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inflation is the standard paradigm for providing the initial conditions for structure formation and cosmic microwave background ( cmb ) anisotropy generation . in the inflationary picture , primordial adiabatic perturbations arise from quantum fluctuations of the _ inflaton _ scalar field which drives the accelerated universe expansion . in the simplest models , the inflaton is assumed to have a shallow potential , thereby leading to a slow rolling of this field down its potential . the flatness of the potential implies that intrinsic non - linear ( hence non - gaussian ) effects during slow - roll inflation are tiny , although non - zero and calculable @xcite . to quantitatively describe the theoretical findings in this framework , let us introduce a useful parameterisation of non - gaussianity according to which the primordial gravitational potential @xmath8 is given by a linear gaussian term @xmath9 , plus a quadratic contribution , as follows ( e.g. @xcite ) : @xmath10 ( up to a constant offset , which only affects the monopole contribution ) , where the dimensionless parameter @xmath0 sets the strength of non - gaussianity . the above mentioned calculation of the amount of non - gaussianity during single - field inflation leads to typical values @xmath11 , much too low to be observable in cmb experiments . however , non - linear gravitational corrections after inflation unavoidably and significantly enhance the non - gaussianity level , leading to values of @xmath12 , almost independent of the detailed inflation dynamics @xcite . an angular modulation of the quadratic term is also found @xcite , so that @xmath0 should be considered as a kernel in fourier space , rather than a constant . the resulting effects in harmonic space might be used to search for signatures of inflationary non - gaussianity in the cmb @xcite . nonetheless , owing to the large values of @xmath13 considered here ( @xmath14 ) we will disregard this complication and assume @xmath0 to be a constant parameter . despite the simplicity of the inflationary paradigm , the mechanism by which adiabatic ( curvature ) perturbations are generated is not yet fully established . in the _ standard scenario _ associated to single - field models of inflation , the observed density perturbations are due to fluctuations of the inflaton field , driving the accelerated expansion . an alternative to the standard scenario which has recently gained increasing attention is the _ curvaton _ mechanism @xcite , according to which the final curvature perturbations are produced from an initial isocurvature perturbation associated to the quantum fluctuations of a `` light '' scalar field other than the inflaton , the so - called `` curvaton '' , whose energy density is negligible during inflation . due to a non - adiabatic pressure perturbation arising in multi - fluid systems @xcite curvaton isocurvature perturbations are transformed into adiabatic ones , when the curvaton decays into radiation much after the end of inflation . another recently proposed mechanism for the generation of cosmological perturbations is the _ inhomogeneous reheating _ scenario @xcite . it acts during the reheating stage after inflation if super - horizon spatial fluctuations in the decay rate of the inflaton field are induced during inflation , causing adiabatic perturbations in the final reheating temperature in different regions of the universe . an important feature of both the curvaton and inhomogeneous reheating scenarios is that , contrary to the single - field slow - roll models , they may naturally lead to high levels of non - gaussianity . large levels of non - gaussianity are also predicted in a number of theoretical variants of the simplest inflationary models . first , generalised multi - field models can be constructed in which the final density perturbation is either strongly @xcite or mildly @xcite non - gaussian , and generally characterised by a cross - correlated mixture of adiabatic and isocurvature perturbation modes @xcite . values of @xmath15 are also predicted in the recently proposed _ ghost - inflation _ picture @xcite , as well as in theories based on a dirac - born - infeld ( dbi)-type lagrangian for the inflaton @xcite . quite recently , there has been a burst of interest for non - gaussian perturbations of the type of eq . ( [ eq : model ] ) . different cmb datasets have been analysed , with a variety of statistical techniques ( e.g. @xcite ) with the aim of constraining @xmath0 . in the last years some authors set increasingly stringent limits on the primordial non - gaussianity level in the cmb fluctuations . using a bispectrum analysis on the cobe dmr data @xcite found @xmath16 . on the same data , @xcite found @xmath17 using spherical mexican hat wavelets ( smhw ) and @xcite using the maxima data set the limit on primordial non - gaussianity to be @xmath18 . all these limits are at the 1@xmath19 confidence level . the most stringent limit to date has been obtained by the _ team @xcite : @xmath20 at @xmath21 cl . consistent results ( an upper limit of @xmath22 at a @xmath2 confidence level ) have been obtained from the wmap data using smhw @xcite . it was shown in @xcite that the minimum value of @xmath13 which can be in principle detected using the angular bispectrum , is around 20 for _ wmap _ , 5 for _ planck _ and 3 for an _ ideal _ experiment , owing to the intrinsic limitations caused by cosmic variance . alternative strategies , based on the multivariate empirical distribution function of the spherical harmonics of a cmb map , have also been proposed @xcite , or measuring the trispectrum of the cmb @xcite . the plan of the paper is as follows : in section [ sect : mapng ] we describe our method to produce the temperature pattern of the cmb in presence of primordial non - gaussianity ; section [ sect : test ] addresses statistical issues to constrain the non - linearity parameter @xmath0 on the basis of wmap data ; finally , in section [ sect : concl ] we draw our conclusions . the non - gaussian cmb maps used in the following analysis have been generated by applying the numerical algorithm introduced by @xcite , hereafter lmm . here we summarise the various steps that define the whole procedure and refer the reader to that paper for further details . the starting point of the method is the simulation , directly in real space , of independent complex gaussian variables @xmath23 , with correlation functions @xmath24 where @xmath25 is the dirac delta function and @xmath26 is kronecker s delta .    the linear potential multipoles @xmath27 ( here @xmath28 ) having the desired correlation properties , as expressed in terms of the primordial ( i.e. unprocessed by the radiation transfer function ) power - spectrum of the gravitational potential @xmath29 , can be obtained by convolving @xmath23 with suitable filter functions @xmath30 : @xmath31 the filters @xmath30 are defined as ( see lmm ) @xmath32 here @xmath33 are spherical bessel function of order @xmath34 . notice that @xmath30 can be pre - computed at the beginning of all simulations and then applied in eq . ( [ eqn : filter ] ) . one more advantage of this approach is that @xmath30 , at fixed @xmath35 , are smooth functions of @xmath36 which differ from zero only in a narrow region around @xmath37 ; as a consequence , the integral in eq . ( [ eqn : real_convolution ] ) can be estimated in a fast way by computing @xmath30 in few points .    at this point the values of the linear potential @xmath38 can be recovered thanks to its expansion in spherical harmonics : @xmath39 the non - gaussian contribution ( modulo @xmath0 ) to the gravitational potential , @xmath40 ( here @xmath41 ) , is obtained directly in spherical coordinates by squaring @xmath42 ; this is then harmonic - transformed by using the healpix package @xcite to get @xmath43 . finally , the linear and non - linear contributions to the total cmb multipoles @xmath44 are obtained by convolving each term with the real - space radiation transfer function @xmath45 , @xmath46 ( see lmm , for the formal derivation ) , @xmath47 notice that also the quantities @xmath45 , which have been numerically estimated by using a modification of the cmbfast code @xcite can be pre - computed and stored for all the simulations of a given model .    in @xcite a different approach to produce non - gaussian maps was adopted . their starting point is the generation on a fourier - space grid of a gaussian field , which is then inverse - fourier transformed and squared to get the non - gaussian part of the gravitational potential in real space . the successive steps involve interpolation on a spherical grid , harmonic transforms and convolution with @xmath45 , to obtain the gaussian and non - gaussian cmb multipole coefficients . in this section , we will estimate @xmath0 from the wmap data using two different approaches . the estimation procedures will be calibrated using non - gaussian maps produced by the method described above . we have replicated the method applied in @xcite ( hereafter mw ) in order to check that the estimates using non - gaussian maps generated with two different methods , give consistent results . we also introduce a new estimator of @xmath0 based on the local curvature properties of the cmb fluctuation field . finally , we observe that the estimates of @xmath0 from the two methods are approximately uncorrelated ; therefore we introduce a combined estimator which reduces the error bars by a factor of about @xmath48 . the local curvature test has been used in the flat limit to test the presence of non - gaussianity due to cosmic strings @xcite , and the extension to the spherical case @xcite , has been used to verify the asymmetries in the wmap data . in this section we will test the power of the local curvature test to detect primordial non - gaussianity . first we review the method . + we consider a cmb temperature map @xmath49 normalised with its standard deviation @xmath19 : @xmath50 the hessian of this map can be written as @xmath51 where the comma denotes the ordinary derivative and the semicolon covariant derivative . in order to evaluate the derivatives it is convenient to go to harmonic - space : @xmath52 and to use the recurrence relations @xcite : @xmath53 twice to obtain the hessian values in every point of the map , as suggested in @xcite . the presence of noise may render the derivatives unstable , but we have checked that the results in this paper do not change significantly when smoothing the map before performing the derivatives . the points of the renormalised map can be classified as :    * _ hills _ where the eigenvalues of the hessian are both positive , * _ lakes _ where the eigenvalues of the hessian are both negative , * _ saddles _ where the eigenvalues have opposite sign . we calculate the proportions of hills , lakes and saddles above a certain level @xmath54 in the normalised map . in this way , we obtain three functions of @xmath54 which in the gaussian case have a known functional form @xcite . deviations of these functions from this gaussian expectation value can be used to detect non - gaussianity .    in order to constrain the @xmath0 parameter our procedure is as follows :    * we generate a set of primordial non - gaussian maps with the method described in section [ sect : mapng ] with the wmap best fit power spectrum and different @xmath0 ; * we convolve with a beam and add noise corresponding to the given experiment ; * we apply the _ kp0 _ galaxy and point source mask ; * we degrade the maps to the resolution of _ nside _ = 256 where the derivatives are performed ; * we count the densities of hills , lakes and saddles for each map and each level @xmath54 with an extended mask to avoid the instabilities of the derivatives close to the boundaries of _ kp0 _ ( for details of the extensions , see @xcite ) . in this way , we obtain the form of the hill , lake and saddle densities as a function of @xmath0 ; * we repeat the last three points for the data of the given experiment . * for each simulation as well as for the data , we construct a @xmath55 as follows @xmath56 where @xmath57 $ ] , @xmath58 and @xmath59 are the hill and lake densities and @xmath60 are the threshold values used , given by @xmath61 and @xmath62 . the maximum threshold was determined in such a way as to obtain sufficient statistics for the lake proportions which go to zero at high thresholds . the covariance matrix @xmath63 with elements @xmath64 is evaluated on the basis of 1000 gaussian simulations . the distribution of the hill and lake densities at each threshold has been found to be close to gaussian , justifying the above form of the @xmath55 . * the estimate of @xmath0 is obtained for each simulation and for the data by minimising this @xmath55 with respect to @xmath0 . * bayesian error bars are derived by constructing the likelihood @xmath65 . integrating this likelihood with respect to the parameter , we obtain the approximate bayesian `` credible regions '' ( see section [ sect : stati ] ) . * the frequentist error bars are derived by the histogram of the estimates @xmath66 from simulations and defining the @xmath67 and @xmath2 levels as the limits within which @xmath68 and @xmath69 of the estimates fall . note that we will perform the estimates ( 1 ) using only the diagonal part of the covariance matrix and ( 2 ) including the full covariance matrix . we will later show that this makes a huge difference when considering the error bars , and that care has to be taken when approximating the covariance matrix to be diagonal . the results of these simulations are shown in the left panel of figure  [ fig : hillfnl ] , where we can see the effect for the different values of @xmath0 on the hill density . we applied this procedure to the publicly available wmap data . we co - added the ( foregrounds subtracted ) maps from the three wmap frequency channels q , u and w following the procedure in @xcite . in figure [ fig : chi2data ] we show the @xmath70 for the data around its minimum in the two cases , with and without the off - diagonal elements of the covariance matrix . when using only the diagonal parts of the covariance matrix we estimate @xmath71 with a @xmath68 and @xmath69 credible region equal to @xmath72 and @xmath73 , respectively , in good agreement with previously released estimates @xcite . we also include proper ( frequentist ) confidence intervals , which turn out to be @xmath74 and @xmath75 , respectively . the relationship between the two approaches is discussed in subsection [ sect : stati ] . when including the full covariance matrix , we estimate @xmath76 with a @xmath68 and @xmath69 approximate credible region equal to @xmath77 and @xmath78 respectively , while the frequentist constraints turn out to be @xmath79 and @xmath80 . we observe that when not including the full covariance matrix , there is a huge difference in error bars between the bayesian / frequentist approaches . including the full matrix , this difference is smaller but persists . the much bigger frequentist error bars can be explained as follows : when assuming that the thresholds are uncorrelated , huge deviations from the expected value at many thresholds provide strong evidence against the model to be tested . on the other hand , taking into account the fact that the thresholds are highly correlated , the evidence becomes weaker and the expected model can still be consistent with the observations . note further that bayesian error bars tend to be lower than in the frequentist case in the above estimates from the wmap data ; the same phenomenon occurs in simulated maps . + as a final remark , we note that equation ( [ eq : chicur ] ) can be exploited to implement a goodness - of - fit test for our specification of noise and foreground features . more precisely , we compared the best fit value @xmath81 obtained for the wmap data with @xmath82 from 200 monte carlo simulations of non - gaussian cmb maps with @xmath83 . when using the diagonal covariance matrix , the observed value corresponds to the @xmath84 quantile , thereby suggesting good agreement between wmap data and our simulated models . when we adopt the full covariance matrix , the observed value is smaller than @xmath85 of the simulations , that is , the local curvature statistics on wmap data are closer to their expected values than the great majority of simulated maps . note however that we are performing four similar tests on the data ( local curvature , wavelets with and without covariance matrix ) and therefore a @xmath85 deviation from simulations is not significant . wavelets are a very flexible tool used in connection with cmb data for denoising @xcite , extracting point sources @xcite , and detecting non - gaussianity @xcite . to test the consistency of our maps of primordial non - gaussian models with those of @xcite we have repeated the method for estimating @xmath0 described in mw . the basics steps for the test are the following ( for more details , see mw ) :    * we generate a set of primordial non - gaussian maps with the method described in section [ sect : mapng ] with the wmap best fit power spectrum and different @xmath0 ; * we add noise and convolve with a beam corresponding to the given experiment ; * we apply the _ kp0 _ galactic cut , leaving the point sources unmasked ; * we degrade the maps to the resolution of _ nside _ = 256 ; * we obtain the wavelet coefficients by convolving each map with the spherical mexican hat wavelet ( smhw ) given by : @xmath86 ^ 2 \left[2-\left(\frac{y}{r}\right)^2e^{-y^2/2r^2}\right],\ ] ] using the scales @xmath87 given in mw ( 14 , 25 , 50 , 75 , 100 , 150 , 200 , 250 , 300 , 400 , 500 , 600 , 750 , 900 , 1050 arcmin ) ; * we use only the coefficients outside an extended _ kp0 _ mask ( for details of the extensions , see @xcite and mw ) to obtain the skewness as a function of @xmath0 . the result is shown in the right panel of figure  [ fig : hillfnl ] ; * we finally repeat the last three steps for the data of the given experiment . * for each simulation as well as for the data , we construct a @xmath55 as follows @xmath88 where the elements of the vector @xmath89 are @xmath90 where @xmath91 is the skewness of the wavelet coefficients on scale @xmath92 . the covariance matrix @xmath63 with elements @xmath64 is evaluated on the basis of 1000 gaussian simulations . the distribution of the skewness at each scale is close to gaussian , justifying the above form of the @xmath55 . * for each simulations and for the data we estimate @xmath0 by minimising this @xmath55 with respect to the parameter . * bayesian error bars are found by constructing the likelihood @xmath65 . integrating this likelihood with respect to the parameter , we obtain the approximate bayesian credible regions ( see section [ sect : stati ] ) . * the frequentist error bars are derived by the histogram of the estimates @xmath66 from simulations and defining the @xmath67 and @xmath2 levels as the limits within which @xmath68 and @xmath69 of the estimates fall . again , the test was performed on the co - added ( foreground cleaned ) q+v+w wmap map . the plot of the @xmath55 for the data is shown in figure  [ fig : chi2data ] . we estimate the bayesian credible region , considering only the diagonal part of the covariance matrix ( the results presented in mw were obtained using the same approximation ) , for @xmath0 to be @xmath93 at the 1@xmath19 level and @xmath94 at the 2@xmath19 level , in agreement with the values derived in mw ( see again the discussion in section [ sect : stati ] ) . considering the full covariance matrix , we estimate @xmath0 to be @xmath95 at the 1@xmath19 level and @xmath96 at the 2@xmath19 level as for the curvature , we have also calculated the frequentist confidence intervals which in the first case turn out to be quite larger , @xmath97 and @xmath98 respectively , while considering the full covariance matrix we find for the error bars on @xmath0 @xmath99 and @xmath100 . again the bayesian credible regions are generally smaller than the frequentist confidence levels , but considering the full covariance matrix , the difference is reduced . note further that the bayesian credible regions are slightly underestimated when approximating the covariance matrix to be diagonal . as before , we also implemented a goodness - of - fit test based on eq . ( [ eq : chiwav ] ) ; we found that the values on the wmap data corresponds to the @xmath101 and @xmath102 quantile obtained from 200 simulations with @xmath103 when using the diagonal and full covariance matrices , respectively . again , this suggests that our specification of noise and foreground masks provides a reasonable approximation to the experimental settings of wmap . in this section we apply the combined procedure introduced in @xcite to improve the constraints on @xmath0 . first of all , we estimated the correlation between the two estimators @xmath104 and @xmath105 by 200 monte carlo simulations and found @xmath106 suggesting that the correlation between the estimators is less than @xmath107 . then , we can lower the error bars on @xmath0 by combining the two statistics . more precisely , we use the estimates obtained with the full covariance matrix and we evaluate the frequentist confidence intervals on the following statistic : @xmath108 using the combined test on the wmap data , we estimate @xmath109 , with the constrains at @xmath7 and @xmath2 levels of @xmath110 and @xmath111 , respectively .      in this subsection we present a brief discussion on the evaluation of ( frequentist ) confidence intervals and ( bayesian ) credible regions on @xmath0 . let us denote by @xmath112 our estimate of the non - linearity parameter obtained by minimising eqs . ( [ eq : chicur ] ) and ( [ eq : chiwav ] ) : also , let us denote by @xmath113 the likelihood function . as well known , an @xmath114-level confidence interval for @xmath0 based upon the observation @xmath112 is the set of all values @xmath0 such that @xmath115 note that the integral is taken with respect to the estimate @xmath116 that is , we are viewing @xmath117 as the probability density of our estimator . in words , we include in the confidence interval all the values @xmath0 such that the probability to get an estimated parameter as @xmath112 or further away is at least as large as @xmath118 . more clearly , a value is included provided it does not entail that _ observing what we observed _ is less probable than @xmath118 . of course , in the special case where the distribution of @xmath116 is gaussian with mean @xmath0 and variance @xmath119 which does not depend on @xmath0 ( or , in general , is symmetric under the exchange of @xmath0 and @xmath120 ) , we have @xmath121 by the symmetry of the previous expression with respect to @xmath122 . this justifies the common practice to derive confidence intervals by integrating the likelihood . rigorously speaking , however , this is no longer justified if the integrand is not symmetric with respect to an exchange of @xmath0 with @xmath116 ( for instance if @xmath19 is not constant with respect to @xmath0 ) . one may then try to justify the integration of the likelihood by a bayesian viewpoint , by assuming a flat prior and viewing @xmath123 as a posterior density function . the resulting set , however , should not be labelled as confidence interval ( which is a frequentist concept ) : it is a bayesian credible region , which will depend in general on the choice of the prior . it is occasionally stated that this dependence is overcome by the choice of a flat prior . the latter is claimed to be non - informative by definition : indeed , no physicist would consider a priori equally likely that @xmath0 lies in @xmath124 $ ] rather than it is in @xmath125 $ ] ( say ) ; thus a flat prior , although unphysical , is usually justified as a _ panacea _ to get objective results . this argument is to some extent misleading , though , as it can be shown by standard counterexamples . take for instance eq . ( [ eq : model ] ) , and assume for brevity s sake that @xmath126 ( otherwise duplicate our argument by symmetry ) . then eq . ( [ eq : model ] ) can be rewritten as @xmath127 that is @xmath128 . of course , from the physical point of view there is no reason to prefer the alternative specification in eq . ( [ eq : model ] ) to ( [ 1bis ] ) . now let us assume we impose a `` non - informative '' ( flat ) prior on @xmath129 , the posterior probability becomes @xmath130 and the credible sets are thus obviously affected , although we are working with exactly the same model as before and we are claiming to have used no a priori information . in short , flat priors are simply shifting the choice from the form of the prior probability to the form of the statistical parametrisation ; the latter , moreover , is not due to physical considerations , but simply to computational convenience .    with this in mind , we will now use @xmath120 estimated from 200 simulations ( with the experimental settings used above ) to investigate whether the distribution of @xmath120 as a function of the model @xmath0 is symmetric with respect to an exchange of @xmath0 and @xmath120 . in figures  ( [ fig : contcur ] ) , ( [ fig : contcurcov ] ) we show a contour plot of this function for the curvature and wavelet test , respectively . when not considering the full covariance matrix , this function is markedly skewed for the wavelet test , but even more so for the curvature test ( see fig . [ fig : contcur ] ) . this may explain the big difference between the confidence intervals and bayesian credible regions for the wavelets and the even bigger difference for the curvature test when correlations are neglected . on the other hand , when correlations are properly taken into account , the model fit is better and the function becomes considerably more symmetric ( see fig . [ fig : contcurcov ] ) . this is also reflected in the fact that the difference between bayesian and frequentist error bars is markedly reduced in this case . in this paper we have used simulated cmb maps of primordial non - gaussian models , generated with the algorithm described in @xcite to estimate the non - linear coupling parameter @xmath0 from the wmap data . as all other estimates of @xmath0 in the literature have been based on non - gaussian cmb maps generated by a different approach @xcite , one of the aims of this paper was to check whether a different way of generating cmb maps with the same kind of primordial non - gaussianity gives a consistent estimate of @xmath0 . in order to perform a direct test of consistency , we applied the estimator of @xmath0 using smhw presented in @xcite on the wmap data . when neglecting the scale - scale correlations , we find that our estimate of @xmath0 is in full agreement with theirs and thus that the two ways of generating primordial non - gaussian maps give fully consistent results . we have also presented a new method to estimate @xmath0 based on the local curvature of the cmb fluctuation field . our estimate of @xmath0 with this method on the wmap data is consistent with estimates of @xmath0 using other approaches . moreover , we point out the importance of including the full covariance matrix in the @xmath55 when estimating @xmath0 with these two methods . we also discuss the difference between frequentist confidence intervals and bayesian credible regions on the estimate of @xmath0 . we show that the difference between these two methods for finding error bars can be huge when not including the full covariance matrix , particularly for the curvature estimator where correlations between thresholds are important . for the wmap data , the bayesian credible region expands whereas the frequentist confidence intervals shrink when taking into account correlations in the covariance matrix . we find further that the bayesian credible regions are in general smaller than the frequentist confidence intervals . we conclude that care must be taken when approximating the confidence intervals on @xmath0 using the integral of the likelihood with respect to the parameter . including the full covariance matrix , we find with the curvature method @xmath3 at the @xmath2 level using frequentist confidence intervals , and @xmath1 using bayesian credible regions . similarly for the wavelet method , @xmath131 with frequentist confidence intervals and @xmath132 . finally , we show that the two methods provide approximately uncorrelated estimates of @xmath0 ; this observation naturally suggests an improved estimator which combines the two methods . with this combined test , we find @xmath5 and @xmath6 at the @xmath7 and @xmath2 level respectively . we are grateful to an anonymous referee for insisting on including the full covariance matrix . we acknowledge the use of the legacy archive for microwave background data analysis ( lambda ) . support for lambda is provided by the nasa office of space science . partial financial support from inaf ( progetto di ricerca `` non - gaussian primordial perturbations : constraints from cmb and redshift surveys '' ) is acknowledged . fkh acknowledges financial support from the cmbnet research training network . we acknowledge use of the healpix @xcite software and analysis package for deriving the results in this paper . this research used resources of the national energy research scientific computing center , which is supported by the office of science of the u.s . department of energy under contract no . de - ac03 - 76sf00098 .
we test the consistency of estimates of the non - linear coupling constant @xmath0 using non - gaussian cmb maps generated by the method described in ( liguori , matarrese and moscardini 2003 ) . 
 this procedure to obtain non - gaussian maps differs significantly from the method used in previous works on estimation of @xmath0 . nevertheless , 
 using spherical wavelets , we find results in very good agreement with ( mukherjee and wang 2004 ) , showing that the two ways of generating primordial non - gaussian maps give equivalent results . 
 moreover , we introduce a new method for estimating the non - linear coupling constant from cmb observations by using the local curvature of the temperature fluctuation field . 
 we present both bayesian credible regions ( assuming a flat prior ) and proper ( frequentist ) confidence intervals on @xmath0 , and discuss the relation between the two approaches . 
 the bayesian approach tends to yield lower error bars than the frequentist approach , suggesting that a careful analysis of the different interpretations is needed . using this method 
 , we estimate @xmath1 at the @xmath2 level ( bayesian ) and @xmath3 ( frequentist ) . 
 moreover , we find that the wavelet and the local curvature approaches , which provide similar error bars , yield approximately uncorrelated estimates of @xmath4 and therefore , as advocated in ( cabella et al . 
 2004 ) , the estimates may be combined to reduce the error bars . in this way 
 , we obtain @xmath5 and @xmath6 at the @xmath7 and @xmath2 level respectively using the frequentist approach . 
 [ firstpage ]    cosmic microwave background - cosmology : theory - methods : numerical - methods : statistical - cosmology : observations