of comuter imlemetatio, a "class" is usually a iterval of the tye a x b, but this has o imortace here. To a class we attribute a "value", which is usually the ceter of the iterval. This choice, also, has o imortace here. Simly seaig, we may cosider our samle as a list of distict values x,..., x (so is the umber of observed classes) with,..., beig the umber of reetitios for each value x,..., x resectively. Let : this is the size of the samle. Let us observe that there are situatios where the umber of ossible classes is uow, or has oly a useless uer boud. For istace, if you sell a ew device, ad wat to ow the umber of failures i a give year, the oly boud you ow is the total umber of devices sold (but of course you exect that ot all of them will brea dow!). If you go to a ew regio, where you have ever bee, ad as the questio : what is the law of robability for the umber of cows i a farm, you have o a riori idea of the ossible rage (ca it be 000? or 0 000? or more?). I the static situatio, all you ow is a list of values ; you ow othig about the way they have bee obtaied. The tool we ow describe allows us to build a robability law from the samle, givig i articular a cofidece iterval for the robability of each value : if this cofidece iterval is arrow for most values, this is satisfactory. This gives a very comlete descritio of the iformatio cotaied i the samle, icludig the ucertaities about it. IV. The static situatio We wat to evaluate the "true" robability of each value x,,..., robability is uow: we have oly the estimate observed o the samle. So itself be treated as a radom variable, of which we wat to fid the law. A classical estimate is:. This "true" should However, this estimate is correct oly asymtotically that is whe : this is the emirical law of large umbers. A correct estimate, valid for all, is: Iformatio coected with a samle. Berard Beauzamy, May 009 4

5 as we will see below., But i fact, the geeral theory reseted i [BB], Chater 4, 9 (multiomial case) gives more tha estimates for each. Ideed, oe ca build a desity fuctio for the ule,...,. This desity fuctio is : f (,..., ) c () where,..., satisfy 0 for all,, ad c is a ormalizatio costat (so that the itegral of f is ). Let us exlai the meaig of this formula. I our samle, the value x has bee observed times ad we would lie to estimate the robability of this value. This robability is uow, so, as we said, it should be treated as a radom variable. Formula () gives the desity of this radom variable, or, more exactly, the joit desity of the ule,..., of radom variables. The ormalizatio costat c ca be easily comuted ([BB], Chater 4, 9) ; its value is : c I(,..., ) with :!! I(,..., ), where. ( )! So we have a very exlicit situatio : the samle gives birth to a robability law o the robabilities of each value, ad the amout of iformatio will be derived from this robability law. For istace, we will derive a robability law for ad the more cocetrated this robability law is, the more certai is. If this robability law for was a Dirac, say for istace at 0., this would mea that is certaily equal to 0.. If this was true for all, we would be sure that our samle characterizes comletely the exerimet : the samle would be erfect. So we see that, for ay samle, the cocetratio of the law f (,..., ) characterizes correctly the amout of iformatio i the samle. We first comute exlicitly the margial law of ay of the 's. Iformatio coected with a samle. Berard Beauzamy, May 009 5

6 . Comutig the margial law for We have, by defiitio : f (,..., ) I(,..., ), 0,...,, with :!! I(,..., ), ad. ( )! We defie, for ay ositive itegers ab, : f ab, ( x) a x ( x) I( a, b) b with as before : ab!! I( a, b) ( ab)! We recall that each fuctio f ab, is a desity fuctio : see [BB], Chater 4. The we have : Proositio. The margial law of each is the desity f., Proof of Proositio. We give it for i order to simlify the otatio ; it is idetical for the others. We comute : I d (the ormalizatio costats will be cosidered at the ed) We have : Set t I d with 0t. The : Iformatio coected with a samle. Berard Beauzamy, May 009 6

7 ... ( ) 0 I t t dt ad thus, u to some ormalizatio costat : ext : The same comutatio gives : ad more geerally :... I I d... I j j j... j I j... j The margial law is obtaied for j, that is j. We fid the law :... f ( ) Let us ow tae care about the ormalizatio. Recall that if : a b Iab, x ( x) dx, 0 where ab, are ositive itegers, the : ab!! I( a, b). ( ab)! ([BB], Chater 4, 5, Lemma ). This gives the fial formula : f( ) I(, ) () We observe that, if the umber of classes is strictly above, this law does ot coicide with the law we would have, dividig ito classes: the first oe ad the rest. This law would have as a desity: f ( ), I(, ) Iformatio coected with a samle. Berard Beauzamy, May 009 7

8 ([BB], Chater 4, 5, Proositio ).. Comutig the exectatios ad the variaces We recall that the exectatio of f, ( x) c x ( x) is 5.B, Proositio 3). ([BB], Chater 4, The exectatio of f( ) I(, ) is therefore : E( ), ad the same for the other : E( ) (3) We see that this quatity is differet from / ; both coicide oly asymtotically, whe. We observe also that is ever equal to 0, eve if 0 : a value which has ever bee observed may be see i the future. The sum of exectatios is of course equal to : E( ) By [BB], Chater 4, 5.E, Proositio 5, the variace of f is :, ( )( ) ( ) ( 3), with relaced by. So, for each,..., ( )( ) ( ) ( ) (4) Let us comute the sum of all variaces. We fid : ( )( ) ( ) ( ) that is : Iformatio coected with a samle. Berard Beauzamy, May 009 8

9 ( ) ( ) ( ) or : Set. We get a simle formula : ( ) ( ) ( ) ad, with S ad S, we ca write : S S S ( S) However, this global variace is ot a good idicator ; it gives the same weight to all elemetary terms, ad, moreover, we are iterested i searate estimates for all 's. So, let us come bac to elemetary variaces. We have the followig elemetary roositio : Proositio. For each, for fixed ad, the variace, as a fuctio of, is icreasig whe 0 ad decreasig whe. The maximum value, obtaied for, is,max. The miimum value, 4( ) obtaied for 0, is ( ) ( ) ( ),mi The roof of this roositio follows immediately from formula (4). We observe that the ( )( ) value for is ( ) ( ), which is usually bigger tha. ( ) Here is the grah of for 000, 0 :. Iformatio coected with a samle. Berard Beauzamy, May 009 9

10 Figure : Grah of the variace of each term As a corollary, we see that the variace, for each, teds to zero whe the samle icreases. We observe that the variace of each term does ot eed to decrease whe the samle icreases (that is, is relaced by ). Let us see this o a examle, whe is relaced by. The coditio: ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) is equivalet to : ( )( ) ( ) or : ( )( ) ( ) ( ) 3( ) 3 ( ) ( ) 3 ( ) which is ot true i geeral. Asymtotically, whe, the behavior of is easy to obtai : Iformatio coected with a samle. Berard Beauzamy, May 009 0

11 Proositio 3. Whe, ( ), where lim. 3. Deedece uo the umber of classes We observe that the umber of classes aears i all these estimates. If we restrict ourselves to the values which aeared, this is erfectly clear ad legitimate. However, as the examles below will show, we ofte wat to tae ito accout values which have ot aeared but might, if the samle was bigger. For istace, if i some river we observed, for some ollutat, a cocetratio of 0.3 g/l ad i aother a cocetratio of 0.5 g/l, there is o reaso that a cocetratio of 0.4 g/l might ot exist somewhere. Also, there is a questio about the uer limit ad the lower limit : should we sto at g/l, 0 g/l, ad so o? Aother examle is about temeratures. If our exerimet deals with the temeratures that oe ca observe i Paris, what extremes should we tae? Of course, we should ot restrict ourselves to the temeratures which have already bee observed, sice warmer oes ad colder oes are also ossible. There is o stadard, theoretical aswer to this questio. The list of ossible classes deeds o the roblem ; it has to be set both looig at the existig (historical) data, ad the hysics of the roblem. 4. Cofidece iterval for each Sice we have a robability law for, ad sice we ow its exectatio ad variace, we ca deduce a cofidece iterval, usig Bieaymé-Chebycheff's iequality, for each, uder the form : P X that is, with our reset otatio : P We observe that, usually, whe the law of the variable is exlicitly give (as it is the case here), Bieaymé-Chebycheff's iequality is ot the best way to obtai cofidece itervals : exlicit comutatios of the itegrals give better results i geeral. But here, it turs out that this iequality gives more recise itervals tha the oes which were derived (by exlicit, but aroximate comutatios) i [BB], Chater 4,. Iformatio coected with a samle. Berard Beauzamy, May 009

12 We wat a cofidece iterval for choice). This meas that :, with cofidece 0.95 (this is a arbitrary Therefore, we have to tae : So the iterval : I is a - cofidece iterval for. This is true i geeral ; however, this iterval may be quite large. We will say that the samle is satisfactory is most of these itervals are small eough, amely if they fit withi a give recisio. Fix a recisio /0 (this choice is arbitrary). We will say that the iterval I above fits withi the recisio if it is of the form ( ) x,( ) x. This is the case if the coditio : that is : is satisfied. We defie : w (the letter w stads for "width" of the iterval). Usig the above defiitios of we get : ad w ( ) ad we wat : w (C) We have obtaied : Iformatio coected with a samle. Berard Beauzamy, May 009

13 Proositio 4. - We have : P ( ) ( ) as soo as : ( )( ) ( C ) This coditio is a fortiori satisfied as soo as : ( C ) Proof of Proositio 4 : the coditio ( C) is simly a rehrasig of coditio ( C ), usig the defiitio of w ; the fact that ( C ) is stroger tha ( C ) is obvious. We ote that coditio ( C ) does ot deed o or ; this is a absolute coditio, which says that if there are eough observatios i the -th class, the there is a high robability to have a shar estimate o. For istace, if 0. (0% recisio) ad 0.95 (95% cofidece), we fid 000. It meas that, o matter how large the total samle is, o matter how may classes there are, if a class has more tha 000 observatios, a 0 % recisio will be obtaied uo with 95% robability. Such a estimate, valid for all ad, is very coservative (this is due to the use of the iequality of Bieaymé-Tchebycheff). So later we will fid more recise meas to determie whether the iformatio is sufficiet. I fact, we do ot eed coditio ( C ) to hold for all 's. We will be satisfied if it holds for most of them. Fix a "satisfactio idex" 0.96 (this is arbitrary ; we too it differet from ). The we will be satisfied with satisfactio idex if coditio ( C ) holds for a set,..., with. for which codi- I ractice, oe will chec the coverse, amely that the sum of all tio ( C ) is ot satisfied is at most. So, fially, we eed three cocets i order to defie global satisfactio : The umber (here 0.96), which idicates the roortio of classes which are correctly hadled from the samle. Here, we wat 96 % of the classes. The umber which deotes the cofidece iterval. We are sure, with robability 0.95 that each will be i the chose iterval. Iformatio coected with a samle. Berard Beauzamy, May 009 3

14 The recisio which relates the size of the iterval to a recisio of the measuremet. It says that the iterval will be of the form ( ),( ). 5. Asymtotic estimates for the cofidece iterval for each We ow determie exlicitly the set where the desity of each is small ; these estimates imrove uo the oes give i [BB], Chater 4. They are oly asymtotic estimates, valid whe. I order to simlify the otatio, let be ay of the. Let f( x ) be the desity of. We have : 's ad let be the corresodig Proositio 5. The maximum value of f is tae for x ; its value is : max f( x) x ( ) For ay 0, outside the iterval I, with : the fuctio f satisfies f( x) Fially, for ay 0 Log ( )., o the iterval I, / with : ( ) the fuctio f satisfies f ( x) dx.moreover, at the edoits : ( ) f( ) ( ) I Proof of Proositio 5. Whe,, ad the desity f( x ) give by Proositio, simlifies to : with : f ( x) ( x) x ( x) ( x) I(,( ) ) Iformatio coected with a samle. Berard Beauzamy, May 009 4

15 Usig Stirlig's formula, we have : ( )!(( ) )! I(,( ) ) ( )! ad therefore : ( ) ( ) ( ) ( ) ( ) ( ) ( x) x ( x) ( ) ( ) ( ) Sice the maximum of the fuctio ( ) h x x ( x) is obtaied for x, we see that the maximum is ( x) is also obtaied for x ad the maximum value is : ( ) max ( x) x. ( ) Whe, the fuctio ( x) is more ad more cocetrated aroud its mea, which is (sice 0 ). Therefore, let x, with 0 whe, ad let us evaluate ( x). We have : ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ad the same way we obtai : ( ) ( ) ( ) So fially : ( ) ( ) () Iformatio coected with a samle. Berard Beauzamy, May 009 5

16 Let 0 ; the coditio ( ) is equivalet to : or, with : / ( ) / We observe that : / ( ) / / Log ( ) / This roves the secod art of our Proositio. The third art follows immediately from Bieaymé-Tchebycheff's iequality, sice : with m ad m m ( ) relaced by. f ( x) dx P m X m ( ). The estimates at the edoits come from (), with 6. A first examle : ollutio i rivers The followig examle comes from a cotract we had i 008 with the Water Agecy "Artois-Picardie" i Frace. It deals with the cocetratio i the ollutat H4. There are 04 measure oits, ad at each oit the cocetratio i H4 is measured. Amog these 04 measure oits, there is oly oe where the cocetratio is 0.03 g/l (to be uderstood as i the class ). So, with our revious otatio, we have, 04. The umber of ossible values is : 000 (basically, from 0 to 0 g/l). We are iterested i the robability desity of, to be i the class g/l. This desity is : 0 f( ) I(, 0) Iformatio coected with a samle. Berard Beauzamy, May 009 6

17 with :! 0! I(,0) 03! 0 03 The exectatio is : 04 3 E( ) ad the variace : ( )( ) ( ) ( ) ow, we use Bieaymé-Chebycheff's iequality, uder the form : P We wat this robability to be 0.95 (this is a arbitrary choice). So we eed to tae such that : 0.95 which gives : The quatity is egative, which gives o iformatio, sice a cocetratio must be ositive. So, usig oly the boud, we get that : with robability So we obtai the followig result : the robability, for a give measure statio, to be i the class H is with a 95% cofidece. This amout of ollutio (very low cocetratio) is exected to be extremely rare. Similar results may be obtaied for all classes. Iformatio coected with a samle. Berard Beauzamy, May 009 7

18 7. A secod examle : Trais beig late Durig the year 008, we had a cotract with the "Réseau Ferré de Frace" (Frech Railways) ; it dealt with the statistics of the delays for the trais. Data are as follows : for a give delay x, i miutes, betwee ad 600 maximum, we ow the umber of trais havig this delay, amog all Frech trais i a give year. The total 600 is therefore the umber of trais which, durig that year, exerieced a delay betwee ad 600 miutes. With the above otatio, we fix 0.90 (90% cofidece iterval) 0. (recisio 0%). We fid that coditio ( ) C, that is w, is satisfied for 84 % of the trais ; more recisely those which have delay 6 mi, ad is ot satisfied for those which have loger delays. What does this mea i ractice? We cosider here the delays of the trais as a radom variable (this delay deeds o may factors, which are ot erfectly ow). Our statemet meas that, if we wat to ow the law of this radom variable, the samle comig from the year 006 is satisfactory (u to the recisio aouced) for trais with delays 6 mi ad is isufficiet for trais with loger delays. I other words, for trais with loger delays, the estimate we get from oe year of observatio is too vague, because such trais are ot umerous eough ; i order to imrove the samle, more years are ecessary. 8. Proerties of the recisio width w Recall that : w ( ) defies the size of the iterval aroud each. We omit the subscrit ad we write w(,, ) to idicate that this umber deeds o the total size of the samle, o the umber of classes ad o the umber of realizatios i that articular class. We have the followig results : Proositio 6. - The umber w(,, ) decreases (that is, we have a smaller iterval) if a ew measuremet aears i that articular class, that is : w(,, ) w(,, ) The roof of this statemet is obvious, sice the umerator does ot chage, whereas the deomiator icreases. Iformatio coected with a samle. Berard Beauzamy, May 009 8

19 Proositio 7. - The umber w(,, ) icreases if a ew measuremet aears i aother class, that is : Ideed, this statemet meas : ( ) ( ) or : which is equivalet to : which is satisfied. w(,, ) w(,, ) Proositio 8. - The umber w(,, ) icreases if a ew (emty) class aears, that is : w(,, ) w(,, ). The roof is the same as for Proositio 7 above. These roerties are easy to uderstad : a ew measuremet i the first class imroves the estimate uo ; all other situatios mae it worse. Our coclusio i this aragrah is that the robability law f (,..., ) gives a satisfactory iformatio about the samle : we may coclude if the samle is sufficiet or ot, or, more secifically, if some classes are sufficiet or ot. We observe that the recisio of the measuremet devices aears i this defiitio, but i a hidde maer : it aears i the umber of classes. If the measuremet is very recise, the umber of classes is high ; if the recisio is low, so is the umber of classes. I our examle regardig trais, the recisio is oe miute ; therefore we have 600 classes for delays betwee miute ad 0 hours. If the rage was the same, but with recisio 0 secods, we would have 6 times more classes. Iformatio coected with a samle. Berard Beauzamy, May 009 9

20 V. The dyamical situatio The dyamical situatio seems much more satisfactory: we see the samle growig. So we may decide to sto whe the total samle cotais hardly more iformatio tha the samles we collected earlier. Let, as before, be the umber of classes. Let,..., be the successive sizes of the samle. Let X i, be the radom variable which idicates whether, at the i -th trial, we fell ito the -th class or ot : so X, if we fell i the -th class, or 0 otherwise. We have : i X i,, for all i (this meas that we have oly oe result for each trial) i X i, amog trials. The i,, for all, where is, as before, the umber of occurreces of the -th class X are ideedet Beroulli radom variables. The robability PXi, is uow, it is the same for all i : this is what we wat to estimate. We will show how to do it for the first class ad it will be the same for all. Therefore, i the sequel, we dro the subscrit ad sea simly of istead of. The same way, we write simly X istead of X i,. i The X i are ideedet radom variables with same law. Each variable. We set 0 ( ), so 0 is the variace of all the X i 's. X i has mea X X At the -th ste, we loo at the quatity Y ; it idicates the roortio of successes of the first class amog the first trials. By the emirical law of large umbers, Y whe. Let y be the realizatio of Y o our samle (the observed values). The the quatity : y y ca be called the variace of the samle of the y,..., y. This is a realizatio of the radom variable : S Y Y Iformatio coected with a samle. Berard Beauzamy, May 009 0

21 Of course, whe, S 0, both almost everywhere ad i law ; this is simly due to the fact that Y for the a.e. covergece ad follows from the Cetral Limit Theorem for the covergece i law. The oe may tae the followig decisio rule : whe the variace of the samle y,..., y is small eough, we cosider that we have reached the limit with roer recisio, ad we tae the estimate : y y. The questio is : is this decisio rule correct, ad what ucertaity does it give uo the y y estimate for? Or, i other words, how do we estimate the differece? Asymtotically, whe, this decisio rule is correct, sice each Y follows a ormal law with arameters is ad the robability : 0,. The the law of Y Y is exlicit ; its exectatio Y Y P ca be comuted exlicitly. But all this is true oly asymtotically! If ow we wat to obtai a correct result for a give value of, we may roceed as follows : Proositio 9. If ( ), withi a cofidece level, we have the estimate :. Proof of Proositio 9. We comute exlicitly : P Y. Sice Y X X, ad the i X are ideedet r.v., we get : 0 ( Y ), Iformatio coected with a samle. Berard Beauzamy, May 009

22 ad therefore, by Bieaymé-Tchebycheff : 0 PY. For give 0, this robability is small whe is large eough. Therefore, for most realizatios of Y, the observed value must be close to. Let x x be the observed realizatio. If, we would have a realizatio of Y far from (sice this realizatio is o ), which is very uliely : this has robability at most. So we see that : with robability at least. If we wat a cofidece level (say 0.95 ), the we get the estimate : ( ) (5.) We observe that aears i this estimate : oe eeds to ow at least a lower boud for i order to aly it (for istace, oe must ow that 0.). This roves our Proositio. ow, if we use Y,..., Y (ad ot just Y ), we will imrove uo this roositio, as follows : Proositio 0. If : Log( ) ( ) the, with a cofidece level, we have the estimate :. Proof of Proositio 0. We comute : Y Y P Iformatio coected with a samle. Berard Beauzamy, May 009

23 Y Y First, we eed to comute. We have : Y Y X j X j j ad therefore : 0 Y Y j j j j Set u ( ). The, Bieaymé-Tchebycheff's iequality gives : j j Y Y u( ) P 0 Let ow y y Let be ow the observed realizatio of as above shows that : Y Y. The same reasoig ( )( ) with robability u. If we wat a cofidece level, we get the estimate : u ( ) ( ) Usig a Taylor exasio of u ( ), we obtai the sufficiet coditio : Log( ) ( ) (5.) which is better tha (5.). Of course, it still cotais the arameter. This cocludes the roof. Iformatio coected with a samle. Berard Beauzamy, May 009 3

24 Refereces [IRS-SCM] Ae-Laure Weber, Ae archer, icolas Péi, Frédéric Huyh, Pierre Fu (IRS), Laure Le Brize, Olga Zeydia, Berard Beauzamy (SCM SA), Imlemetatio of a exerimetal desig to evaluate the codes used to determie the erichmet of uraium samles. Paer reseted by IRS ad SCM at the "Euroea Safeguards Research ad Develomet Associatio" meetig, May 007. [BB] Berard Beauzamy : Méthodes robabilistes our l étude des héomèes réels, ISB : , Editios de la SCM, mars 004. [BBOZ] Berard Beauzamy et Olga Zeydia : Méthodes robabilistes our la recostructio de doées maquates,isb : , Editios de la SCM, avril 007. Iformatio coected with a samle. Berard Beauzamy, May 009 4