Bayesian Inference With Log-Fourier Arrival Time Models and Event Location Data

Transcription

1 Bayesan Inference Wth Log-Fourer Arrval Tme Models and Event Locaton Data Tom Loredo May 1990; Sep Nov INTRODUCTION In these notes we descrbe Bayesan calculatons wth models that supply Bayesan counterparts to frequentst perod searchng wth the Raylegh and Zn 2 statstcs, and we outlne how to analyze event arrval tme data that ncludes a locaton for each event on a detector wth a known pont spread functon. Many of these results are from my notebook of a few years ago, but I suppose nothng s true nowadays untl t s n TEX! The reader s presumed to be famlar wth the materal n 2 and 3 of Gregory & Loredo (1992; hereafter GL). In partcular, our startng pont s the arrval tme lkelhood functon gven by equaton (3.5) of that paper. For the record, ths lkelhood functon s not our nventon; t s wdely known n frequentst studes of nhomogeneous Posson processes. Before we start, let s summarze what s dfferent about the Bayesan approach to arrval tme seres analyss. There are two key dfferences between Bayesan methods and ther frequentst counterparts. Frst, Bayesan methods must assume a partcular functonal form (usually wth several free parameters) for the possble perodc sgnal. Frequentst methods nstead try to reject a constant model for the sgnal, wthout explct reference to a specfc famly of perodc models. Ths would seem to be a serous drawback to Bayesan methods, snce we usually don t know the shape of the perodc lghtcurve we are tryng to detect. However, from the Bayesan pont of vew the choce of statstc to be used n a frequentst test mplctly corresponds to a model choce. Ths s because dfferent statstcs correspond to dfferent deas about what departures from unformty one wants to be most senstve to, whch s smply another way of thnkng of a perodc model. Ths s recognzed nformally n frequentst lterature n the acknowledgment that dfferent choces of statstc are most senstve to underlyng sgnals of dfferent shapes (e.g., the χ 2 statstc can detect sgnals wth one or more narrow peaks well, whle the Raylegh statstc can detect smooth, snusodal sgnals well). It s recognzed more formally n consderatons of the power of a partcular frequentst test: ts ablty to accurately detect a perodc sgnal of specfed shape when such a sgnal s present. We have made somethng of an ndustry dentfyng partcular models for whch a Bayesan calculaton leads to consderaton of a functon of the data drectly related to a common frequentst statstc. Ths makes the mplct assumptons of a frequentst procedure explct, so that ts approprateness can be more easly judged, and modfcatons pursued. These notes offer an example of ths. The second key dstncton between Bayesan and frequentst methods s how a statstc s used to determne whether a perodc sgnal s present or not. In the frequentst approach, we evaluate the statstc at many frequences, and note the largest observed value. We then [MEMO! Not for crculaton! [vonmses.tex: 9/4/ :02

2 calculate the chance probablty for the statstc beng as large as the maxmum value or larger, assumng that the sgnal s constant; f ths probablty s small, the constant sgnal model s rejected. Ths calculaton s sgnfcantly complcated by the need to examne many perods (and possbly many values of other parameters, such as the phase assgned to t = 0), and the number and locaton of the perods must be specfed wthout consderaton of the data. These aspects of a frequentst analyss ntroduce a troublng subjectvty n the results. In contrast, the Bayesan approach requres that one ntegrate, rather than maxmze, a nonlnear functon of the test statstc over the perod range beng searched (and over any other parameters n the model). As wth the frequentst test, the range to search must be subjectvely specfed. However, the number and locaton of perods searched s rrelevant; as many perods should be used as s necessary to accurately compute the requred ntegrals. The result s penalzed for the unknown parameters automatcally and objectvely by the averagng performed by the ntegraton process. Also, the result s condtonal on the one observed data set; the frequentst probablty s the probablty, not only of the observed data, but of all worse data sets as well. Another dstncton worth mentonng s that the Bayesan approach forces us to clearly dstngush the tasks of sgnal detecton and parameter estmaton. In partcular, the Bayesan approach dentfes a nonlnear functon of the test statstc whose ntegrals gve straghtforward probablty statements about the allowed ranges ( credble regons ) of unknown parameters, such as the frequency. Okay, now that all (well, most) of the propaganda s out of the way, let s move on to the calculatons! 2. THE LIKELIHOOD FUNCTION AND MODEL CRITERIA We presume we have a parameterzed model rate functon, r(t), whch we wll use to model data consstng of N arrval tmes, t, over some observng nterval of total duraton T. We use T to desgnate both the total lve-tme duraton of the observatons, and the set of ntervals (possbly separated by gaps) that comprse the observatons, so that T stands for an ntegral over all observng ntervals (they need not be contguous). For perodc models, the parameters for r(t) wll typcally nclude an ampltude, A; a frequency, ω (or equvalently a perod, P = 2π/ω); a phase, φ; and possbly some other parameters, S, that parameterze the lghtcurve shape. These ngredents, together wth Posson assumptons, gve the arrval tme lkelhood functon n equaton (3.5) of GL. We reproduce t here, gnorng the rrelevant t factors (they do not depend on the parameters, and cancel out n Bayes s theorem): [ N L(A, ω, φ, S) = exp dt r(t) r(t ). (2.1) T =1 Bayesan calculatons requre ntegrals over parameter space. Such ntegrals arse when we normalze dstrbutons, fnd margnal dstrbutons for nterestng subsets of parameters, or compare rval models. Calculatng these ntegrals can be dffcult. In the context of searchng for perodctes n arrval tme data, any model wll have at least four parameters: A, ω, φ, and at least one shape parameter. The lkelhood functon usually vares wldly wth ω, so any ntegrals requre evaluaton of equaton (2.1) at many frequences. At each [MEMO! Not for crculaton! 2 [vonmses.tex: 9/4/ :02

3 of these frequences we have to ntegrate over the other parameters. Ths gets out of control pretty quckly! So the game we have to play n Bayesan modelng of arrval tme data s to try to fnd physcally useful models that allow at least some of the requred ntegratons to be done analytcally. In the next secton, we show that the ampltude can always be ntegrated analytcally. So our model choce should be guded by focussng on the other parameters. The GL model s cute n that t s a model wth many S parameters and thus capable of descrbng a dverse set of shapes for whch ntegrals over all of the S parameters can be done analytcally. Only the ω and φ ntegrals need to be done numercally. Here we wll take a complementary approach, dentfyng models for whch φ ntegrals can be done analytcally. We note n passng that cases where the perod s known or s well-constraned a pror (e.g. n searchng for X-ray pulsatons at a known rado pulsar perod) can be sgnfcantly more tractable than cases where we must search for an unknown perod. In the former case, t may be possble and nterestng to consder more complcated models than one could otherwse consder because the ω dmenson s elmnated, or at least small. But even n such cases t s useful to have computatonally effcent models, such as those dscussed here and n GL. 3. THE AMPLITUDE PARAMETER 3.1. Normalzed Models One parameter the ampltude s common to all models, and t turns out that we can take care of t once and for all, as we show n ths secton. We can always wrte a perodc model rate n the form, r(t) = Aρ(ωt φ), (3.1) where ρ(θ) s a perodc functon wth perod 2π. Wthout loss of generalty, we can also requre that the ampltude parameter be the average rate, A 1 P P dt r(t). (3.2) Together, these two equatons mpose a normalzaton constrant on ρ. To see ths, change varables n equaton (3.2) to θ = ωt φ, and use the fact that ω = 2π/P. Then equaton (3.2) becomes, 2π dθ ρ(θ) = 2π, (3.3) or, equvalently, 0 P dt ρ(t) = 1. (3.4) That s, ρ must be normalzed as f ρ/2π were a probablty dstrbuton n phase. We ll see n a mnute why ths s a convenent parameterzaton; roughly, t lets A control the expected number of events, and ρ descrbe the shape of the lghtcurve. (Actually, our man conclusons wll hold for any normalzng constant for ρ, as long as t does not depend on the model parameters.) [MEMO! Not for crculaton! 3 [vonmses.tex: 9/4/ :02

4 Now plug ths form nto the lkelhood functon. For a contguous nterval, T, the ntegral n the exponent of equaton (2.1) s, T dt Aρ(ωt φ) = AP ωt φ dθ ρ(θ) 0 2π φ AT. (3.5) To get the last lne, we used the fact that the ntegral dvded by the 2π factor up front s just equal to the number of perods covered by T ; multplyng by P thus gves T. It s approxmate because T may not be an ntegral number of perods long, so the last part of the ntegral wll nclude only part of one perod. The ntegral of ρ/2π over a fracton, f, of a perod s not generally equal to f, hence the approxmate result. But so long as T P, the error we make n ths fnal fractonal perod wll be small compared to the total ntegral, because the total ntegral wll contan many perods. So from now on we ll assume we are nterested n perods such that many perods are contaned n the data. Ths regme s very nce because the exponental factor n the lkelhood functon then depends only on A, and not on any other parameters. For smplcty we ve shown ths for a contguous nterval; but t also holds for noncontguous ntervals of total duraton T P, so long as the holes n the observng tme do not lne up n phase (such algnment could happen for a strong perodc sgnal observed by a detector wth bad dead tme; but we probably don t need statstcs to detect pulsatons n such data!). The lkelhood functon n the many-perod regme s then, L(A, ω, φ, S) = [A N e AT ρ(ωt φ). (3.6) Now we can see what ths parameterzaton buys us: the lkelhood factors nto a part that depends only on A and a part that depends on all the other parameters (f ρ were not normalzed, the rght hand sde of equaton (3.5) would be a functon of ω, φ, and S that would appear n the exponent n the lkelhood). Ths means that so long as our pror smlarly factors (that s, knowledge of A tells us nothng about the other parameters a pror, and vce versa), nferences about A wll be ndependent of those about ω, φ, and whatever other parameters we have, and vce versa. Snce ths s true, let s take care of the ampltude nferences now, once and for all, so that n the rest of these notes we can focus on the other parameters Inferrng the Ampltude Let s presume the pror for the parameters factors n the way just stated, so that p(a, ω, φ, S M) = p(a M) p(ω, φ, S M), (3.7) where M s the background nformaton specfyng the model, ts relaton to the data (.e. the Posson assumptons we used to derve the lkelhood functon), and anythng else we mght know about the model (e.g. constrants on the perod from other observatons). To nfer all of the parameters of a model, we use Bayes s theorem to calculate the full jont posteror for the parameters, p(a, ω, φ, S D, M) = p(a M) p(ω, φ, S M) p(d M) L(A, ω, φ, S). (3.8) [MEMO! Not for crculaton! 4 [vonmses.tex: 9/4/ :02

5 Here p(d M) s the global lkelhood for the model, gven by, p(d M) = da dω dφ ds p(a, ω, φ, S M) L(A, ω, φ, S) = da p(a M)A N e AT dω dφ ds p(ω, φ, S M) ρ(ωt φ).(3.9) It s merely a normalzaton constant for purposes of nferrng parameters; but t plays a much more mportant role n comparng alternatve models (e.g. comparng a model wth a perodc sgnal to one wthout). It s the hardest ntegral we ll eventually need to calculate. Equaton (3.8) gves the mplcatons of the data for all of the model parameters. A summary of the mplcatons for the ampltude alone s gven by the margnal dstrbuton for A, obtaned by ntegratng the jont posteror over all the other parameters. Thanks to the factorzaton of the lkelhood functon n equaton (3.6), we can calculate the margnal for A analytcally for any model n normalzed form as follows: p(a D, M) = dω dφ ds p(a, ω, φ, S D, M) 1 = p(d M) p(a M)AN e AT dω dφ ds p(ω, φ, S M) ρ(ωt φ) = p(a M)AN e AT da p(a M)A N. e AT (3.10) The factorzaton of the lkelhood and pror lets us obtan ths result wthout even requrng us to know how to ntegrate over (ω, φ, S), because the requred ntegral cancels. The margnal posteror for A depends on the pror, as t must. However, the dependence s weak provded that the pror s nonzero near A = N/T (where the lkelhood peaks) and does not vary rapdly (.e., on a scale N/T ) there. Moreover, snce the A parameter s common to all models (ncludng nonperodc ones), the pror for A s completely rrelevant for choosng among competng models (assumng t s the same for all models). We demonstrated these facts n Loredo (1992) and n GL usng a flat pror wth a sharp cutoff at some maxmum rate, A max. Here, we repeat some of the calculatons wth an exponental pror, both to further demonstrate the nsenstvty to the form of the pror, and because the algebra s partcularly smple wth ths conjugate pror. Mke West (1992) has suggested ths pror for smlar Posson problems. So we presume our pror nformaton ncludes a nonzero expectaton value for A, whch we denote A 0. Then the pror for A s p(a M) = 1 A 0 e A/A 0. (3.11) You can verfy that ths s normalzed, and that A = A 0. It wll be more convenent to wrte ths pror n terms of the tmescale τ 0 = 1/A 0, so that p(a M) = τ 0 e Aτ 0. (3.12) Usng ths pror, the normalzaton constant n the denomnator of equaton (3.10) s, τ 0 da A N e A(T +τ0) = 0 τ 0 N!. (3.13) (T + τ 0 ) N+1 [MEMO! Not for crculaton! 5 [vonmses.tex: 9/4/ :02

6 Thus the margnal posteror for A s, ( p(a D, M) = 1 + τ ) N+1 0 T (AT ) N T N! [ ( exp AT 1 + τ 0 T ). (3.14) In the lmt that A 0 (so τ 0 0), the pror flattens and becomes vanshngly small everywhere. But the posteror remans fnte and perfectly well behaved, because the τ 0 factor n the pror also appears n the normalzaton constant, and thus cancels out. The τ 0 = 0 lmt s, T (AT )N p(a D, M) = e AT. (3.15) N! Ths s just of the form of a Posson dstrbuton N, multpled by T so that t s normalzed wth respect to A rather than N. Consdered as a functon of A, t s a Gamma dstrbuton. Ths lmtng form s the same as that found n our earler papers wth a lmtng flat pror, as t should be snce as A 0, the exponental pror becomes flat. The mode of equaton (3.14) s the value of A that makes the dervatve wth respect to A vansh; t s, Â = N 1 T 1 + τ. (3.16) 0 T The posteror expectaton value for A s found by multplyng equaton (3.14) by A and ntegratng; the result s, Ā = N T 1 + τ. (3.17) 0 T Ths dffers only slghtly from the mode provded that N 1. A measure of the wdth of the margnal posteror s provded by the posteror standard devaton, whch s, N σ A = T 1 + τ. (3.18) 0 T All of these summares of the margnal posteror dffer from the nfnte A 0 lmt by factors of (1 + τ 0 /T ). Thus, so long as τ 0 T (or A 0 1/T ), our posteror estmates of A are nsenstve to the precse pror nformaton about A Elmnatng the Ampltude We have so far focused attenton completely on A. But n practce, t s the other parameters that are usually of greater nterest, partcularly the perod or frequency. In the remander of these notes we wll focus on the margnal posteror for all parameters other than A, whch we call the jont margnal dstrbuton. The jont margnal dstrbuton s, p(ω, φ, S D, M) = da p(a, ω, φ, S D, M) = C 1 p(ω, φ, S M) (3.19) ρ(ωt φ), where the normalzaton constant s gven by, C = dω dφ ds p(ω, φ, S M) ρ(ωt φ). (3.20) [MEMO! Not for crculaton! 6 [vonmses.tex: 9/4/ :02

7 To get equaton (3.19), we cancelled the ntegral over A n the numerator wth an dentcal ntegral factor n the denomnator. Ths s possble only because of the factored form of the lkelhood and pror, and t holds exactly for any ndependent pror for A. Equaton (3.19) wll be the focus of attenton n the remander of these notes. We need to ntegrate t over φ and S to nfer the frequency. We also need to evaluate C both to normalze nterestng dstrbutons, and to enable us to calculate the global lkelhood, whch we need to compare models. To see the relevance of C to the global lkelhood, note that we can use equaton (3.13) to compute one of the factors n the global lkelhood gven n equaton (3.9), gvng p(d M) = = τ 0 N! (T + τ 0 ) N+1 dω dφ ds p(ω, φ, S M) ρ(ωt φ) Cτ 0 N!. (3.21) (T + τ 0 ) N+1 Thus f we can compute C for any model, we can trvally compute the global lkelhood for that model. In the followng sectons, we dscuss models for whch ntegrals over φ can be performed analytcally, sgnfcantly smplfyng the resultng numercal calculatons. These models also bear close relatonshps wth common frequentst statstcs used for perod detecton n arrval tme seres The Constant Model But before we dscuss perodc models, we note that we have all the ngredents we need to completely treat the smplest nonperodc model: a constant model. Ths model has only one parameter, the unknown DC ampltude, so that r(t) = A, (3.22) correspondng to ρ(t) = 1. The lkelhood functon s smply L(A) = A N exp( AT ), and estmates of A and ts uncertanty are just as were gven above. Fnally, the global lkelhood for ths model s the ntegral n the denomnator of equaton (3.10), whch we evaluated n equaton (3.13), so that, τ 0 N! p(d M 0 ) =, (3.23) (T + τ 0 ) N+1 where M 0 denotes the nformaton specfyng the constant model. From equatons (3.23) and (3.21), we see that the Bayes factor n favor of a partcular perodc model over the constant model s smply, p(d M) B M,0 = C. (3.24) p(d M 0 ) Thus, up to the pror odds, the odds n favor of a perodc model s gven by C. Note that all dependence on the pror for A has dropped out, because ths parameter s common to both models. [MEMO! Not for crculaton! 7 [vonmses.tex: 9/4/ :02

8 4. THE LOG-SINUSOID MODEL To detect and characterze a perodc sgnal, one s frst mpulse mght be to consder a snusodal model rate, or a Fourer seres of harmonc snusods. Such models work well for the analyss of tme seres consstng of samples of a sgnal contamnated wth addtve nose to whch we assgn a Gaussan probablty dstrbuton. Bretthorst (1988) has studed such models n depth wth Bayesan methods, and shown how they are related to frequentst methods that rely on the DFT of the data. Two aspects of arrval tme seres modelng conspre to argue aganst ths choce. Frst, we must model an event rate, not a sgnal ampltude. An event rate must be nonnegatve everywhere. Thus a smple snusod s not a vald model; we must nstead consder a functon lke, r(t) = A[1 + f cos(ωt φ), (4.1) where a DC offset component s added (the pulsed fracton, f, s bound between 0 and 1). Second, n the Gaussan case, the lkelhood functon s an exponental of a sum of the sgnal model evaluated at the sample tmes. In our Posson arrval tme lkelhood functon, we have nstead a product of rates evaluated at event tmes. As a result, models whch are analytcally tractable n the Gaussan case can be ntractable n the Posson case. In fact, t s possble to use the model rate of equaton (4.1), and analytcally margnalze wth respect to φ. However, the constant term necessary to make equaton (4.1) everywhere nonnegatve leads to a very large number of terms n the product of event rates, growng roughly lke 3 N. Thus, analytcal margnalzaton does not help us wth ths model. In fact, ths model s more tractable f any requred margnalzatons over φ are performed numercally rather than analytcally. Snce r(t) must be everywhere nonnegatve, and snce we d lke products of r(t) to have a smple form, t makes more sense to model the logarthm of the rate as a Fourer seres. We begn n ths secton by consderng a sngle snusod, takng r(t) e κ cos(ωt φ). (4.2) Our frst task s to cast ths model nto normalzed form, ntroducng a ρ functon that s normalzed accordng to equaton (3.3). Ths task s smplfed by notng that, 2π 0 dθe κ cos θ = 2πI 0 (κ), (4.3) where I n (κ) denotes the modfed Bessel functon of order n. Thus the normalzed rate proportonal to an exponentated snusod s, ρ(t) = 1 I 0 (κ) eκ cos(ωt φ). (4.4) Ths rate model has a sngle smooth pulse whose peak s at phase φ and whose shape s controlled by one shape parameter, κ, that jontly governs the duty cycle and the peakto-background rato. When κ = 0, the shape s flat; as κ, the shape becomes a δ-functon at phase φ. For large but fnte κ, the lghtcurve has a Gaussan shape near ts peak wth a standard devaton of 1/ κ. The peak-to-background rato s e 2κ. We restrct [MEMO! Not for crculaton! 8 [vonmses.tex: 9/4/ :02

9 κ to be 0 wthout loss of generalty, snce a change n sgn of κ can always be accounted for by a change n φ of π. We sometmes call ths model the von Mses model, because ρ/2π s a common dstrbuton n the statstcs of drectonal data on a crcle known as the von Mses dstrbuton. It s a crcular generalzaton of the Gaussan dstrbuton. But we wll usually use a more descrptve name, callng equaton (4.4) the log-snusod model. We wll denote the nformaton specfyng ths model by the symbol M 1. To complete the specfcaton of ths (or any) model, we must specfy prors for the model parameters. We wll presume that the pror factors as the product of ndependent prors for ω, φ, and κ, so that We assgn a unform pror for φ; p(ω, φ, κ M 1 ) = p(ω M 1 ) p(φ M 1 ) p(κ M 1 ). (4.5) p(φ M 1 ) = 1 2π. (4.6) Ths ntutvely appealng flat pror can be formally justfed f we nsst that our conclusons be ndependent of the choce of the orgn of tme. We use the same pror for ω that we used n GL: p(ω M 1 ) = 1 ω log(ω h /ω lo ). (4.7) The functonal form of ths pror comes from demandng ndependence of our conclusons wth choce of tme scale, and also s form-nvarant f we change varables from ω to P (ths s more an ssue of prncple than of practce; results wll not change drastcally n most cases f a flat pror s used). The pror range, however, s subjectve. It wll have lttle effect on our nferences about ω (provded any detected frequency les n the pror range!) but t wll affect the results of sgnal detecton calculatons. Ths cannot be helped, and actually s qute reasonable; even frequentst sgnal detecton results depend on the search range. Fnally, we need to assgn a pror for κ. For lack of anythng better, we ll just use a flat pror up to some cutoff, κ max : p(κ M 1 ) = 1 κ max. (4.8) If there s a perodc sgnal present, the value of κ max wll have neglgble affect on our nferences about κ or ω. However, t wll have an mportant effect on sgnal detecton results, just as does the frequency range. The problem s: how do we choose κ max? It should come from our pror expectatons for how small a duty cycle we would consder reasonable. Ths s a subjectve aspect of the analyss as s the choce of frequency range to search, and even the choce of model to consder. We ll skrt the ssue here, and smply keep κ max as an unspecfed constant n the equatons. But n an actual sgnal detecton applcaton, we ll have to worry explctly about κ max. As a fnal note on ths subject, we pont out that the tradtonal Raylegh test has no counterpart to κ, but that t s known to be poor at detectng sgnals that are not roughly snusodal. We conjecture that we could fx κ = 1 (.e. use a δ-functon pror) and duplcate ths feature of the Raylegh test; allowng κ to vary may mprove the ablty to detect peaker lghtcurves than snusods. We ll fnd further evdence n support of ths conjecture n the followng secton. Wth the rate model and all the prors specfed, we can now fnally do some calculatons! [MEMO! Not for crculaton! 9 [vonmses.tex: 9/4/ :02

10 4.1. The Jont Margnal Dstrbuton The margnal posteror for ω, φ, and κ s gven by substtutng equaton (4.4) nto equaton (3.19): [ p(ω, φ, κ D, M 1 ) = C 1 1 p(ω, φ, κ D, M 1 ) [I 0 (κ) N exp κ cos(ωt φ), (4.9) where C 1 s the normalzaton constant gven by evaluatng equaton (3.20) for model M 1. The sum n the exponent can be smplfed by usng two trgonometrc denttes, cos(ωt φ) = cos(φ) cos(ωt ) + sn(φ) sn(ωt ) (4.10) = S(ω) cos(φ φ ), where and [ 2 [ 2 S 2 (ω) = cos(ωt ) + sn(ωt ), (4.11) tan(φ ) = sn(ωt ) cos(ωt ). (4.12) The jont margnal can thus be wrtten, p(ω, φ, κ D, M 1 ) = C 1 1 p(ω, φ, κ D, M 1 ) [I 0 (κ) N exp [ κs(ω) cos(φ φ ). (4.13) The only parameter that S depends on s ω, and ts functonal form may look famlar: S/ N s the Raylegh statstc, and 2S 2 /N s sometmes called the Raylegh power or Fourer power of the tme seres. For future reference, we note two other ways of wrtng S(ω): 2 S 2 (ω) = e ωt j j = N + 2 (4.14) cos[ω(t j t ). It s clear from the frst lne that S 2 N 2, wth equalty only when all events are separated from each other by exact nteger multples of a perod. The second lne shows that, roughly speakng, S 2 counts the number of pars of events separated by an nteger number of perods. j 4.2. Estmatng and Elmnatng the Phase The φ constant defned by equaton (4.12) s the most probable value for φ, condtonal on partcular values of ω and κ. To see ths, note that the dervatve of equaton (4.13) wth respect to φ s proportonal to the product of sn(φ φ ) and the rght hand sde of ths equaton. The dervatve thus vanshes at φ = φ and at φ = φ ± π. Snce φ appears n the jont margnal only n the cosne n the exponental, multpled by postve quanttes, t s clear that the former root s a maxmum, and the latter a mnmum, provng our asserton. From equaton (4.12), we see that φ depends only on ω, and not on κ. Thus the locaton of the mode s ndependent of κ, although the wdth of the dstrbuton about the [MEMO! Not for crculaton! 10 [vonmses.tex: 9/4/ :02

11 mode depends on κ. Snce cos(φ φ ) 1 (φ φ ) 2 /2 for φ near φ, equaton (4.13) s approxmately Gaussan as a functon of φ near the mode, wth a standard devaton of, 1 σ φ =. (4.15) κs(ω) The Gaussan approxmaton wll be good when κs 1. Our next task s to elmnate φ from the jont posteror. Usng equaton (4.3), we can perform the necessary ntegral of equaton (4.13) over φ analytcally, gvng us the margnal dstrbuton for ω and κ: p(ω, κ D, M 1 ) = dφ p(ω, φ, κ D, M 1 ) = C1 1 p(ω M 1 )p(κ M 1 ) I (4.16) 0[κS(ω) [I 0 (κ) N. Ths s one of our man results. It shows that the Raylegh statstc s a suffcent statstc for estmatng ω and κ n the context of the log-snusod model: the varaton of S wth ω tells us how to estmate the frequency; and the actual value of S at any gven ω tells us how clumped the arrval tmes are, gvng us an estmate of κ. The former fact s recognzed n the Raylegh test, but equaton (4.16) tells us exactly how to massage the Raylegh statstc mathematcally n order to make probablty statements about ω. There s no counterpart to κ n the formulaton of the Raylegh test, so the way that equaton (4.16) extracts nformaton about the clumpness of the arrval tmes s unque to the Bayesan formulaton Estmatng and Elmnatng κ Equaton (4.16) s the jont margnal for both ω and κ. Here we study the κ dependence of ths dstrbuton. It s of nterest, both for estmatng κ condtonal on ω, and for ntegratng over κ. We need to ntegrate κ both to fnd the margnal for the most nterestng parameter the frequency and to calculate the value of C 1, needed to normalze dstrbutons and to compare the log-snusod model wth compettors. We cannot go much further analytcally, but we wll close ths secton by developng some of the analytcal background needed to deal wth κ numercally. We dscuss estmaton of κ and ntegraton of κ together, because these are not really separate tasks n practce: we need to know about the shape of the dstrbuton n κ n order to ntellgently ntegrate t, and ths knowledge can be expressed as estmates of κ and ts uncertanty. We begn by examnng the qualtatve behavor of p(ω, κ D, M 1 ) regarded as a functon of κ, wth ω fxed. The frst thng to note s that the functonal dependence on κ s completely determned by the magntude of S, and does not depend on any other nformaton about the dstrbuton of the events n tme. Ths compresson of nformaton nto S s a computatonal advantage of ths model. However, t may be a descrptve dsadvantage, snce there s clearly other nformaton about the lghtcurve shape n the dstrbuton of arrval tmes. Ths s partly why we wll consder generalzatons of the log-snusod model n the followng secton. To get a qualtatve sense of the varaton of equaton (4.16) wth κ, note that the modfed Bessel functons have the followng asymptotc behavor: ( x ) [ n 1 2 n! I n (x) + x2 e x 2πx [1 4n2 1 4 (n+1)! 8x, as x 0;, as x. (4.17) [MEMO! Not for crculaton! 11 [vonmses.tex: 9/4/ :02

12 In partcular, I 0 (x) 1 for x 1. For small κ (more precsely, for κn 1), p(ω, κ D, M 1 ) C 1 1 [ 1 + κ2 4 (S2 (ω) N). (4.18) Thus p(ω, κ D, M 1 ) s flat and nonzero at κ = 0, ncreases wth κ f S(ω) > N, and decreases wth κ f S(ω) < N. For large κ (more precsely, for κs 1), p(ω, κ D, M 1 ) p(ω M 1) C 1 κ max S(ω) (2πκ) (N 1)/2 exp [κ(s(ω) N). (4.19) Snce S N, the exponent s negatve, so p(ω, κ D, M 1 ) decreases exponentally for large κ (except for the pathologcal case where all event separatons are nteger numbers of perods, whch corresponds to κ = anyway). These asymptotc behavors mply that the most probable value of κ s ˆκ = 0 f S(ω) < N, correspondng to no perodcty; and that there s a nonzero mode only f S(ω) > N. To locate the mode, we can calculate the partal dervatve of equaton (4.16) wth respect to κ, usng the fact that I 0 = I 1. It s a bt smpler to work wth Λ = log p. Denotng the partal dervatve wth respect to κ by κ, we fnd, κ Λ = S I 1(κS) I 0 (κs) N I 1(κ) I 0 (κ). (4.20) Snce I 1 (0) = 0, ths always vanshes at κ = 0, as we antcpated from equaton (4.18). Ths wll be the mode f S < N. But when S > N, there wll be another nonzero root, whch we denote by ˆκ, because Λ must eventually fall at large κ. Ths root can be found by settng equaton (4.20) equal to 0 and solvng for ˆκ numercally. A reasonable frst guess for ts locaton wll help us fnd t. The asymptotc expressons are a place to start. For small κ, equaton (4.17) gves, I 1 (x) I 0 (x) x ( ) 1 x2. (4.21) 2 8 Usng ths n equaton (4.20), and settng the result equal to 0 gves, ˆκ 2 8 S2 N S 4 N. (4.22) Ths expresson wll probably not be useful very often, as t apples only when ˆκN 1, whch corresponds to very small values of κ when N s large. More useful s the large κ lmt. We can see from equaton (4.19) that the κ dependence of the ω κ margnal s of the form of a Gamma dstrbuton, κ (N 1)/2 exp[κ(s N). The mode of ths functon gves an alternatve estmate for ˆκ, N 1 ˆκ 2[N S(ω). (4.23) In practce, we should probably compute both estmates and use the one that s self consstent as our frst guess. That s, we use equaton (4.22) when that estmate satsfes ˆκN 1, and equaton (4.23) when that estmate satsfes ˆκS 1. [MEMO! Not for crculaton! 12 [vonmses.tex: 9/4/ :02

13 Once ˆκ s known, a measure of the accuracy of the estmate can be found by fttng a Gaussan to the peak. The mean of the Gaussan wll be ˆκ, and ts varance wll be nversely proportonal to the second dervatve of Λ evaluated at ˆκ: σ 2 κ = 1 2 κλ(ˆκ). (4.24) Usng the result that I 1 (x) = I 2(x) + I 1 (x)/x, the second dervatve can be calculated from equaton (4.20), gvng, 2 κλ = [ S2 I 2 (κs) + 1 I 0 (κs) κs I 1(κS) I2 1 (κs) N [ I 2 (κ) + 1 I 0 (κs) I 0 (κ) κ I 1(κ) I2 1 (κ). (4.25) I 0 (κ) To summarze, gven a choce for ω, we can estmate κ as follows. If S N, ˆκ = 0 (.e. the data favor no pulsatons at ths ω). Otherwse, we must fnd ˆκ numercally by settng equaton (4.20) equal to 0 and solvng for ˆκ, say, usng the Newton-Raphson method wth equaton (4.22) or equaton (4.23) as a startng guess. A measure of the uncertanty n κ can be found by evaluatng equaton (4.25) at ˆκ and usng ths to calculate σ κ wth equaton (4.24). More commonly, we wll not have a partcular ω n mnd, and wll nstead need to ntegrate out κ n order to dentfy probable choces for ω (or whether any frequency s ndcated at all). Two procedures suggest themselves. Frst, snce κ 0 and p(ω, κ D, M 1 ) falls exponentally at large κ, Gauss-Laguerre quadrature wll probably work effcently, partcularly when S< N. If we have a code that gves abscssas, x, and weghts, w, so that, dxf(x)e x w f(x ), (4.26) 0 (e.g. the gauleg subroutne n Numercal Recpes), then a change of varables from κ to x = κ(s N) lets us wrte, p(ω D, M 1 ) w e x p(ω, κ = x /(S N) D, M 1 ). (4.27) The change of varables matches the asymptotc behavor dsplayed n equaton (4.19) to the Gauss-Laguerre formula, hopefully allowng us to use a small number of ponts to accurately estmate the ntegral. If S N, there s lkely to be a large peak at ˆκ that wll make Gauss-Laguerre quadrature naccurate (unless we use an unreasonable number of abscssa values). Unfortunately, ths s the most nterestng case, snce t corresponds to strong evdence for a perodc sgnal. One approach s to use Gauss-Hermte quadrature after a transformaton to x = (κ ˆκ)/σ κ. But I am suspect of ths approach, snce κ s bounded on one sde of ˆκ and ts probablty eventually falls exponentally, and not Gaussanly, on the other sde. A related alternatve s to wrte, p(κ) = f(κ) + p(ˆκ)g[(κ ˆκ)/σ κ, (4.28) where p(κ) denotes p(ω, κ D, M 1 ) wth ω fxed, G(x) = exp( x 2 /2), and f(κ) s thus the dfference between the functon we want to ntegrate and a Gaussan ft to ts peak. Then [MEMO! Not for crculaton! 13 [vonmses.tex: 9/4/ :02

14 the desred ntegral of p s gven by the ntegral of p(ˆκ)g and the ntegral of f = p p(ˆκ)g. The former ntegral s trval; t s p(ˆκ)σ κ 2π. The f ntegral can be done n two parts. One part s for κ < 0, arsng from the Gaussan beng nonzero for negatve κ; ths ntegral s proportonal to an error functon. The other part s p p(ˆκ)g for κ 0, whch can probably be evaluated accurately wth Gauss-Laguerre quadrature, snce the peak s subtracted off. Only some test calculatons can reveal whether ths procedure wll work n practce. Fnally, we note that there wll only be sgnfcant evdence for a perodc sgnal, and thus meanngful estmates for ω and κ, when S s large. Thus hgh accuracy n the κ ntegral s not requred for values of ω that gve small values of S, snce ether there s no evdence for a perodc sgnal at any ω, or there s strong evdence at some other ω, and ntegrals n that regon of ω wll domnate the total ntegral over all parameter space Estmatng the Frequency As wth κ, we cannot do much analytcally wth regard to ω. Plotted as a functon of ω, I 0 [κs(ω) s usually very complcated. Even the peaks can have a complcated shape. However, we can get some dea of how a Bayesan frequency estmate wll compare wth a nave estmate based on the wdth of the Raylegh peak as follows. Let ˆω denote the frequency that maxmzes S(ω); ths wll be very near the most probable frequency value (t wll not be precsely the mode because of the /ω dependence of the pror; t s the maxmum lkelhood value). Let Ŝ = S(ˆω) be the value of S at ˆω. Near ths peak, S wll be approxmately parabolc, S(ω) Ŝ [ 1 (ω ˆω)2 2δ 2, (4.29) where δ s the negatve recprocal of the second dervatve of S wth respect to ω at the peak; ths s a smple measure of the wdth of the Raylegh peak that would equal the half-wdth f S were exactly parabolc. Let s consder the large ˆκŜ lmt, for whch the margnal for ω and κ s gven by equaton (4.19), and for whch ˆκ s gven by equaton (4.23). In ths lmt, the ω dependence of the margnal s approxmately Gaussan, wth p(ω, κ = ˆκ D, M 1 ) exp The standard devaton of ths Gaussan s, [ ˆκŜ (ω ˆω)2 2δ 2. (4.30) σ ω = δ ˆκŜ δ 2(N Ŝ) Ŝ(N 1). (4.31) Snce n ths lmt ˆκŜ 1, from equaton (4.23) we see that the factor multplyng δ s 1. Thus the wdth of the posteror n ω can be drastcally smaller than the wdth of the Raylegh peak, because of the nonlnear processng done by the Bessel functon to convert S nto a probablty densty. Very smlar conclusons were found by Bretthorst n hs study of Gaussan spectrum analyss, where he often estmated frequences wth uncertantes orders of magntude smaller than the wdth of the DFT peak (Bretthorst 1988). [MEMO! Not for crculaton! 14 [vonmses.tex: 9/4/ :02

15 4.5. Sgnal Detecton The fnal quantty we need to calculate s the global lkelhood for M 1, p(d M 1 ). As shown n equaton (3.21), f we can calculate C 1, we can calculate the global lkelhood and any nterestng odds ratos. The C 1 ntegral s gven by equaton (3.20), but for the log-snusod model we can perform the φ ntegral analytcally, as n equaton (4.16). Thus we have, C 1 = 1 κ max log(ω h /ω lo ) dω dκ I 0[κS(ω) ω[i 0 (κ) N. (4.32) The ntegrand s the same functon (up to the constant C 1 ) whose numercal quadrature over κ we dscussed n the prevous subsecton; all those comments apply here. In fact, we have to do the ntegrals n equaton (4.32) frst, because they determne C 1. So all we need to dscuss s the quadrature over ω. There s no partcularly cute or ntellgent way to code the ω ntegral. If you plot the result of the κ ntegral as a functon of ω, you wll see a functon that vares wldly wth ω. Only a brute force approach guarantees accurate estmaton of the ntegral of such a complcated functon. The one savng grace s that, f there s evdence for a sgnal, the value of the ntegral s typcally much larger than t would be n the absence of a sgnal, and s completely domnated by the area under one or a few very narrow peaks. A good ntegraton strategy s to do the ntegral wth the trapezod rule over the entre search range, wth a step sze chosen, not to guarantee accuracy of the ntegral over every bump and wggle, but only to make sure each bump s sampled about twce. That s, make the step sze about half the scale of varaton of the functon wth ω. Along the way, note the locaton of the M largest peaks (wth M 10). Then go back, subtract off the contrbutons of the peaks to the orgnal ntegral from the ntal, crude grd, and redo the peaks wth a much larger number of ponts per peak than was n the orgnal grd. Provded we can do ths ntegral n a reasonable amount of tme, we have everythng we need n order to draw conclusons about the evdence for a log-snusod sgnal n the data. The Bayes factor n favor of M 1 over the constant model s smply B 1,0 = C LOG-FOURIER MODELS Wthout dong any numercal calculatons wth real (or smulated!) data, we can antcpate some weaknesses of the log-snusod model. Some of them were alluded to n the prevous secton. The basc problem s that we have only one shape parameter. Even f we suspect that the sgnal we are tryng to detect has a sngle peak per perod (as does the log-snusod lghtcurve), a sngle parameter smply cannot descrbe the possble shapes we mght expect. In partcular, κ determnes both the duty cycle and the peak-to-trough rato of the lghtcurve. But n general, there s no reason to suspect these characterstcs to be lnked. In partcular, a source emttng all of ts sgnal n a sngle peak (.e., wth a 100% duty cycle) could have almost any observed peak-to-trough rato, dependng on the background rate n the detector. Thus as a mnmum we would lke to have separate control of the background level and the pulse wdth. We emphasze that these problems are not unque to the Bayesan treatment. The prevous calculaton elucdates the model assumptons mplct n frequentst use of the Raylegh statstc, and weaknesses of the log-snusod model should also be present n frequentst analyses usng the Raylegh statstc. Indeed, t s wdely known that the Raylegh [MEMO! Not for crculaton! 15 [vonmses.tex: 9/4/ :02

16 statstc s poor for detectng narrow pulses, although I am not aware of publshed studes elucdatng how ts senstvty depends on background level as well as pulse wdth. These problems encourage us to consder models wth two or more shape parameters. One way to go s to add an explct background term to the rate, wrtng r(t) = B + A I 0 (κ) eκ cos(ωt φ), (5.1) where B s a parameter that partly determnes the background rate. Unfortunately, addng a term outsde of the exponent destroys the analytcal smplcty of the log-snusod model. For example, ntegrals over φ no longer have a smple analytcal form. Ths leads us to consder addng terms nsde the exponent. Here we wll add addtonal snusodal terms, each wth ther own κ and φ, and wth harmonc frequences. Ths corresponds to modelng log(r) wth a Fourer seres. Ths s reasonable because log(r), unlke r tself, s not constraned to be postve. In the rest of ths secton we work out some of the detals arsng n analyss wth ths model, whch we call the log-fourer model. We wll fnd n the course of the analyss that, just as the Raylegh statstc arose naturally as the suffcent statstc for the log-snusod model, so somethng lke the Z 2 n famly of statstcs (a harmonc sum of S values) wll arse as approxmate suffcent statstcs for a subclass of log-fourer models. The Bayesan calculaton wll thus elucdate the model assumptons mplct n the use of harmonc sums, and tell us how to convert them nto probablty statements about the exstence of a sgnal and ts frequency and shape. It wll also suggest some generalzatons to Z 2 n, although these may not be computatonally feasble for many datasets. But before gong on, we mght antcpate some drawbacks to ths model. Frst, although addng addtonal κ coeffcents greatly enlarges the accessble range of lghtcurve shapes, there s no smple relatonshp between the coeffcents and smple lghtcurve characterstcs (such as background level or pulse wdth). We have ganed more control over the lghtcurve shape, but t s complcated control. Second, as n the log-snusod case, ntegratons over the κ coeffcents must be done numercally. Thus addng only one or two more terms can drastcally ncrease the computatonal burden. A possble fx to ths s to smply fx all κ values (say, at κ = 1), and let only the phases vary. Such a model s numercally tractable and descrbes a wde array of shapes, but may not descrbe the physcally relevant array of shapes, partcularly f the detector happens to have a sgnfcant background rate. We show below, however, that ths constrant s mplctly assumed n tests that use smple harmonc sums, lke the Z 2 n statstc. So although we won t be able to go too far wth ths model analytcally, we can at least treat a constraned verson of t that may be useful as a Bayesan counterpart to Z 2 n. These dscouragng remarks are offered as motvaton to the reader to come up wth alternatve models that combne numercal tractblty wth physcal relevance. There s a lot of potental for progress here, f only someone can come up wth a clever enough model The Normalzed Log-Fourer Model The log-fourer normalzed rate wth H harmoncs s, ρ(t) = [ 1 H I(κ, φ) exp κ α cos(αωt φ α ), (5.2) α=1 [MEMO! Not for crculaton! 16 [vonmses.tex: 9/4/ :02

17 where κ denotes the set of κ α coeffcents; φ denotes the set of phases, φ α ; and I(κ, φ) s a normalzaton constant gven by, [ 2π H I(κ, φ) = dθ exp κ α cos(αθ φ α ). (5.3) 0 α=1 Two serous weaknesses of ths model are the lack of an analytcal expresson for I(κ, φ), and the explct dependence of ths ntegral on the phases. Ths forces us to do an extra numercal ntegral for every value of κ consdered, and prevents rgorous analytcal margnalzaton of the phases. We d love to learn of any cute trcks that let us quckly (analytcally?!) evaluate the necessary ntegrals! For prors, we wll use the same prors that were used n for the log-snusod model: flat prors for phases, a Jeffreys pror for ω (.e., 1/ω), and ndependent flat prors for each of the κ α. The complexty of the model makes the physcal sgnfcance of the pror for κ unclear; we choose ths pror smply for defnteness and smplcty. We denote the nformaton specfyng a log-fourer model wth H harmoncs by M H. When H = 1, we recover the log-snusod model dscussed prevously The Jont Margnal Dstrbuton Pluggng ths model nto equaton (3.19), we fnd the jont margnal dstrbuton for ω, all phases, and all κ α : [ p(ω, φ, κ D, M H ) = C H p(ω, φ, κ M H ) [I(κ, φ) N exp κ α cos(αωt φ α ), α (5.4) where C H s the normalzaton constant found by ntegratng the other factors over all 2H + 1 parameters. For each α term n the exponent, we can make the same trgonometrc substtutons that were made n equatons (4.10) through (4.12), so that the jont margnal can be cast nto the form, [ p(ω, φ, κ D, M H ) = C H p(ω, φ, κ M H ) [I(κ, φ) N exp κ α S(αω) cos(φ α φ α). α (5.5) Here S s exactly the same functon as n equaton (4.11), only here t s evaluated at several harmonc frequences. Also, the values of φ α are gven by H ndependent versons of equaton (4.12), one for each harmonc An Approxmate Treatment of a Constraned Model If I(κ, φ) dd not depend explctly on φ, we could straghtforwardly fnd the most probable phases and the margnal for ω and κ, usng the same methods as n the prevous secton. The most probable phases would be the φ α values, and the margnal for κ and ω would be proportonal to α I 0[κ α S(αω). We wll proceed by makng a poorly justfed assumpton (acknowledgng that there may be a yet-to-be-found reasonable justfcaton). We wll smply set φ α = φ α, and study the jont margnal condtonal on ths assumpton, whch we call the phase-condtoned margnal. To the extent that I(κ, φ) vares weakly wth φ, ths corresponds to condtonng on the [MEMO! Not for crculaton! 17 [vonmses.tex: 9/4/ :02

18 best-ft value of φ. I haven t been able to prove that I(κ, φ) has weak enough dependence on φ to justfy ths (I suspect t doesn t), but even f t does, ths alone does not justfy the approxmaton. We have to worry about possble correlatons of other parameters wth φ. In partcular, t s clear that the wdth of the jont margnal n ω, and the locaton and the wdth n κ, depend on φ. So we expect ths approxmaton not to affect our best-ft frequency estmates, but to lead us to underestmate our uncertanty for ω. The approxmaton corrupts our κ estmates to an unknown degree. Fnally, the approxmaton also corrupts our sgnal detecton results by an unquantfed amount. Despte these drawbacks, we pursue ths approxmaton for two reasons. Frst, n the Gaussan spectrum analyss case, we know that a smlar assumpton underles use of the Lomb-Scargle perodogram. Ths s clear from the least-squares dervatons n Lomb (1976) and Scargle (1982), where an unknown phase s smple set equal to ts least squares value, wth no account taken of the uncertanty of ths phase. In fact, the equatons are essentally the same as ours, so we are condtonng on least-squares phase estmates, whch are often okay n the Gaussan case, but may not be okay n the Posson case. (By the way, Bretthorst (1988) provdes a Bayesan counterpart to the Lomb-Scargle work that explctly and analytcally accounts for the phase uncertanty.) Second, we wll fnd that ths approxmaton, combned wth a constrant on κ, leads to a condtoned margnal whose suffcent statstc s a harmonc sum smlar to Zn. 2 Thus ths approxmaton wll gve us nsght nto what s beng mplctly assumed when we use Zn. 2 The phase-condtoned margnal s, [ p(ω, φ = φ, κ D, M H ) = C H p(ω, φ, κ M H ) [I(κ, φ ) N exp κ α S(αω). (5.6) If we further constran the model by settng κ α = 1, then the condtoned margnal for ω s gven by, p(ω, φ = φ, κ = 1 D, M H ) e z H, (5.7) where, z H = α α S(αω). (5.8) Ths statstc s smlar to Z 2 n wth n = H, whch n our notaton s gven by, Z 2 H = 2 N S 2 (αω). (5.9) α We can clarfy the relatonshp between the two of these statstcs by notng that, z 2 H = N 2 Z2 H + α S(αω)S(βω). (5.10) β α That s, zh 2 has the quadratc S terms consdered n Z2 H, plus addtonal blnear S terms. Recall that S 2 s a sum of squared cosnes and snes of the data ponts, as shown n equaton (4.11). Thus f we take our data set, cut t n half n tme, and calculate ZH 2 for each half, the value of ZH 2 for the entre data set s gven smply by summng the two values (up to a constant factor). On the other hand, the blnear term n equaton (5.10) ndcates that zh 2 for the whole data set s not proportonal to the sum of z2 H over parts of the data [MEMO! Not for crculaton! 18 [vonmses.tex: 9/4/ :02

19 set. Essentally, ZH 2 computes an ncoherent sum of Fourer power at harmonc frequences, whle z nstead sums ampltudes rather than powers, and thus contans phase nformaton not used by ZH 2. We mght summarze what we have learned about Zn 2 as follows. Frst, t s condtoned on partcular values for the phases that are good estmates for the phases n the Gaussan (least squares) case, but that may not be good estmates n the Posson case. Ths condtonng gnores the effects that phase uncertanty has on other parameters. Second, Zn 2 may tactly assume fxed harmonc ratos (but we have not found Bayesan estmates for κ α, and these may lead to a smlarly weghted sum). Fnally, Zn 2 does not use all of the phase nformaton that s used n a Bayesan analyss wth a log-fourer model. We thus suspect that the Bayesan analyss wll more easly detect real sgnals of varous shapes than does Zn; 2 but further calculatons are requred to justfy ths suspcon. Ths motvates us to more fully treat the log-fourer model, relaxng some of the assumptons made here. But ths wll have to wat for a later memo (and for some nspraton regardng the ntegrals!). 6. INCLUDING POSITION INFORMATION We have presumed all along that the data consst only of the arrval tmes of the events. However, many nstruments also provde postonal nformaton for each event, so that the data consst of trples, (x, y, t ). If there s a background rate, the poston nformaton can help us dstngush possble source events from background events. Currently, ths nformaton s only crudely ncorporated, by throwng out all data outsde of a specfed radus from a known or best-ft poston. Somethng more sophstcated, whch weghts the ponts accordng to the pont spread functon, s lkely to be better. We outlne n broadbrush the Bayesan approach here. Smlar (but not explctly Bayesan) consderatons are beng pursued by Deter Hartmann and Larry Brown at Clemson Unversty. We don t go nto too much detal regardng specfc models, because a full analyss of such data gets computatonally burdensome pretty quckly. It thus may not be practcal for searchng for possble sgnals of unknown frequency. However, f we are searchng for a sgnal of known frequency (such as a rado pulsar counterpart), or f we have detected a sgnal and obtaned a frequency estmate from a smpler method, the lkelhood functon descrbed here may be qute usable. Wth poston nformaton avalable, the relevant model rate must be a functon of poston as well as tme; we denote t by r(x, y, t). As already noted, the poston nformaton s mportant because t helps us dstngush a possble sgnal from background. Thus from the outset we wll explctly decompose r nto background and sgnal parts. We wll presume the background rate s constant n tme (t s easy, at least n prncple, to generalze to tmedependent cases) and that t s known as a functon of poston; we denote the background rate per unt tme per unt x y area by b(x, y). It may have unknown parameters that we must nfer from separate background measurements or from a pror assumptons about the sgnal+background data (e.g., that the background has a known spatal dstrbuton, and that some annulus contans only background and thus sets ts scale). But for smplcty we assume here that t s perfectly known. We presume the pont spread functon for the nstrument s known, so that that a source wth flux Φ(t) n drecton n produces a count rate at detector poston (x, y) equal to k(x, y n)φ(t). In ths notaton, then, k has unts of physcal area per x y area. Thus [MEMO! Not for crculaton! 19 [vonmses.tex: 9/4/ :02