document: one of the most important astrophysical phenomena still lacking an explanation is the origin of the celestial gamma - ray bursts ( grb ) . 
   these are powerful flashes of gamma - rays lasting from less than one second to tens of seconds , with isotropic distribution in the sky . 
   they are observed above the terrestrial atmosphere with x gamma  ray detectors aboard satellites @xcite . 
   thanks to the bepposax satellite @xcite , afterglow emission at lower wavelengths has been discovered @xcite and we now know that at least long ( @xmath17s ) grb s are at cosmological distances , with measured red shifts up to 4.5 ( see , e.g. , review by djorgovski @xcite and references therein ) . among the possible explanations of these events , which involve huge energy releases ( up to @xmath18 erg , assuming isotropic emission ) , the most likely candidates are the collapse of a very massive star ( hypernova ) and the coalescence of one compact binary system ( see , e.g. , reviews by piran @xcite and mszros @xcite and references therein ) . in both cases emission of gravitational waves ( gw ) is expected to be associated with them ( e.g. ref . 
   @xcite ) . according to several models , the duration of a gw burst is predicted to be of the order of a few milliseconds for a variety of sources , including the coalescing and merging black holes and/or neutron star binaries . 
   therefore gw bursts can be detected by the present resonant detectors , designed to detect gw through the excitation of the quadrupole modes of massive cylinders , resonating at frequencies near 1 khz .    at the distances of the grb sources ( @xmath19 gpc ) , 
   the gw burst associated with a total conversion of 1 - 2 solar masses should have amplitude of the order of @xmath20 . 
   the present sensitivity for 1 ms gw pulses of the best gw antennas with signal to noise ratio ( snr ) equal to unity is @xmath21 ( see e.g. ref . 
   @xcite ) , which requires a total conversion of one million solar masses at 1 gpc . however , although detection of a gravitational signal associated with a single grb appears hopeless , detection of a signal associated with the sum of many events could be more realistic . 
   thus we launched a program devoted to studying the presence of correlations between grb events detected with bepposax and the output signals from gravitational antennas nautilus and explorer . 
   searching for correlation between grb and gw signals means dealing with the difference between the emission times for the two types of phenomena . 
   furthermore , there is also the fact to consider that the time difference can vary from burst to burst . in the present analysis we use an algorithm based on cross - correlating the outputs of two gw detectors ( see @xcite ) , thus coping with the problem of the unknown possible time difference between grb and gw bursts , and also of the unmodelled noise . 
   the rome group operates two resonant bar detectors : explorer @xcite , since 1990 , at the cern laboratories , and nautilus @xcite , since 1995 , at the infn laboratories in frascati .    0.1 in    . 
   main characteristics of the two detectors . 
   @xmath22 indicates , for each detector , the two resonant frequencies and @xmath23 indicates the bandwidth . 
   the relatively larger bandwidth of explorer is due to an improved readout system . [ 
   cols="^,^,^,^,^,^,^,^ " , ]     [ finale ]    the agreement between the values of the simulated input signals and the values calculated using eq . 
   ( [ uppere ] ) shows that our model is correct . 
   having presented the experimental method and the model for the averaged correlation at zero delay time @xmath24 , we can infer the values of gw amplitude @xmath25 consistent with the observation . 
   we note that , using eqs . 
   [ uppere ] and [ snrc1 ] , energy @xmath26 is related to the measured cross - correlation @xmath24 by e_0=t_eff ( ) ^1/4 ( ) ^1/2 [ eq : relazione ] hence , the data are summarized by an observed average squared energy @xmath27 , at @xmath28 standard deviation from the expected value in the case of noise alone , as calculated with the aid of eq . 
   ( [ eq : sigmar_e ] ) where we put @xmath29 . 
   the standard deviation , expressed in terms of squared energy , is obtained from eq . 
   ( [ eq : sigmar_e ] ) , in the case @xmath30 , which gives @xmath31 . 
   according to the model discussed above , in the case of gw signals of energy @xmath32 , we expect @xmath33 to be a random number , modeled with a gaussian probability density function around @xmath34 with a standard deviation @xmath35 : f(e_0 ^ 2|e^2 ) , where @xmath32 is the unknown quantity we wish to infer from the observed value of @xmath26 , given in eq . 
   [ eq : relazione ] . 
   this probability inversion is obtained using bayes theorem ( see , e.g. , @xcite for a physics oriented introduction ) : f(e^2|e_0 ^ 2 ) f(e_0 ^ 2|e^2 ) f_(e^2 ) [ eq : bayes ] where @xmath36 is the prior probability density function of observing gw signals of squared energy @xmath34 . 
   in fact , we are eventually interested in inferring the gw s amplitude @xmath25 , related to the energy @xmath32 by eq . 
   ( [ upperh ] ) . 
   therefore we have a similar equation : f(h|e_0 ^ 2 ) f(e_0 ^ 2| h ) f_(h ) [ eq : bayesh ] where @xmath37 is obtained by a transformation of @xmath38 . as prior for @xmath25 we considered a uniform distribution , bounded to non negative values of @xmath25 , obtained from eq . 
   [ eq : bayesh ] , i.e. @xmath39 is a step function @xmath40 . 
   this seems to us a reasonable choice and it is stable , as long as other priors can be conceived which model the  positive attitude of reasonable scientists 
   ( see ref . 
   @xcite ) . , but rather the probability per decade of @xmath25 , i.e. researchers may feel equally uncertain about the orders of magnitudes of @xmath25 . 
   this prior is known as jeffreys prior , but , in our case , it produces a divergence for @xmath41 in eq . 
   [ eq : bayesh ] , a direct consequence of the infinite orders of magnitudes which are equally believed . to get a finite result we need to put a cut - off at a given value of @xmath25 . 
   this problem is described in depth , for example , in @xcite and in @xcite . 
   [ foot:1 ] ]    the probability density function of @xmath25 is plotted in fig . 
   [ finaleh ] . 
   the highest beliefs are for very small values , while values above @xmath42 are practically ruled out . from fig . 
   [ finaleh ] we obtain an expected value and standard deviation for @xmath25 of @xmath43 and @xmath44 , respectively , which fully account for what is perceived as a null result .    in these circumstances , we can provide an upper limit , defined as value @xmath45 , such that there is a given probability for the amplitude of gw s to be below it , i.e. _ 0^h(ul)f(h 
   |e_0 ^ 2)h = p_l , with @xmath46 the chosen probability level . 
   results are plotted in fig.[fighh ] . 
   for example , we can exclude the presence of signals of amplitudes @xmath47 with 95% probability . 
   probabilistic results depend necessarily on the choice of prior probability density function of @xmath25 . 
   for example , those firmly convinced that gw burst intensities should be in the @xmath48 region would never allow a 5% chance to @xmath25 above @xmath49 . 
   therefore , in frontier research particular care has to be used , before stating probabilistic results . 
   the bayesian approach , thanks to the factorization between likelihood and prior , offers natural ways to a present prior - independent result 
   . the simple idea would be just to provide the likelihood for each hypothesis under investigation , in our case @xmath50 . 
   more conveniently , it has been proposed ( @xcite-@xcite ) to publish the likelihood rescaled to the asymptotic limit , where experimental sensitivity is lost completely ; @xmath51 , in our case . 
   indicating with @xmath52 this rescaled likelihood , we have ( h ) = . 
   [ eq : rbur_def ] in statistics jargon , this function gives the bayes factor of all @xmath25 hypotheses with respect to @xmath53 . 
   in intuitive terms , it can be interpreted as a `` relative belief updating ratio '' or a `` probability density function shape distortion function '' , since from eq . 
   ( [ eq : bayesh ] ) we have = . 
   [ eq : bayesf ] in the present case we get , numerically ( h ) = a h 0 , where @xmath54 is the rescaling factor , @xmath55 and @xmath56 the result is given in fig . 
   [ loglinrh ] ,    where the choice of the log scale for @xmath25 is to remember that there are infinite orders of magnitudes where the value could be located ( and hence the problem discussed in footnote [ foot:1 ] ) . 
   interpretation of fig . 
   [ loglinrh ] , in the light of eqs . 
   ( [ eq : rbur_def])-([eq : bayesf ] ) , is straightforward : up to a fraction of @xmath57 the experimental evidence does not produce any change in our belief , while values much larger than @xmath57 are completely ruled out . 
   the region of transition from @xmath52 from 1 to zero identifies a _ sensitivity bound _ for the experiment . 
   the exact value of this bound is a matter of convention , and could be , for example , at @xmath58 , or @xmath59 . 
   we have @xmath60 and @xmath61 , respectively . 
   note that these bounds have no probabilistic meaning . in any case 
   , the full result should be considered to be the @xmath52 function , which , being proportional to the likelihood , can easily be used to combine results ( for independent datasets the global likelihood is the product of the likelihoods , and proportional constants can be included in the normalization factor ) . 
   note that the result given in terms of scaled likelihood and sensitivity bound can not be misleading . 
   in fact , these results are not probabilistic statements about @xmath25 and no one would imagine they were . on the other hand , 
   confidence limits , which are not probabilistic statements on the quantity of interest , tend to be perceived as such ( see e.g. @xcite and references therein ) . 
   using for the first time a cross - correlation method applied to the data of two gw detectors , explorer and nautilus , new experimental upper limits have been determined for the burst intensity causing correlations of gw s with grb s . analyzing the data over 47 grbs , we exclude the presence of signals of amplitude @xmath62 , with 95% probability , with a time window of @xmath63 . with the time window of @xmath64 
   s , we improve the previous gw upper limit to about @xmath65 .    the result is also given in terms of scaled likelihood and sensitivity bound , which we consider the most complete and unbiased way of providing the experimental information .    in a previous paper @xcite 
   we had given more stringent upper limits , but this was under the hypothesis that the gw signals always occur at the same time with respect to the grb arrival time . 
   here , instead , we only require that the time gap between the grb and the gw burst be within a given time window . 
   similar comparison can be made with the auriga / batse result @xcite , where an upper limit `` @xmath66 with c.l . 
   95% '' is estimated under the assumption that gw s arrive at the grb time within a time window of @xmath3 s.    finally , we remark that this method can be applied for any expected delay between grb and gw , with appropriate time shifting of the integration window with respect to the gbr arrival time , according to the prediction of the chosen model . 
   we thank f. campolungo , r. lenci , g. martinelli , e. serrani , r. simonetti and f. tabacchioni for precious technical assistance . 
   we also thank dr . 
   r. elia for her contribution . 
   given two independent detectors , let @xmath67 and @xmath68 be the measured quantities , which are the ( filtered ) data in our case . 
   we introduce the variables @xmath69 and @xmath70 where @xmath71=t_{eff}$ ] and @xmath72=t_{eff}$ ] . 
   we recall that the cross - correlation function is r()= [ crossa ] with summation extended up to the number of independent samples @xmath73 . 
   we calculate @xmath74 . 
   thus @xmath75 . 
   we easily verify that @xmath76=0 $ ] . in the absence of any correlated signal we also have @xmath77=0 $ ] . 
   let us calculate the variance of @xmath78 . 
   we notice that when squaring the numerator of eq.[crossa ] and taking the average , the cross - terms vanish if the @xmath73 data are independent from each other . 
   then , since also @xmath79 and @xmath80 are independent variables , we obtain    _ 
   r^2== = [ siga ] the previous considerations still apply to the cumulative cross - correlation @xmath81 , obtained by averaging @xmath82 independent @xmath78 . 
   the final variance for the cross - correlation @xmath81 is _ r^2= [ sigma ]    let us now consider an energy signal @xmath83 on both detectors at the same time 
   . the expected value of the cross - correlation @xmath78 at @xmath84 will be positive . in the case of @xmath73 independent data 
   the signal @xmath83 will appear in one datum only and we have : e[r(=0)-r(0)]== [ segnalea ] where we put @xmath85 . we note that , in eq.[segnalea ] , @xmath86=0 $ ] also when a signal is present . 
   p. astone , c. buttiglione , s. frasca , g.v . 
   pallottino and g. pizzella , il nuovo cimento 20,9 ( 1997 ) p.astone et al . 
   `` gravitational astronomy '' pag 189 + ed . 
   d.e.mcclelland and h.a.bachor , world scientific 1991            g. d agostini _ bayesian reasoning in high - energy physics : principles and applications _ cern 99 - 03 ( 1999 ) this report and related references can be found at the url : http://www-zeus.roma1.infn.it/  agostini / prob+stat.html          g. dagostini , _ overcoming priors anxiety _ , invited contribution to the monographic issue of the revista de la real academia de ciencias on bayesian methods in the sciences , ed . j.m . 
   bernardo ( 1999 ) physics/9906048 . 
   g. dagostini , proc . 
   xviii international workshop on maximum entropy and bayesian methods , garching ( germany ) , july 1998 , v. dose , w. von der linden , r. fischer , and r. preuss , eds , kluwer academic publishers , dordrecht , 1999 , physics/9811046 .
summary: data obtained during five months of 2001 with the gravitational wave ( gw ) detectors explorer and nautilus were studied in correlation with the gamma ray burst data ( grb ) obtained with the bepposax satellite . during 
   this period bepposax was the only grb satellite in operation , while explorer and nautilus were the only gw detectors in operation . 
   no correlation between the gw data and the grb bursts was found . 
   the analysis , performed over 47 grb s , excludes the presence of signals of amplitude @xmath0 , with 95% probability , if we allow a time delay between gw bursts and grb within @xmath1 s , and @xmath2 , if the time delay is within @xmath3 s. the result is also provided in form of scaled likelihood for unbiased interpretation and easier use for further analysis .    _ 
   @xmath4 istituto nazionale di fisica nucleare infn , rome _ + _ @xmath5 university of rome `` tor vergata '' and infn , rome 2 _ + _ @xmath6 ifsi - cnr and infn , rome _ + _ 
   @xmath7 university of laquila and infn , rome 2 _ + _ @xmath8 iess - cnr and infn , rome _ + _ @xmath9 university of rome `` la sapienza '' and infn , rome _ + _ 
   @xmath10 istituto nazionale di fisica nucleare infn , frascati _ + 
   _ @xmath11 university of ferrara and iasf - cnr , bologna _ 
   + _ @xmath12 university of ferrara , ferrara _ + _ @xmath13 university of ferrara , ferrara and ita `` i. calvi '' , finale emilia , modena _ + _ @xmath14 university of rome `` tor vergata '' and infn , frascati _ + _ @xmath15 ifsi - cnr and infn , rome 2 _ + _ @xmath16 ifsi - cnr and infn , frascati _ 
   +   +