since ancient times , people have struggled with how best to draw a map .the dawn of the age of discovery and its great seafaring voyages brought with it an urgent growth in cartographic investigation and prompted the development of differential and projective geometry .but it has been long recognized that all maps distort certain surface features , so that the question of which map to use is answered by which features one wants to represent and in which manner they should be represented .conformal maps such as the mercator projection , prominent for use in navigation at sea , locally preserve angles and shapes at the expense of distorted areas - making greenland look bigger than africa and south america .other typical map projections preserve areas or distances between two points at the expense of distortions in other features .there are an infinite number of possible map projections , an infinite number that function as a compromise between distortions in two desired features , and an infinite number that do not preserve any particular feature .gauss theorema egregium of 1828 illuminated the reason behind the inability to find a distortionless map , showing that any map between surfaces of different curvature will always introduce distortions .  in more recent times, there has been an interest in mimicking certain aspects of gravitating systems in laboratory settings .for example , similarities between the space - time metric and the navier - stokes equations for a flowing fluid show that a sonic or surface wave horizon in the flowing fluid could provide an experimentally accessible surrogate for the study of near - horizon relativistic effects including hawking radiation .  light propagation through dielectric media is another system that could serve as such an analog space - time .the connection between light propagation in gravitational systems and light propagation in dielectric media was recognized in the early days of general relativity when it was observed that the relativistic deflection of starlight by a massive object such as the sun could instead be explained by an appropriate distribution of refractive media around the object .it was subsequently shown by gordon that the correspondence also holds in the other direction ; that light propagation in certain refracting media can instead be mathematically described as light propagation through a curved vacuum space - time with an `` optical metric '' that is determined by the optical properties of the medium .later , plebanski identified the effective permittivity and permeability of an arbitrary space - time metric , and these results were used by de felice to describe a dielectric representation of schwarzschild space - time , arguably the first definite description of an analog black hole . in the last decade , developments in metamaterials have opened the possibility of physically realizing artificial media possessing some of the unusual dielectric properties required to construct a dielectric analog space - time and has sparked a resurgence of interest in the subject .dielectric analog space - times are obtained as a projection of a curved vacuum manifold into a dielectric medium residing in a space - time of different curvature , usually flat minkowski space - time , in very much the same manner that earth s curved surface is cartographically projected onto a flat plane .gauss theorema egregium applies equally to this modern concept of space - time map making ; consequently , any dielectric analog space - time must necessarily introduce distortions into its representation of light propagation through the vacuum .this fact seems to be underappreciated in much of the literature , which has largely focused on the form of the optical metric and the coordinate description of optical rays through the analog .the majority of investigations in the subject have followed the footsteps of plebanski and de felice in identifying the dielectric properties of the analog and the optical metric with the vacuum metric , an identification that could mislead to the assumption that light propagation through the analog faithfully emulates the behavior of light propagating through the vacuum .  by analyzing the kinematics of a congruence of light in the analog ,we show that the plebanski analog fails to emulate physically measurable characteristics such as the fractional rate of expansion of a transverse cross section of the congruence .thus the plebanski choice of projection map is actually quite similar to the mercator projection in that it preserves the coordinate description of a ray but introduces distortions into the evolution of the cross - sectional area of a congruence . as with cartography ,there are an unlimited number of analog projections , and a more general mapping of vacuum space - times into analog representations was introduced in ref . . using this ,we demonstrate how an expansion - preserving projection map may be constructed , such that the expansion of a congruence is faithfully represented in the analog at the expense of distortions to the optical metric and the coordinate path of light rays .we also provide a more detailed analysis of the covariance properties of dielectric analog space - times and show that the analog system lacks covariance in the sense that two analogs representing two different coordinate descriptions of the vacuum space - time are not themselves related by the same , or even an associated , coordinate transformation .  furthermore , plebanski s identification of a space - time metric with effective dielectric properties has been widely adopted as a basis for the interpretation of transformation optics ( to ) , and to has been correspondingly invoked to undertake studies in analog space - times .we argue that while dielectric analog space - times and to are indeed very closely related , there are subtle differences that make the analog space - times approach unsuitable as a rigorous foundation for transformation optics . aside from conceptual issues , there are three problems with using analog space - times as the basis for to .first , the lack of any unique or canonical projection from the vacuum to the analog makes this approach to to ill - defined .second , any useful formulation of to should be fully covariant , while analog space - times suffer from covariance issues .third , the fact that no analog projection can faithfully replicate all aspects of light propagation in the vacuum means that the to analog simply does not function as expected .this paper is organized as follows .section [ sec : review ] briefly reviews the covariant formulation of electrodynamics in dielectric media .section [ sec : dielctricanalogs - review ] reviews a rigorous theory of dielectric analog space - times and studies their covariance properties . in sec .[ sec : lightpropagation ] we study the kinematics of light propagation through the analog , with a particular focus on the transverse expansion , and show how an expansion - preserving analog projection map may be constructed for a particular choice of congruence . in sec .[ sec : misconceptions ] we compare analog space - times with to , arguing that while they are closely related they are sufficiently , though subtly , distinct to have important consequences .concluding remarks are given in sec .[ sec : conclusion ] . throughout the paperwe provide examples and illustrative calculations in terms of the friedmann - lamatre - robertson - walker ( flrw ) analog space - time .the metric signature is chosen to be and the speed of light in vacuum is set to .we wish to describe electrodynamics inside dielectric media residing in a possibly curved space - time with metric .we use a tensorial , exterior calculus formulation of electrodynamics in dielectric media , the principal notions of which are reviewed here . a more complete introduction to tensorial electrodynamicsmay be found in , e.g. , refs . , while a more complete discussion of our application to electrodynamics in media may be found in ref . .  in this formulation ,components of the vector field pairs and are seen to be components of the field strength and excitation 2-forms ( antisymmetric second - rank covariant tensors ) and . in a locally flat cartesian frame , and have the matrix representations and   pseudo - riemannian -dimensional manifolds such as are naturally endowed with an operator called the hodge dual that for takes a -form to an -form . for the purposes pursued herein , we only require the hodge dual to act on 2-forms . in the case of a four - dimensional space - time , the hodge dual maps a 2-form into another 2-form and has the following index expression : hence , the components of in a local cartesian frame are   maxwell s homogeneous and inhomogeneous equations are  [ eq : maxwell ]   where denotes the exterior derivative and is the charge - current 3-form , though in what follows we assume .a complete solution of maxwell s equations requires the specification of a constitutive relation between the fields and .it has been shown that such a constitutive relation for linear media has the form  where contains all the information about the medium such as permeability and permittivity , while separately contains information about the space - time geometry . in component form, the constitutive relation reads the dielectric tensor is independently antisymmetric under index exchange on either or , and thus has a maximum of 36 independent parameters . in practice ,further symmetries may be imposed based on thermodynamic or energy conservation arguments that reduce the number of independent components to 15 .the coordinate - independent vacuum is uniquely defined as the trivial dielectric such that .component - wise expansion of the constitutive eq .( [ eq : g = starfcompon ] ) in a local cartesian frame recovers the vectorial relations {\bar{\bar{\gamma}}}{^c } \vec{b } ,  \quad\vec{h}=\bar{\bar\mu}^c\vec{b } + \tensor[^e]{\bar{\bar{\gamma}}}{^c}\vec{e}\ ] ] which can be put into the traditional relations {\bar{\bar{\gamma}}}{}\vec{h } , \quad\vec{b}=\bar{\bar\mu}\vec{h}+ \tensor[^e]{\bar{\bar{\gamma}}}{}\,\vec{e}\ ] ] where the `` traditional '' matrices of permittivity , permeability , and magnetoelectric couplings and in eq .( [ eq : constitutive2 ] ) , are related to the `` covariant '' quantities of eq .( [ eq : constitutive1 ] ) as follows :  [ eq : material ] {\bar{\bar{\gamma}}}{^c}\right)\left(\bar{\bar\mu}\right)\,\left(\tensor[^e]{\bar{\bar{\gamma}}}{^c}\right ) , \label{eq : mu - epsilon } \\ \tensor[^h]{\bar{\bar{\gamma } } } { } & = \left(\tensor[^b]{\bar{\bar{\gamma}}}{^c}\right)\,\left(\bar{\bar{\mu}}\right),\quad \tensor[^e]{\bar{\bar{\gamma } } } { } = -\left(\bar{\bar{\mu}}\right)\,\left(\tensor[^e]{\bar{\bar{\gamma}}}{}\right ) .\label{eq : gamma}\end{aligned}\ ] ]  thus in a locally flat frame , the components of may be identified as the usual permeability and permittivity , but the meaning of these concepts is less clear in the general curvilinear coordinates of curved space - times where the metric contributes nontrivially to the constitutive relation .equations ( [ eq : maxwell ] ) and ( [ eq : g = starf ] ) provide a description of electrodynamics in linear , dispersionless , lossless media residing in possibly curved space - time manifolds . by selecting a map from one such manifold to another ,the behavior of electromagnetic fields in one manifold may be identified with electromagnetic fields in another manifold such that maxwell s equations are satisfied in the target manifold . in particular , on - shell electromagnetic fields in an arbitrary vacuum space - time may be identified with on - shell solutions in a nonvacuum dielectric residing in minkowski space - time , thereby enabling a flat space - time dielectric representation of a curved space - time in much the same way that the curved surface of the earth may be mapped into a flat cartographic representation .but as the map - makers of old discovered , there is no canonical choice of projection map . indeed , since the curved space - time and the flat space - time are not isometric , gauss theorema egregium demands that not all features of the curved space - time can be simultaneously faithfully represented in the flat space - time , and the choice of projection map is informed by what features of the curved space - time one wishes to faithfully represent .choosing a coordinate identity as the projection map recovers plebanski s early results that have been used almost exclusively in the study of dielectric analogs ever since .is projected into a dielectric analog residing in minkowski space - time .electromagnetic fields on are projected to by the pullback of . ]consider the curved vacuum space - time manifold with metric .the dielectric properties of this vacuum space - time are uniquely described by .we wish to project on - shell solutions of maxwell s equations in into on - shell solutions of maxwell s equations in a dielectric medium residing in minkowski space - time with metric , as in fig .[ fig : analogmap ] . for this purpose , define the _ analog projection map _   the electromagnetic fields and are projected into the analog with the pullback of , which is denoted . mapping on - shell solutions from to requires satisfaction of the two conditions  [ eq : g = pll ]   equations ( [ eq : g = plla ] ) and ( [ eq : g = pllb ] ) can be solved for the analog medium ( setting ) , resulting in  where is equivalent to the jacobian matrix of the projection .capital latin indices refer to the vacuum manifold , while lower case greek indices refer to the minkowskian analog .equation ( [ eq : chi - general ] ) gives the parameters of a dielectric material in minkowski space - time that supports the electromagnetic field configuration specified by the projection . in this sense, the medium emulates the behavior of electromagnetic fields in the curved vacuum space - time .choosing , the coordinate identity map , then and expanding the constitutive eq .( [ eq : g = starf ] ) for this trivial analog medium in minkowski space - time and converting to the representation of eq .( [ eq : constitutive2 ] ) leads directly to plebanski s results for the effective permeability , permittivity , and magnetoelectric couplings of a vacuum space - time  {\bar{\bar{\gamma}}}{}=\left(\tensor[^h]{\bar{\bar{\gamma}}}{}\right)^t=-\epsilon_{ijk}\frac{g_{0j}}{g_{00}},\ ] ] where is the purely spatial part of the vacuum space - time metric and are the time - space components of the metric .if the permeability and permittivity are isotropic , then the bianisotropy vector evident in the magnetoelectric coupling may be identified with a characteristic velocity of the medium note that eqs .( [ eq : plebanskiepsilon])-([eq : bianisotropy ] ) do not conserve index type since they are not a covariant representation of the medium .analog space - times are not covariant in the sense that they depend on both the choice of coordinates in and the choice of projection , and a coordinate transformation in does not correspond to a coordinate transformation in the analog .on the one hand it is straightforward to see that plebanski s analog model , based on the projection , leads to analogs that depend on the choice of coordinates in . given two coordinate descriptions of and employing on each , two analog mediums are calculated and found to be physically different objects rather than different coordinate descriptions of the same physical object .for example , consider flrw space - time in comoving coordinates and physical coordinates , related by .if both the comoving and physical coordinate descriptions of flrw are projected into the spherical coordinates of minkowski space - time with , then are both already described in the same minkowski spherical coordinates and can not be related by a coordinate transformation . in particular , suppose is the jacobian matrix of a coordinate transformation in the vacuum space - time .then eq .( [ eq : trivialanalog ] ) would become where the unprimed index on is equivalent to an index in the analog because of the fixed projection map . if the first two factors of could be pulled through the leading operator such that the acted on as in eq .( [ eq : trivialanalog ] ) , then eq . ( [ eq : plebanskinoncovariance ] )would show that is related to by a coordinate transformation .but this is clearly not possible so we conclude that analog space - times based solely on are not covariant under coordinate transformations .   and may be related by , but is not associated to the coordinate transformation . the analog model is not covariant under coordinate transformations in . ]  on the other hand , since the projection map is freely specifiable , we may find related projections and such that the same physical analog is produced , but this is not the same thing as a coordinate transformation in the analog .let be a coordinate transformation in with jacobian matrix , and let be an independent coordinate transformation in minkowski space - time with jacobian matrix , as depicted in fig .[ fig : commdiagram ] . given the analog projection , given by eq .( [ eq : chi - general ] ) , which may be related to the primed quantities by inserting two factors of , this can be written as with or in terms of index - free matrix multiplication since , this says that by choosing to be the related projection that makes fig .[ fig : commdiagram ] commutative , and will represent the same physical analog medium .however , the coordinates in the analog are still completely independent of the coordinates in .one could choose and we see that the action of is simply to undo the coordinate transformation , which returns us to the original configuration in the original analog coordinates .since the projection is freely specifiable , may be represented by an infinite number of physically inequivalent analog media in minkowski space - time .the projection into the analog is not unique and there does not exist a canonical choice of . indeed , as we have seen , a single choice of is not even possible if one wants different coordinate descriptions of a vacuum manifold to have the same physical analog representation .however , this wide freedom in the choice of also implies that the analog model only has meaning when interpreted with respect to the chosen map .furthermore , since a curved vacuum space - time can not be isometrically mapped into a minkowskian analog , not all features of light propagation in the curved manifold can be simultaneously emulated by the analog . in subsequent sectionsit is shown that the choice of projection map should be informed by which feature of light propagation one wishes to emulate .   to illustrate the physical inequivalence of analogs obtained using with different coordinate descriptions of the same space - time , we consider flrw space - time in conformal , comoving , and physical coordinates , and find that the corresponding analogs are not related by a coordinate transformation .friedmann - lamatre - robertson - walker space - time is often described by comoving coordinates with the line element in this form it may be seen that a spatial hypersurface of the flrw space - time expands or contracts with time through the scale factor and may be used to model an expanding or contracting universe . from eq .( [ eq : trivialanalog ] ) , and eqs .( [ eq : g = starfcompon])-([eq : material ] ) , the corresponding trivial dielectric analog of flrw space - time in comoving coordinates is given by  [ eq : comovmat ]   thus for this coordinate representation , the analog model is an isotropic but time - dependent medium .  alternatively , flrw is actually a conformally rescaled form of minkowski space - time . defining the conformal time the flrw space - time now has the line element and we find that the resulting analog is just empty minkowski space - time  [ eq : confmat ]   this result highlights an important feature of dielectric analog space - times in that ,as far as the light cone is concerned , a dielectric analog actually represents an equivalence class of conformally related space - times .the reason is that the operators in eq .( [ eq : chi - general ] ) are conformally invariant and so is also conformally invariant .thus given a coordinate description that is conformally related to minkowski space - time we should not be surprised that the analog is just minkowski space - time .  in the comovingcoordinates introduced above , a spatial hypersurface is seen to expand even though the spatial coordinate points themselves remain fixed .this implies that the physical distance between two points of fixed comoving coordinates is actually increasing [ or decreasing depending on ] with time .the physical distance to an object at fixed comoving coordinate is an observer would naturally interpret the behavior of objects at fixed comoving coordinates as moving away from each other with recessional velocities that are proportional to their distance - a rule known as hubble s law .this hubble flow may be conveniently described by the physical coordinates with line element the same procedure as used before gives the physical coordinates analog as  [ eq : physmat ]   which is a complicated anisotropic magnetoelectric medium .  in each of these cases ,the target manifold has been specified as minkowski space - time with spherical coordinates .it is therefore clear that plebanski s approach with analog projection map results in different physical analog models for different coordinate descriptions of the same vacuum space - time .from the time of eddington and gordon , through plebanski and to the present , the study of dielectric analog space - times has been largely focused on coordinate descriptions of the geometric optics ray trajectories of light through the analog , and it appears to have been widely assumed that this coordinate description suffices as a faithful representation of light propagation through the curved space - time .however , a coordinate description of ray trajectories in the analog is of limited value because the coordinates in and the analog coordinates refer to different metrics .so even if rays in the analog pass through points with the same coordinate labels as rays in , they can not simultaneously replicate all kinematical aspects of light propagation in , such as the fractional expansion , shear , and vorticity of a congruence ; quantities that carry characteristic information about the space - time .for example , the fractional expansion of a null congruence tells us about the presence of singularities , trapped surfaces , and horizons , and so would need to be properly analyzed for any analog that claimed to replicate such features .in fact , it turns out that the ability of an analog model to faithfully emulate these characteristic features is quite limited .we focus on the fractional expansion of a congruence of light rays in the analog , and find that while a projection may be found such that the expansion of a congruence in the analog matches the expansion of a congruence in , the required map is specific to the chosen congruence ; any other congruence in the analog will fail to have the correct expansion .the discussion of light rays falls within the realm of geometric optics .to arrive at the geometric optics limit in the analog , one may begin by assuming a solution for the 4-vector potential and . inserting this into maxwell s equations ( [ eq : hom ] ) and( [ eq : inhom ] ) and taking the stationary phase approximation leads to  where  is the background minkowski metric , and is the wave vector . the existence of nontrivial solutions to eq .( [ eq : eikonal ] ) requires .it turns out that this determinant condition is satisfied identically , so the subsidiary condition must be satisfied , where the indices only run over the purely spatial components .  for impedance - matched media such as that obtained by projection from a vacuum space - time , will always be of the form where can be thought of as a second - order polynomial in , and is some factor that does not depend on . the condition will be everywhere satisfied on a solution curve if and only if , so the condition becomes the condition .for impedance - matched media , can always be written as which is of the same form assumes in vacuum space - times , but where would be replaced with the space - time metric .thus , in media the tensor is often referred to as the `` optical metric , '' but should be understood as containing information on both the medium parameters and background metric . since is a function on the cotangent bundle that vanishes everywhere on a solution curve , then the exterior derivative of also vanishes , , leading to hamilton s canonical equations which may be integrated to find the ray trajectories .   before showing the failure of the analogto correctly replicate kinematical aspects of light propagation in , we first examine some examples where an analog generated by has potentially interesting properties . consider the projection between flrw space - time in comoving coordinates and the spherical coordinates of minkowski space - time .the optical metric is , so one might naively expect light propagation in the analog to fully mimic light propagation in the vacuum .the wave vector for a future - directed radially propagating ray in the analog is from we get in which the positive ( negative ) sign corresponds to outgoing ( ingoing ) rays .the tangent to the curve is found from the first of hamilton s equations ( [ eq : hamilton ] ) to be meanwhile , the second of hamilton s equations ( [ eq : hamilton ] ) provides an evolution equation for  thus we have a single equation for , but from the first of hamilton s equations we know that .the solution for is straightforwardly and substitution into eq .( [ eq : v - analog ] ) yields for an outgoing ray , one deduces   if light commences propagating radially inward from all points in the analog at some time , then the largest radius in the analog from which an observer at has received a signal at time is the above relation replicates the most common definition of the particle horizon - the most distant point in the universe from which we have received light since the big bang .furthermore , if we consider the frequency of two successive pulses , separated temporally by an interval , propagating in the analog and reaching an observer , then eq .( [ eq : dr / dt ] ) gives the frequency shift between the two pulses by for a detailed derivation of frequency shift and particle horizon from eq .( [ eq : dr / dt ] ) , see for example section 10 of ref .note that redshifts in the analog are facilitated by the time dependence of the medium parameters .the above results show that at least some of the basic and well - known effects observed in flrw space - time can be replicated in the analog .  before moving on ,it is worth pointing out some generic features of light propagation in the analog .first , for both ingoing and outgoing rays , indicating that light rays in the analog are time - like .second , shows that the light rays are not geodesic with respect to the background minkowski space - time , nor is it possible to find a reparametrization of the curve so as to make it geodesic .third , is valid in the analog , but it is not true that .despite the fact that the projection results in an optical metric , light propagation in the analog does not faithfully replicate all aspects of light propagation in the vacuum .the reason is that although the optical metric is identical to the vacuum metric , most physical measures of light propagation in the analog are actually made with respect to the background minkowski metric , such as the kinematical decomposition of a congruence .this decomposition provides physically meaningful measures such as the fractional rate of expansion of the cross section of a laser beam , and in curved space - times is instrumental in determining the existence of trapped surfaces and horizons . as such ,we focus on the fractional rate of expansion of a radial congruence of light propagating through the analog , and find that projecting with does not reproduce the vacuum rate of expansion .furthermore , we find that although it is possible to construct a projection tailored such that the expansion is preserved , it can only be preserved for a single choice of congruence at a time .preserving the expansion for a different choice of congruence requires a different projection .   before analyzing the expansion of a congruence in the analog , we first find the expansion in the vacuum space - time that we are attempting to emulatewe again consider future - directed radially propagating rays . in the vacuum , these rays are null geodesics for which and .it may be shown that the evolution of a congruence centered on the curve is governed by  .the expansion is the fractional rate of change of the transverse cross section of the congruence , which is equal to the trace of the projection of into the transverse subspace .the light cone at every point is spanned by both outgoing and ingoing rays , meaning that the transverse subspace is only two dimensional .one may find a projection operator from the inherited metric on the transverse subspace  in the literature , is typically introduced as an `` auxiliary '' null vector such that .the expansion of the vacuum congruence about is from which we find the expansion of radially outgoing and ingoing rays to be , respectively ,  [ eq : expansion - null - general - frw ]    by contrast , we have said that rays in the analog follow nongeodesic time - like curves with respect to the background metric , so it is tempting to examine the kinematics of light in media with the formalism for time - like congruences .however , with the advent of metamaterials that offer a great degree of control over light propagation , we find the need for an analysis that can track a congruence that smoothly varies from null to time - like , and even to space - like . the light cone at every point is still defined by a set of two linearly independent rays - left / right or ingoing / outgoing rays .thus the transverse subspace is still two dimensional , as would be expected for say , the cross section of a laser beam .the projection operator defined by eq .( [ eq:2-metric - null ] ) made use of the fact that in the vacuum , a relation that does not hold in the analog .it is therefore necessary to be cognizant of index placement in the analog , and to construct a more general projection operator .it was previously mentioned that could be considered as a second - order polynomial in .thus at every point we actually have two solutions , corresponding to ingoing and outgoing rays .let the pairs and denote the tangent vector and wave covector for outgoing and ingoing rays , respectively , which satisfy  [ eq : conditions - on - the - pairs ]   with these fields , the projection operator generalizes to  one may readily verify that , and also that .since is orthogonal to both and the transverse subspace is still only two dimensional , which is confirmed by the trace .this construction of follows directly from maxwell s equations in media and is equally applicable to congruences that are null , time - like , or space - like with respect to the background metric .  using this projection operator to calculate the transverse fractional expansion of a congruence in the analog with and as in eq .( [ eq : v - analog - new ] ) , one finds which disagrees with the vacuum expansion found in eqs .( [ eq : expansion - null - general - frw - a ] ) and ( [ eq : expansion - null - general - frw - b ] ) .we have found that an analog model based on fails to correctly mimic the fractional rate of expansion of the congruence despite the fact that the optical metric and the coordinate description of light curves are the same as in the original space - time .since the projection is freely specifiable , it is natural to ask whether a different choice of projection could reproduce the correct expansion rate at the expense of the optical metric and coordinate description of curves .  the quantity governs the evolution of the separation between the curve with tangent and a nearby curve in the congruence .let be the projection of into the transverse subspace , thereby describing the purely transverse evolution of the congruence .the fractional expansion is the trace of , so by calculating in both the vacuum and the analog we may find a condition that determines an expansion - preserving projection .expanding the covariant derivative , we have in vacuum , and in the analog , where capital latin indices refer to the vacuum space - time , lowercase greek indices refer to the analog in minkowski space - time , while and are the christoffel symbols for and , respectively .let be the sought - after analog projection map , with associated jacobian .the ingoing and outgoing rays in the analog are related to those in by  [ eq : transform - vacumm - to - analog - u , v ]   from which it follows that the projection operator transforms as and finally , one can recast the analog in eq .( [ eq : b - transverse - analog ] ) in terms of quantities in as {\gamma}{^m_{ns}}u^s\right],\ ] ] with   the above relations show that is not quite the same as a simple mapping of from the vacuum to the analog since is not necessarily the same as .taking the trace of eqs .( [ eq : b - transverse - vacuum ] ) and ( [ eq : transform - vacumm - to - analog - b - analog ] ) we are able to compare and as quantities in . one finds  [ eq : traces - of - b - vacuum , analog ] , \label{eq : traces - of - b - vacuum}\\ { ^a}\theta=\bar b\indices{^\alpha_\alpha}=h\indices{_m^n}\left[u\indices{^m_{,n}}+{^*}\gamma\indices{^m_{ns}}u^s\right ] .\label{eq : traces - of - b - analog}\end{aligned}\ ] ]  hence , in order to have an analog that correctly replicates the expansion of a congruence , we must satisfy the condition ^s=0.\ ] ] now , if one wished to have an analog that correctly replicates the expansion for all congruences then one would have to satisfy the stronger condition which says that pulls from the minkowski background of the analog into .however , by gauss theorema egregium such a map can not exist because the curved space - time and the flat minkowski background of the analog are not isometric .therefore , we have found that a single map can not satisfy eq .( [ eq : constraint - on - the - analog ] ) for all congruences .the best one can hope for is to satisfy eq .( [ eq : constraint - on - the - analog ] ) for a given choice of congruence .  as an example , we seek the map that will preserve the expansion of the radially outgoing congruence with tangent vector given by eq .( [ eq : v - analog - new ] ) , e.g.  [ eq : arbitrary - congruence ]   given the symmetries of the space - time and the congruence , we suspect that the desired projection from flrw to the analog has the form with the associated jacobian since we denote both by .equation ( [ eq : constraint - on - the - analog ] ) provides the differential equation which has the solution where is an integration constant .thus , for the particular choice of congruence made above we have found an expansion - preserving map . for any other choice of congruence we would have to go through the same procedure , for which the outcome would be the specification of a totally different analog projection map .one of the pioneering formulations of transformation optics was based on eqs .( [ eq : plebanskiepsilon ] ) and ( [ eq : magnetoelectric ] ) ; in other words , on plebanski s analog space - times model using as the projection .however , we argue that , while closely related , analog space - times are not the same thing as to .the typical explanation of to is that one starts in minkowski space - time and performs a coordinate transformation which results in a new metric describing a curved space - time ; this curved space - time metric is then matched to a dielectric medium via eqs .( [ eq : plebanskiepsilon ] ) and ( [ eq : magnetoelectric ] ) .this explanation is problematic because the space - time curvature is a diffeomorphism invariant , so it is not possible to use a coordinate transformation to generate a curved space - time and associated metric .one could instead argue that the mechanism being employed is a map into a curved space - time that has the desired properties , and a subsequent projection into an analog medium .such an argument actually bypasses the initial coordinate transformation because it implies that there exist curved vacuum manifolds that possess all possible variations of light propagation , and it is merely a matter of identifying the desired vacuum manifold and projecting to an analog .but we have just shown that analog space - times suffer from three serious issues that make them a poor choice as the basis for to .  1 .there is no canonical choice of projection map , so any formulation of to based on this approach is not unique . 2 .analog space - times are not covariant under coordinate transformations , while any useful formulation of to should be covariant .most concerningly , analog space - times do not fully replicate the behavior of light propagation in the vacuum .thus any _ transformation medium _ obtained as the analog of a space - time with the desired light propagation would simply fail to work as expected .  .an initial distribution of electromagnetic fields on are transformed by the pullback of , and only have support in . ]  instead , it has been shown that a more consistent and fully covariant formulation of to can be obtained through what has been termed a _ corporeal transformation _ as depicted in fig .[ fig : to ] .the key difference is that we begin and end in the same space - time manifold rather than projecting from one manifold to a different manifold .the dielectric medium can be solved for in exactly the same manner as finding an analog space - time , but eq . ( [ eq : chi - general ] ) becomes instead where the only change is that everything refers to the same coordinate system on the same manifold .physically this makes more sense . for all intents and purposes we can not engineer the space - time ; instead , the space - time remains fixed and we simply introduce the presence of dielectric media to manipulate the fields , an operation described by the corporeal transformation .the corporeal transformation is also not a coordinate transformation since the introduction of media should not have anything to do with coordinates ; thus this basis for to is fully covariant under coordinate transformations .this formulation resolves the three major issues enumerated above .first , since there is no additional projection map involved , the transformation medium is unique for a given corporeal transformation .second , this formulation of to is fully covariant under coordinate transformations , as depicted in fig .[ fig : tocommdiagram ].   in the vacuum configuration corresponds to the same coordinate transformation in the transformation media .thus to is fully covariant under coordinate transformations . ]  in particular , given a coordinate transformation with jacobian matrix in the vacuum , the corporeal transformation , metric , and hodge dual can all be reexpressed in the new coordinates as , , and , and it follows that so unlike the case for analog space - times , a coordinate change in the initial to vacuum corresponds to the same coordinate change in the transformation medium .third , since to only refers to a single space - time there is no question about which metric kinematical aspects of light propagation are referred to . the kinematics of a congruence in the transformation medium follows exactly from the corporeal transformation and is not compromised by an intermediary step in a nonphysical curved space - time .two final remarks about the formulation of to discussed here are that the initial configuration need not be vacuum , and that it allows one to do to in curved space - times , not just in minkowski space - time .by examining the kinematical properties of dielectric analog space - times , we have shown that the projection of a curved space - time into a dielectric analog residing in a space - time of different curvature , such as minkowski space - time , is limited in its ability to faithfully represent all aspects of light propagation in .this is reminiscent of the challenge faced by cartographers when representing a curved surface on a flat sheet , and follows directly from gauss theorema egregium which forbids distortion - free mappings between nonisometric spaces .although such mappings are well defined , with tensorial field quantities mapped with well - defined differential geometry operations , analog space - times are found to be noncovariant under coordinate transformation in the sense that the analog model is coordinate dependent .two analogs representing two different coordinate descriptions of the same space - time are not necessarily the same physical device described in concomitantly related coordinates .relying solely on the plebanski model of coordinate identification projection would result in two completely different physical devices representing the same space - time .this helps to highlight the fact that there is no canonical choice of projection map ; an infinite number of projections are possible , and which projection one chooses to employ must be guided by which features of vacuum light propagation one wishes to exhibit in the analog .this is nicely illustrated by looking at an analog of flrw in different guises .it was found that plebanski s model of flrw could reproduce some notion of the particle horizon and redshift , but completely failed to adequately represent the kinematical properties of a congruence of light rays , in particular the expansion of a transverse cross section of the congruence .it is possible to construct a projection map such that the expansion is preserved for a given choice of congruence at the expense of distortions in the optical metric and coordinate description of light rays , but not for all congruences simultaneously .last , the lack of canonical choice of projection , issues with covariance under coordinate transformations , and the simple fact that the analog does not fully represent light propagation in the vacuum , all cast doubt on the viability of the dielectric analog space - times approach as a basis for transformation optics . instead, these issues can be simultaneously resolved by simply assuming that to is formulated entirely as corporeal transformations on a single manifold .such transformations do not touch the metric or the coordinates and only act to actively modify fields ; these have a straightforward interpretation as the insertion of a field - modifying medium .45ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] link:\doibase 10.1007/bf02345020 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.46.1351 [ * * ,  ( ) ] link:\doibase 10.1103/physrevd.51.2827 [ * * , ( ) ] link:\doibase 10.1103/physrevlett.106.021302 [ * * , ( ) ] _ _ ( , ) * * ,  ( ) link:\doibase 10.1103/physrev.118.1396 [ * * ,  ( ) ] link:\doibase 10.1007/bf00758153 [ * * , ( ) ] link:\doibase 10.1109/tap.1962.1137809 [ * * ,  ( ) ] link:\doibase 10.1121/1.388724 [ * * ,  ( ) ] link:\doibase 10.1103/physrevlett.76.4773 [ * * ,  ( ) ] link:\doibase 10.1103/physrevlett.76.2480 [ * * ,  ( ) ] link:\doibase 10.1063/1.365625 [ * * , ( ) ] link:\doibase 10.1103/physrevd.62.044044 [ * * ,  ( ) ] link:\doibase 10.1103/physrevlett.88.061101 [ * * ,  ( ) ] link:\doibase  10.1103/physrevlett.99.183901 [ * * ,  ( ) ] link:\doibase 10.1063/1.3184594 [ * * , ( ) ] link:\doibase 10.1103/physrevd.82.124021 [ * * ,  ( ) ] link:\doibase 10.1364/oe.18.015183 [ * * ,  ( ) ] link:\doibase  10.1063/1.3485819 [ * * ,  ( ) ] \doibase doi : 10.1016/j.physleta.2010.03.061 [ * * ,  ( ) ] \doibase  doi : 10.1016/j.physletb.2011.01.016 [ * * ,  ( ) ] link:\doibase 10.1088/2040 - 8978/13/2/024004 [ * * ,  ( ) ] link:\doibase 10.1103/physrevb.85.235122 [ * * ,  ( ) ] link:\doibase 10.1007/s10714 - 013 - 1644 - 4 [ * * ,  ( ) ] link:\doibase 10.1103/physrevd.91.124035 [ * * ,  ( ) ] \doibase  http://dx.doi.org/10.1016/j.physleta.2015.10.043 [ * * ,  ( ) ] link:\doibase 10.1126/science.1125907 [ * * ,  ( ) ] link:\doibase 10.1126/science.1126493 [ * * ,  ( ) ] link:\doibase 10.1088/1367 - 2630/8/10/247 [ * * ,  ( ) ] \doibase doi :  10.1016/j.photonics.2007.07.013 [ * * ,  ( ) ] _ _ ( , , ) _ _ ( , , ) _ _ ( , , ) in link:\doibase 10.1016/b978 - 0 - 12 - 394297 - 5.00005 - 2 [ _ _ ] , vol . , ( , ) , p.  link:\doibase 10.1088/2040 - 8978/13/2/024008 [ * * ,  ( ) ] link:\doibase 10.1088/2040 - 8978/13/5/055105 [ * * , ( ) ] http://dx.doi.org/10.1017/cbo9780511809033[__ ] ( , ) . link:\doibase 10.1088/2040 - 8978/13/2/024007 [ * * , ( ) ] http://dx.doi.org/10.1017/cbo9780511606601[__ ] ( , ) . http://dx.doi.org/10.1007/978-3-319-19240-6[__ ] ( , ) link:\doibase 10.1103/physreva.92.013834 [ * * , ( ) ]link:\doibase 10.1103/physreva.82.053801 [ * * ,  ( ) ] link:\doibase 10.1088/2040 - 8978/14/1/015102 [ * * ,  ( ) ]

Abstract: it is commonly assumed that if the optical metric of a dielectric medium is identical to the metric of a vacuum space - time then light propagation through the dielectric mimics light propagation in the vacuum . however , just as the curved surface of the earth can not be mapped into a flat plane without distortion of some surface features , so too is it impossible to project the behavior of light from the vacuum into a dielectric analog residing in minkowski space - time without introducing distortions . we study the covariance properties of dielectric analog space - times and the kinematics of a congruence of light in the analog , and show how certain features can be faithfully emulated in the analog depending on the choice of projection , but that not all features can be simultaneously emulated without distortion . these findings indicate conceptual weaknesses in the idea of using analog space - times as a basis for transformation optics , and we show that a certain formulation of transformation optics closely related to analog space - times resolves these issues .  _ keywords _ : artificial black hole , transformation optics , metamaterials


in pharmaceutical fields as well as many others , there is great interest in conducting randomized trials with designs that can enable combining external information , such as prior information or expert opinions , with trial data to enhance the interpretation of the findings . inearly landmark works and parmar , spiegelhalter and freedman ( ) provided an interesting illustration of integrating expert opinions with data from cancer trials using a bayesian framework .parmar , spiegelhalter and freedman ( ) noted that the added flexibility to such trials to stop for efficacy , futility or safety can greatly increase the efficiency of clinical research .designs for incorporating external information are also useful in drug development when a pilot study , also known as a _ hypothesis generating study _ ,is conducted with a sample size that may be inadequate for detecting clinically meaningful treatment effects . in this case , relevant information including trial results and expert opinions can be used to help decision makers with whether to proceed with a larger confirmatory study and , if so , how to design it .although the applications have drawn increasing interest in recent years , little attention has been devoted to the special yet commonly seen clinical trials with binary outcomes , partly due to an inaccurate common belief that little is new regarding the trials with binary outcomes . in this paper , motivated by a case study of a clinical trial with binary outcomes in a migraine therapy ,we develop and compare statistical methods which can effectively combine information from clinical trials of binary outcomes with information from surveys of expert opinions .the results show that clinical trials with binary outcomes can behave quite differently .thus , special care is warranted for such trials .the bayesian paradigm has played a dominant role in combining expert opinions with clinical trial data .almost all methods currently used are bayesian ; see , for example , , spiegelhalter , abrams and myles ( ) , and . in the bayesian paradigm , as illustrated in spiegelhalter , freedman and parmar ( ) , a prior distribution is first formed to express the initial beliefs concerning the parameter of interest based on either some objective evidence or some subjective judgment or a combination of the two .subsequently , clinical trial evidence is summarized by a likelihood function , and a posterior distribution is then formed by combining the prior distribution with the likelihood function .although this general bayesian paradigm also applies to the special case of clinical trials with binary outcomes , the simple univariate bayesian approach developed in spiegelhalter , freedman and parmar ( ) for clinical trials with normally distributed outcomes can not be applied directly to clinical trials with binary outcomes .the latter point is contrary to the common belief which we elaborate below.=1  _ univariate bayesian approach_.  consider a clinical trial of binary outcomes with a treatment group and a control group .denote by  , the responses from the treatment group and by , the responses from the control group .assume that the parameter of interest is the difference of the success rates between the two treatments , , and its prior distribution is formed based on some objective evidence and/or some subjective judgment .let and , where and .note that is an estimator of a popular univariate bayesian approach , as seen in spiegelhalter , freedman and parmar [ ( ) , pages 360361 ] , would then treat as the `` likelihood function '' of , and proceed to compute the posterior distribution of , , according to   when the prior is modeled as a normal distribution , the approach involves an explicit posterior and is straightforward .although this univariate bayesian approach has been used in practice , it has in fact a theoretical flaw . strictly speaking , ( [ eqlikeuni ] ) is not a likelihood function of , even in the context of estimated likelihood [ see , e.g. , ] .in particular , in the clinical trial motioned above , a conditional density function solely depending on a single parameter does not exist , and , thus , it is not possible to find a `` marginal likelihood '' of .therefore , is not well defined and any univariate bayesian approach focusing directly on is not supported by the bayesian theory .  the point above is alluded to in the argument made by and that a bayesian approach is not good for `` division of labor '' in the sense that `` statistical problems need to be solved as one coherent whole in a bayesian approach , '' including `` assigning priors and conducting analyses with nuisance parameters . ''this observation suggests that a sound bayesian solution in the current context is a full bayesian model that can jointly model and or their reparametrizations .joseph , du berger and belisle ( ) presented such a full bayesian approach using ( mostly ) a set of independent beta priors for and  . however , the paper focused mainly on the utility of the approach in sample size determination rather than on its general performance in the context of clinical trials with binary outcomes . in the present paper , in addition to the independent beta priors , we broaden the scope of the full bayesian approach to include three more flexible priors , namely , independent hierarchical beta priors , dependent bivariate beta ( bibeta ) priors [ ] and dependent hierarchical bivariate beta priors .we also develop a markov chain monte carlo ( mcmc ) algorithm for implementing these full bayesian approaches , since most resulting posteriors do not assume explicit forms .the full bayesian approaches are theoretically sound , and intuitively would have been expected to provide a systematic solution to the problems in our case study . however , a close examination of the situation with skewed priors reveals a surprising phenomenon in which the estimate derived from the posterior distribution may not be between those from the prior distribution and the likelihood function of the observed data ( details in section [ sec42 ] ) .we shall refer to this phenomenon as the `` discrepant posterior phenomenon . '' to the best of our knowledge , this discrepant posterior phenomenon has not been reported elsewhere .this observation indicates that clinical trials of binary outcomes can behave differently from the normal clinical trails studied in spiegelhalter , freedman and parmar ( ) .it also shows an inherent difficulty in the modeling of trials with binary outcomes , especially if and are potentially correlated .this discrepant posterior phenomenon manifests itself in settings beyond binary outcomes , and it has far reaching implications in bayesian applications in general , as we discuss in section [ sec5 ] .in addition to studying the full bayesian approach , we also propose a new frequentist approach for combining external information with clinical trial data . and argued that a frequentist approach has `` the edge of division of labor '' over a bayesian approach .they illustrated this point by using the example of population quantile , which can be directly estimated in a frequentist setting by its corresponding sample quantile without any modeling effort or involving other ( nuisance ) parameters . in our context , this indicates that we can use a univariate frequentist approach to model directly the parameter of interest  , without having to model jointly the treatment effects . on the other hand , it is clear that a standard frequentist approach is not equipped to deal with external information such as expert opinions , which are not actual observed data from the clinical trials . to overcome this difficulty, we take advantage of the _ confidence distribution _ ( cd ) , which uses a sample - dependent distribution function to estimate a parameter of interest [ see , e.g. , and ] .in particular , we use a cd to summarize external information or expert opinions , and then combine it with the estimates from the clinical trial .this alternative scheme can be viewed as a compromise between the bayesian and frequentist paradigms .it is a frequentist approach , since the parameter is treated as a fixed value and not a random entity .it nonetheless also has a bayesian flavor , since the prior expert opinions represent only the relative experience or prior knowledge of the experts but not any actual observed data .the cd approach is easy to implement and can be a useful data analysis tool for the type of studies considered in the present paper .the main emphasis of the paper is on the study and comparison of the methods for incorporating expert opinions with clinical trial data in the binary outcome setting , and not the methods for pooling together individual expert opinions .the latter have been discussed extensively by .the goal of this research is to raise awareness of the complexity of the practice of incorporating external information .although it draws attention to a difference between bayesian and non - bayesian approaches in practice , it is not meant to either promote or criticize any of the bayesian or frequentist approaches .the rest of this section describes a pilot clinical study in a migraine therapy by _ johnson and johnson , inc . _ in section [ sec2 ] we develop full bayesian approaches with four different priors and implement the approaches through an mcmc algorithm . in section [ sec3 ]we present the alternative approach of frequentist bayes compromise using cds . in section [ sec4 ]we illustrate the approaches discussed in sections [ sec2 ] and [ sec3 ] using the data presented in section [ sec11 ] .we also conduct a simulation study to compare the performance of these approaches in situations where the prior distributions are skewed . finally , we provide in section [ sec5 ] some concluding remarks and discussions .our data are collected from a recent clinical study on patients with migraine headaches .the objective was to determine the potential impact of a preventive migraine therapy , topiramate , on the therapeutic efficacy of the acute migraine therapy , almotriptan .the study consisted of a 6-week open - label phase followed by a randomized double - blind phase that lasted 20 weeks .patients received topiramate during the open - label run - in period that enabled the selection for randomization of patients who could tolerate a dosing regimen of 100 mg and who met the eligibility criteria based on migraine rate .those found eligible were randomly assigned to receive topiramate ( treatment a ) or placebo ( treatment b ) . throughout the study ,almotriptan 12.5 mg was used as an acute treatment for symptomatic relief of migraine headaches .the patients recorded assessments of migraine activity , associated symptoms and other relevant details into an electronic daily diary ( personal digital assistant [ pda ] ) .the numbers of patients in the treatment and the control groups are and , respectively .the slight difference in the group size reflects the dropout of a handful of patients during the double - blind phase .the most common reason for these dropouts was subject choice / withdrawal of consent .few patients discontinued treatment due to limiting adverse event during the double - blind phase ..2d1.2d2.2d2.2d2.2d2.2d2.2d2.2d2.2d1.2@ & + & + & & + & & + * investigator * & & & & & & & & & & & & + 1 & & & 5 & 5 & 10 & 30 & 30 & 15 & 5 & & & + 2 & & & & 3 & 7 & 20 & 25 & 20 & 15 & 7 & 3 & + 3 & & & & & 10 & 15 & 20 & 20 & 20 & 10 & 5 & + 4 & & & 2 & 3 & 5 & 5 & 50 & 20 & 10 & 5 & & + 5 & & & & 5 & 10 & 15 & 15 & 30 & 20 & 5 & & + 6 & & & & & & 10 & 20 & 30 & 30 & 10 & & + 7 & & & & & 5 & 20 & 50 & 20 & 5 & & & + 8 & & & & & & 5 & 50 & 40 & 5 & & & + 9 & & & & & 5 & 5 & 20 & 30 & 20 & 10 & 10 & + 10 & & & & & 5 & 15 & 15 & 20 & 15 & 15 & 10 & 5 + 11 & & & & & & 5 & 10 & 30 & 25 & 15 & 10 & 5 + group mean & & & 0.64 & 1.45 & 5.18 & 13.18 & 27.73 & 25.00 & 15.46 & 7.00 & 3.45 & 0.91 +  the trial objectives and study design were presented at an investigator meeting prior to the start of the study . at the meeting , following the design of parmar , spiegelhalter and freedman ( ) , the study sponsor sought the individual opinions of each investigator expert regarding the expected improvement of treatment a over treatment b for a series of clinical outcomes . for illustrationwe focus on a specific outcome , pain relief at two hours ( pr2 ) after dosing with almotriptan , one of 16 outcomes investigated in the trial . during the meeting ,each expert was asked to use the 12 intervals shown in table [ tab1 ] to assign a `` weight of belief , '' based on his / her experience , in the difference in the percentage of patients expected to achieve pr2 in the two treatment groups . in other words ,each expert was given 100 `` virtual patients '' to be assigned to one of the 12 possible intervals of difference between the two treatments ( from to ) in table [ tab1 ] .table [ tab1 ] shows the belief distributions for each of the 11 experts and the group mean .the histogram in figure [ fig1 ] shows the group means of the 11 experts beliefs of the improvement of treatment a over treatment b.  the histogram in figure [ fig1 ] , derived from the arithmetic means in the last row of table [ tab1 ] , is to be used as a ( marginal ) prior in our bayesian analysis for the improvement of treatment a over treatment b. this practice effectively assumes that the heterogeneity of expert opinions could be averaged out by arithmetic means [ cf .a similar assumption is also used in spiegelhalter , freedman and parmar ( ) and the development of the frequentist approach in section [ sec3 ] .further discussion on heterogeneity among experts can be found in section [ sec5 ] .the goal of our project is to incorporate the information in figure [ fig1 ] , solicited from experts , with the data from the pilot clinical trial , and make inference about the improvement of the treatment effect .findings from the inference are intended for generating hypotheses to be tested in future studies .beta distributions are often conventional choices for modeling the prior of the bernoulli parameter or .they are sufficiently flexible for capturing distributions of different shapes . in particular , we consider four forms of joint beta distributions for the prior of .  * _ independent beta prior_. joseph , du berger and belisle ( ) used independent beta priors to summarize `` pre - experimental information '' about `` two independent binomial parameters '' and as follows : with and . here , are unknown prior parameters ( hyperparameters ) which can be estimated using the method of moments , following and joseph , du berger and belisle ( ) .specifically , for our clinical study , the average treatment effect and its standard deviation , and , can be obtained ( estimated ) from the mean and standard deviation of the histogram of figure [ fig1 ] .based on previous clinical trials [ cf . and ] , the average effectiveness of treatment b and its standard deviation can also be obtained .we can estimate the prior parameters by solving the equations , , and  . *_ independent hierarchical beta prior . _ gelman et al . [ ( ) , chapter 5 ] suggested that hierarchical priors are more flexible and can avoid `` problems of over - fitting '' in bayesian models .we modify their approach to reflect the informative prior in our problem setting with two sets of independent bernoulli experiments .specifically , we still model the prior of independently with , but each , for , assumes two levels of hierarchies as follows : here , is the mean and is the `` sample size '' of , following , and refers to the standard gamma distribution whose shape parameter is and scale parameter is 1 .again , we use the method of moments to estimate the unknown parameters in this ( hyper ) prior distribution . in this case , the first two marginal moments of are and  ] is a confidence distribution ( cd ) for a parameter , if it meets the following two requirements : ( r1 ) for each given , is a continuous cumulative distribution function ; ( r2 ) at the true parameter value , , as a function of the sample , follows the uniform distribution  ] requirement holds only asymptotically , and the continuity requirement on is dropped .also , when it exists , is called a cd density or confidence density .the cd is a function of both the parameter and the random sample .it is also a sample - dependent distribution function on the parameter space , following requirement _r1_. conceptually , it estimates the parameter by a distribution function . as an estimation instrument , it is not much different from a point estimator or a confidence interval .for example , for point estimation , any single point ( a real value or a statistic ) can , in principle , be an estimate for a parameter , and we often impose additional restrictions to ensure that the point estimator has certain desired properties , such as unbiasedness , consistency , etc .the two requirements in definition [ defa ] play roles similar to those restrictions . specifically , _ r1 _ suggests that a sample - dependent distribution function on the parameter space can potentially be used as an estimate for the parameter .the  ] random variables , and is a continuous function from \times[0,1]$ ] to which is monotonic ( say , increasing ) in each coordinate . then, is a combined cd for which contains information from both expert opinions and the clinical trial .one simple choice is with weights and , where is an estimate of the standard deviation of ( specifically , is the standard deviation of the histogram in figure [ fig1 ] in our application , and it is also an estimate of the standard deviation of in the normal case ) . in this case , can be expressed as , and thus gives rise to the following combined cd for :  .  when both and are normal ( or asymptotically normal ) cds , the normal combination in ( [ eqg - weight ] ) is the most efficient in terms of preserving fisher information . in nonnormal cases , singh , xie and strawderman ( ) studied several choices of the function and their bahadur efficiency .but it remains an open question what choice of is most efficient in preserving fisher information in a general nonnormal setting .although we use the simple normal combination in ( [ eqg - weight ] ) in this paper mostly for simplicity , our experience with the numerical studies has shown this combination to be quite adequate in most applications .in fact , in many nonnormal cases , it incurs very little loss of efficiency in terms of preserving fisher information from both and .  in a bayesian approach ,it is a conventional practice to fit a prior density curve to the histogram in figure [ fig1 ] .although this step is not needed in the proposed cd approach , we may sometimes also fit a density function to .for example , we may fit a normal curve to the histogram of figure [ fig1 ] by matching its first two moments , say , mean and variance . in this case, we have a normal cd from the expert opinions . incorporating it with in ( [ eqhtnormal ] ), we have the combined cd or where and this combined cd turns out to be the same as the posterior distribution function obtained from the univariate bayesian approach described in the when the normal prior is used . because bayes s formula requires that we know in the univariate bayesian approach and this conditional density function does not exist in our clinical setting , we have argued that the univariate bayesian approach is not supported by bayesian theory .the cd development , interestingly , provides theoretical support for using the posterior distribution from a non - bayesian point of view if the prior distribution of expert opinions can be approximated by a normal distribution . in this case, the univariate bayesian approach can also produce a result that `` makes sense , '' and practically we can use either the cd approach or the univariate bayesian approach . but this statement is not true in general .we now provide numerical studies to illustrate and compare the bayesian and cd approaches discussed in sections [ sec2 ] and [ sec3 ] . in section [ sec41 ] we focus on the data from the migraine pain study outlined in section [ sec11 ] . in section [ sec42 ]we simulate a skewed distribution of expert opinions and combine the simulated prior information with the clinical trial data .   for the outcome pr2 , the clinical data report that 31 out of 68 patients in the control group and 33 out of 59 patients in the treatment group achieved pain relief at 2 hours .our goal is to incorporate the expert inputs reported in figure [ fig1 ] with these observed outcomes .we apply the full bayesian approach described in section [ sec2 ] to analyze the pr2 data , using each of the four beta priors for .the numerical results are reported in figures [ fig2](a)(d ) and also in the first 12 rows of table [ tab2 ] .the dotted lines in figures [ fig2](a)(d ) indicate the marginal prior density functions , and the dashed lines indicate the ( standardized ) profile likelihood function of based only on the clinical trial data .the solid lines in figures [ fig2](a)(d ) are the marginal posterior distributions of .they are obtained by using the density estimation function _density_( ) in the r software and from 1000 metropolis hasting samples of . in each case and for each of the 1000 replications , the metropolis hastings algorithm is iterated times ( burn - in ) .the acceptance rates are on average , , and , respectively , in ( a ) to ( d ) . for the independent betaprior , the exact formula of the posterior distribution is available , and it is plotted in figure [ fig2](a ) as the dash - dot broken curve ( it is barely visible in the plot , since it is almost identical to the solid curve ) .the close agreement of these two curves for the posterior distribution indicates that the mcmc chain of the metropolis hastings algorithm has generally converged with in this case .in applying the cd approach , we use both the raw histogram in figure [ fig1 ] and the distribution to approximate the prior cd of expert opinions . figures [ fig2](e)(f ) and the last six rows of table [ tab2 ] contain numerical results .the dotted lines in figures [ fig2](e)(f ) indicate the prior cds , the dashed lines indicate the profile likelihood function of based only on the clinical trial data , and the solid lines are for the combined cds for .in this particular example , all six approaches ( four bayesian and two cd approaches ) seem to yield similar posterior or combined cd functions , and thus similar statistical inferences , regardless of which approach is used .although the six marginal posterior or combined cd distributions are slightly different from one another , the difference appears to all fall within the expected estimation error of the density curves .this result is not surprising , since , although skewed , the degree of skewness of the histogram in figure [ fig1 ] does not appear to be great enough to render the normal approximation invalid .in fact , in this case , such a result is expected to hold if the central limit theory is in place for the clinical data of binary outcomes .it is worth noting here that the bayesian approach implemented through an mcmc method is more demanding computationally .the outcome in the previous subsection begs the question of whether there would be a significant difference among the approaches if the distribution of expert opinions were unambiguously skewed , so that the normal approximation is clearly not valid .conventional wisdom suggests that full bayesian approaches based on beta priors , though computationally more intensive , would have advantages due to their flexibility in capturing distributions of various shapes . the cd approaches , allowing skewed priors , may also work .however , the numerical results reveal a surprising finding in the full bayesian approaches .  in this simulation study, we again use the observed clinical data on pr2 , but replace table [ tab1 ] and figure [ fig1 ] of expert opinions with their simulated counterparts , assuming that the underlying prior distribution function of is a bivariate beta distribution , .the marginal means of the distribution are and .thus , the simulated prior represents a treatment effect improvement on average about 75% to 90% , which are similar to those of the real trial in section [ sec41 ] .specifically , we simulate responses of 100 patients for each of the 11 experts from , tally the results in the format of table [ tab1 ] ( not shown ) , and then plot them as a histogram in figure [ fig3 ] . for a direct visual comparison , figure [ fig3 ] includes the curve of the density . also plotted in figure [ fig3 ]is , as a common approach to fitting a skewed distribution , the following fitted log - normal density . here , and are the mean and the standard deviation computed from the histogram , and is a constant used to capture the shift of the log - normal distribution from .   using distribution . ]we apply the same four full bayesian approaches used in section [ sec41 ] to incorporate the simulated expert opinions represented in figure [ fig3 ] with the clinical trial data on pr2 .the four sets of prior parameters used in these four approaches are , , and  , respectively . in the third approach, we directly use the true set of prior parameters ; in the other three , the prior parameters are obtained by the method of moments outlined in section [ sec2 ] . in the simulation example , the metropolis hasting algorithm is again iterated times ( burn - in ) , and it is repeated 1000 times to obtain 1000 independent metropolis hasting samples of in each case .the acceptance rates are on average , , and , respectively .we also apply two cd combination approaches to incorporate the simulated expert opinions in figure [ fig3 ] with the clinical trial data on pr2 .similar to that in section [ sec4 ] , the first cd approach directly uses the raw histogram in figure [ fig3 ] .the second cd approach , in order to have a direct comparison with the bayesian approach using the underlying prior , combines the underlying marginal density function of with the cd from the clinical trial data . of course ,in reality we do not know the underlying prior distribution or the underlying marginal density function of .thus , the second cd approach has only theoretical value . without relying on the underlying cd prior, we also consider the cd approach which combines the fitted log - normal distribution in figure [ fig3 ] with the cd from the clinical trial data .however , since the log - normal curve is evidently a poor fit for the histogram in figure [ fig3 ] , the result for this cd approach , though not too far off , does not seem well justified and is thus omitted .figures [ fig4](a)(d ) show the results on the improvement using the full bayesian approaches , and figures [ fig4](e)(f ) show the results using the cd combination approaches .figure [ fig4 ] adopts the same notation and symbols used in figure [ fig2 ] .again , for the independent beta prior , the posterior density from the algorithm closely matches the one using its exact formula ( dashed - dotted line ) , indicating that the mcmc chain of the metropolis hasting algorithm has generally converged in this case . also , we report in table [ tab3 ] the numerical results from the six approaches : the mode , median , mean and confidence / credible intervals of the marginal priors , the profile likelihood function and the marginal posteriors of .  the cd approaches perform exactly as anticipated .however , examining the modes of the three curves in each of figures [ fig4](a)(d ) , we notice that the mode of the marginal posterior distribution ( solid curve ) lies to the right of the peaks of both the marginal prior distribution ( dotted curve ) and the profile likelihood function ( dashed curve ) .the numerical results in table [ tab3 ] also confirm that the mode , median and mean of the marginal posterior distributions of from all four full bayesian approaches are much larger than their counterparts from the corresponding marginal priors and profile likelihood functions .this discrepant posterior phenomenon is counterintuitive !for instance , if we use the means as our point estimators , we would report from figure [ fig4](c ) that the experts suggest about improvement and the clinical evidence suggests about improvement but , incorporating them together , the overall estimator of the treatment effect is , which is bigger than either that reported by the experts or that suggested by the clinical data .this result is certainly not easy to explain to clinicians or general practitioners of statistics . in any event, it seems worthwhile to investigate further and see what ramifications this intriguing phenomenon may have.=-1  to further examine the phenomenon , we compare the percentiles of the marginal priors , the profile likelihood function and the marginal posterior distributions of the treatment effect in table [ tab3 ] . in each of the four bayesian approaches, the 95% posterior credible interval lies inside the corresponding 95% interval from the prior and has substantial overlap with the corresponding 95% interval from the profile likelihood .but this is not always the case at the 80% and 90% levels , where several posterior credible intervals do not lie within the boundaries of the corresponding intervals from the priors and the likelihood functions .the outcome of whether the posterior credible interval lies within the boundaries of the other two depends on the choice of the credible level .thus , using credible intervals as our primary inferential instrument can not completely avoid the discrepant posterior phenomenon either.=-1  to better understand this phenomenon , we plot in figures [ fig5](a)(d ) the contours of the joint prior distribution , the likelihood function of and the joint posterior distribution of for each of the four full bayesian approaches .we show that certain projections of figures [ fig5](a)(d ) lead to the marginal distributions and plots in figures [ fig4](a)(d ) .as marked in figure [ fig5 ] , the center ( mode ) of each contour plot is on a line ( or ) .varying in produces a family of parallel lines , all making a angle with the horizontal axis .the projections of the three distributions along these parallel lines onto the interval of possible values of , , lead to the plots of marginal distributions in figures [ fig4](a)(d ) .the yellow curves in ( a ) are a posterior contour plot from the exact formula . although the contour plots of the posterior distributions sit between those of the prior distributions and the likelihood function , their projected peaks ( modes ) are more to the upper - left than those of the marginal priors and the profile likelihood function .further investigation indicates that this is a genuine mathematical phenomenon which holds for all four bayesian approaches and not merely an aberration due to some special circumstances .in fact , when the center ( mode ) of a posterior distribution is not in the interval joining the two centers ( modes ) of the joint prior and likelihood functions , as is often the case with skewed distributions ( and even sometimes with nonskewed distributions ) , there always exists a linear direction , say , with some coefficients a and b , along which the marginal posterior fails to fall between the marginal prior and likelihood functions of the same parameter .reparametrization , if done carefully , such as considering joint distribution of or others , may sometimes help hide the discrepant posterior phenomenon on the direction , but can not eliminate it systematically .we have found no discussion of such a geometric finding on marginalization in the bayesian literature .see further discussion in section [ sec5].=-1  .these two - dimensional distributions are projected along the lines of onto the interval of possible values of , , leading to figures [ fig4](a)(d ) .the yellow curves in ( a ) are a posterior contour plot from the exact formula . ]to incorporate expert opinions in the analysis of a clinical trial with binary outcomes in a meaningful way , we have developed and studied several bivariate full bayesian approaches as well as a cd approach .we show that both the bayesian and the proposed cd approaches may provide viable solutions . although the paper focuses on expert opinions in pharmaceutical studies , the methodologies developed here can be applied to incorporating other types of priors or external information , for example , historical knowledge .these methodologies should also be useful in many other fields , including finance , social science studies and even homeland security , where prior knowledge , expert opinions and historical information are much valued and need to be incorporated with observed data in an effective and justifiable manner .  in this paper we have examined and compared both bayesian and cd approaches . although there does not exist the usual theoretical platform for a direct comparison on efficiency or lengths of intervals , the comparison can be summarized in three aspects : empirical results , computational effort and theoretical consideration .the empirical findings from figure [ fig2 ] show that , as long as the histogram of the expert opinions can be well approximated by a normal distribution , all approaches considered in this paper perform comparably , in terms of the posterior distribution or the combined cd and their corresponding inferences .however , if the histogram is skewed , the full bayesian approach may produce the discrepant posterior phenomenon , which is difficult to avoid in theory and difficult to explain in applications .the cd approach avoids such a phenomenon .  in terms of the computational effort ,the bivariate full bayesian approach is demanding since it requires running a large - scale simulation using an mcmc algorithm , while the proposed cd approach is both explicit and straightforward to compute .in addition , the cd approach can directly incorporate the histogram of expert opinions without an additional effort of curve fitting .theoretically , since it is not possible to find a `` marginal '' likelihood of [ i.e. , a conditional density function , any univariate bayesian approach focusing on the parameter of interest is not supported by bayesian theory .a full bayesian solution is to jointly model [ or a reparameterization of the pair ] and , subsequently , make inferences using the marginal posterior of .the full bayesian approaches developed in the paper follow exactly this procedure and are theoretically sound .the proposed cd approach is developed strictly under the frequentist paradigm and is also theoretically sound .unlike the full bayesian approaches , the cd approach can focus directly on the parameter of interest without the additional burden of modeling other parameters or the correlation between and , and thus appears to have some advantage in this application .a surprising finding in this research is the discrepant posterior phenomenon occurring in the full bayesian approaches under skewed priors .although it may be mitigated if the prior is only slightly skewed or is in accordance with the likelihood function , the phenomenon is intrinsically mathematical . how much skewness is required to produce the phenomenon depends on all elements involved , including shapes and locations of both the likelihood and the prior .the reactions to this phenomenon we have encountered thus far fall roughly into two groups .one group views the discrepant posterior as a mathematical truth and , if one has faith in the choice of the prior , one should proceed to make inference using this marginal posterior , even though the outcome is counterintuitive .the other worries about the counterintuitive result and would try to find alternative approaches of a good operating characteristic for the particular problem at hand , even at the cost of abandoning the mathematically solid full bayesian approach in favor of less rigorous approaches such as the univariate bayesian approach described in section [ sec1 ] . in any case, the lesson learned from the bayesian analysis here is that the choice of the prior really matters and it needs to be in some agreement with the likelihood function , which is similar in spirit to what was referred to as `` model dependent '' in .we also consider this a manifestation of an inherent difficulty in modeling accurately the joint effects of the two treatments as reflected in and and their correlation .this difficulty illustrates again the complexity of the practice of incorporating external information in trials with binary outcomes .the discrepant posterior phenomenon is caused by `` marginalization , '' but it is different from the `` marginalization paradox '' discussed in and . in particular , the marginalization paradox in dawid , stone and zidek ( ) refers to the phenomenon that the marginal posterior of obtained from the joint prior and its full model can sometimes be quite different ( `` incoherent '' ) from the posterior obtained by applying the bayes formula directly to its marginal prior and marginal model are consistent ( `` coherent '' ) with the joint prior and the full model .here , represents nuisance parameters .this paradox is different from what we observed here . in our example , it is not possible to have the marginal model , and the discrepant posterior phenomenon in the full bayesian approach is that the estimate derived from the marginal posterior may not be between the estimates from the marginal prior and the profile likelihood function .this is counterintuitive in practical applications .it is worth noting that the discussion and implications of the discrepant posterior phenomenon extend beyond the setting of binary outcomes to any multivariate setting involving skewed distributions .as long as the center ( mode ) of a posterior distribution is not in the interval joining the centers ( modes ) of the joint prior and the likelihood function , there always exists a direction along which the center ( mode ) of the marginal posterior fails to fall between the centers ( modes ) of marginal prior and the profile likelihood function .this phenomenon has implications in the general practice of bayesian analysis .for instance , many researchers in machine learning and other fields routinely draw conclusions solely based on marginal posterior distributions without checking ( or it is very difficult to check ) the validity of such conclusions .the discrepant posterior phenomenon suggests that further care is needed .many methods have been introduced to model `` reported '' expert opinions , account for their potential errors and heterogeneity , and subsequently pool them ; see for an excellent review of this topic .in particular , spiegelhalter , freedman and parmar ( ) described `` arithmetic and logarithm pooling '' as the `` two simplest methods '' for pooling expert opinions , and articulated a `` strong preference '' `` for arithmetic pooling to obtain an estimated opinion of a typical participating clinician . '' the underlying assumption of arithmetic pooling is that the average of the `` observed '' expert opinions is an unbiased representation of the `` true '' prior knowledge .this assumption naturally facilitates the additive error model used in section [ sec32 ] for summarizing `` reported '' expert opinions in a cd .clearly , the modeling principle and development we used to summarize the expert opinions in a cd are similar in spirit to those discussed in and spiegelhalter , freedman and parmar ( ) for bayesian approaches .the modeling framework developed in section [ sec32 ] is sufficiently flexible and can be modified to accommodate various ways of aggregating expert opinions .in particular , it can incorporate weighting schemes to develop a robust method against extreme expert opinions , introduce additional terms to reflect biased opinions or additional uncertainties , or use the geometric mean as a way to pool the expert opinions .some of these extensions ( e.g. , the robust method ) by themselves could be attractive choices to produce priors in the context of traditional bayesian approaches .due to space limitations , we will not pursue these extensions in this paper .  in a different direction , we have also considered modeling the survey data of expert opinions using a traditional random effects approach . in such a model , we provide a regression model for the responses of the 100 `` virtual patients '' of each expert ( as described in section [ sec11 ] ) and add a random effect term to account for the expert - to - expert variation .however , it seems nontrivial to overcome the technical difficulty in making the modeling process free of the number ( 100 ) of `` virtual patients . ''in fact , this difficulty led us to the bootstrap argument in section [ sec32 ] , in which we mimic a potential model of expert exposure to pre - existing experiments .clearly , there remain many challenging issues in modeling the survey data of expert opinions , even for the seemingly simple binary setting .this is part of a collaborative research project between the office of statistical consulting at rutgers university and ortho - mcneil janssen scientific affairs ( omjsa ) , llc .the authors thank dr .karen kafadar , the ae and two reviewers for their constructive comments and suggestions , which have greatly helped improve the presentation of the paper .the authors also thank drs .steve ascher , james berger , david draper , brad efron , xiao - li meng , kesar singh and william strawderman for their helpful discussions and ms. y. cherkas for her computing assistance on a related mcmc algorithm .

Abstract: external information , such as prior information or expert opinions , can play an important role in the design , analysis and interpretation of clinical trials . however , little attention has been devoted thus far to incorporating external information in clinical trials with binary outcomes , perhaps due to the perception that binary outcomes can be treated as normally - distributed outcomes by using normal approximations . in this paper we show that these two types of clinical trials could behave differently , and that special care is needed for the analysis of clinical trials with binary outcomes . in particular , we first examine a simple but commonly used univariate bayesian approach and observe a technical flaw . we then study the full bayesian approach using different beta priors and a new frequentist approach based on the notion of _ confidence distribution _ ( cd ) . these approaches are illustrated and compared using data from clinical studies and simulations . the full bayesian approach is theoretically sound , but surprisingly , under skewed prior distributions , the estimate derived from the marginal posterior distribution may not fall between those from the marginal prior and the likelihood of clinical trial data . this counterintuitive phenomenon , which we call the `` discrepant posterior phenomenon , '' does not occur in the cd approach . the cd approach is also computationally simpler and can be applied directly to any prior distribution , symmetric or skewed .  ,  ,