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**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS**

How To Win With Non-Gaussian Data: Poisson Goodness-of-Fit
A. Connors CHASC and Eureka Scientific; (aconnors@eurekabayes.com) D. A. van Dyk CHASC and University of California, Irvine; (dvd@uci.edu)
Abstract. We propose a new metho d to test for go o dness-of-fit of a mo del for low-count Poisson data. Our approach do es not resemble the usual metho ds of approximations to 2 , but instead explicitly uses the full Poisson distribution. First, we propose to use a simple (Poisson-specific) multiscale mo del to characterize the "mismatch" between a best-fit physical mo del and the data. Next, we embed this multiscale mo del into a probabilistic/likelihood framework (via hierarchical Bayes), allowing us to handle statistical uncertainties. We then use MCMC to map out the shape of the joint posterior probability of all of the unknown parameters. Finally, we note that this is a generalization of a problem with a known solution: whether an additional component of known shape is justified by the data. Hence, when the multiscale structure that we use to mo del the "mismatch" between the data and the physical model accounts for a significant number of counts, and/or when the scale-factor for the best-fit physics mo del is significantly different than one, then the original fit was not "go o d-enough". That is, the fitted multiscale "mismatch" isolates the discrepency between mo del and data. We use mo dels of the Gamma-Ray sky as viewed by CGRO/EGRET as our example. We demonstrate that our method works nicely on several examples, but further work is needed to investigate the metho d's power. The authors and CHASC acknowledge support from NSF (DMS 04-06085) and Chandra X-Ray Center, and AISRP NAG5-12082. It is a particular pleasure to acknowledge the SAMSI06 Special Program in Astrostatistics.

1.

Introduction: Two Tough Problems tough related questions that often arise in astronomical image Good Enough?" and "Is there additional unknown structure in That is, faced with a dataset, and a physical model1 that is to hysicist generally confronts four main questions:

We b egin by discussing two analysis: "Is a fitted model the image (or sp ectrum)?" b e fit to the data an astrop
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Throughout this pap er we use "physical model" to mean a model of the photon source. It is assumed to b e a parametric model based on prior physical information, and of (relatively) low dimension. The "statistical model", on the other hand, is a model of the complete data generating mechanism. It incorp orates b oth the physical model and the statistical distribution of the data including any calibration and measurement uncertainties.

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Question 1: What physical model parameters b est fit the data, and what are their uncertainties; Question 2: What is the evidence for an additional comp onent of known shap e (such as a line in an an energy sp ectrum or a p oint-source in an image); Question 3: Is this b est-fit good enough; and Question 4: If the fit is not good enough how does one quantify the unknown additional structure in the image and the uncertainties in this structure? Most astrophysicists know reasonably principled procedures for Question 1 (exploring the probability space around the maximum likelihood estimate) and Question 2 (such as computing limits on the size of the p ossible additional comp onent, or calibrating the rise in log-likelihood as in Protassov et al. (2001) and the references therein). However outside the Gaussian realm, Questions 3 and 4--is the fitted physical model good enough and how does one quantify additional structure in the image--are tough. Since we will use the structure later, we next sketch four example problems, rather p edantically, from simplest to more troublesome. 1.1. One Bin Problems

Problem 1: One Bin, Gaussian Statistics. In the first problem, data are a rate in a single bin, with a known physical model giving the mean (and previously-measured instrumental characteristics giving the variance). It is straightforward to obtain the tail probability of the data under the Gaussian model. There is a well-known reference distribution (i.e., 2 ) that does not dep end on details of the parameters or the model and that 1 is easy to calculate once ahead of time using a standard table. Further, any extra or missing comp onent can b e estimated simply as the data minus the model (i.e., the residual), whether one prefers a mean, a mode, or a median, since they are the same. Unfortunately, the simplicity of this analysis is unique to this simple case. Problem 2: One Bin, Poisson Counts. In this case, the data are again in a single bin, with the known physical model giving the mean. But now the "rate" is a discrete variable, i.e., the Poisson-distributed counts in the bin (or detector) of known size (or known "effective area"). Once again, the computation of a tail probability is straightforward (if more tedious--it must b e calculated rather than looked up; see Bevington 1969, Loredo 1992). One has a choice of estimator for b oth the physical model and the excess structure. The Bayesian p osterior mean, mode, and median, for example, are not the same. Nonetheless the procedures are conceptually simple (if numerically more tedious). 1.2. Multi-Bin Problems

With more bins, we can model shap e or structure in how the flux varies among the bins as well as the total flux. At the same time, the two tough questions, Questions 3 and 4, require more complex treatment.


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Problem 3: Multiple Bins, Gaussian Statistics. Presume one has a sp ectral energy distribution or image, with indep endent Gaussian measurements and heteroscedastic errors (i.e., the variances are known ahead of time, but differ among the energy channels or image pixels). Question 1: Parametrized physical model--best fit and uncertainties. As Lampton, Margon and Bowyer (1976) and Cash (1976) suggest, one finds the mode (or modes) by maximizing the log-likelihood function via one of a numb er of standard numerical optimization methods (as in Press et al 1992). The region around the mode can then b e mapp ed. The corresp ondence b etween drop in log-likelihood and interval coverage is assumed (via the Central Limit Theorem) to b e given by 2 regardless of the details of the model. Alter natively, a Bayesian procedure maximizes and maps out the p osterior probability of the parameters given the model and data. The probability that a parameter is within a certain range ("credible region") is then calculated directly (as in Loredo 1992 and references therein). Question 2: Evidence for an additional component of known shape. Since a parametric function for the p ossible additional comp onent is known, one builds on the procedure describ ed ab ove for one bin in two ways. First, one investigates the limits on the total flux (or equivalent) of the unknown comp onent by computing confidence or credible intervals. Second, one investigates whether the rise in log-likelihood up on the addition of this comp onent signal it is statistically significant. (We remind astrophysicists that for many applications, often including p oint sources and sp ectral lines, the mapping b etween the rise in log-likelihood and the chance significance cannot b e compared to a pre-computed reference distribution such as 2 , but must b e calibrated via appropriate Monte Carlo; Protassov et al. 2002) Question 3: Is the best fit good enough? One can compute a 2 on degrees of freedom, and compare it to a previously tabulated "reference distribution" (i.e., 2 ) by invoking the Central Limit Theorem (e.g., Eadie et al. 1971 and Bevington 1969). There is nothing technically incorrect ab out this simple, p opular procedure. However 2 is almost always an approximation to the true distribution. Hence any statistic that relies on its tail (where the differences would b e the most pronounced) should b e viewed with skepticism. Also, Loredo (1992) p oints out that occasionally one should expect data that are not a "good fit" to one's model, by chance. Question 4: Quantify unknown additional structure and its uncertainties. If the physical model is not sufficient to describ e the data, the difference b etween the data and the physical model, i.e., the residual, is a reasonable estimate of additional unknown comp onent. Uncertainties can b e computed by appropriately scaling the pixel-wise observational variances. If something is known ab out the structure of the additional comp onent (such as local smoothness) one could gain additional p ower by using a model that formalizes this knowledge. However there is nothing actually wrong with simply subtracting the physical model from the data, as is usually done.


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Problem 4: Multiple Bins, Poisson Counts. This is the same as Problem 3, but with Poisson data, i.e., discrete counts. Question 1: Parametrized physical model--best fit and uncertainties and Question 2: Evidence for an additional component of known shape. The procedures for these questions are essentially the same as for the Gaussian case (Cash 1979; Protassov et al. 2002). Question 3: Is the best fit good enough? Here is our challenge. In the Poisson case, there is no pre-computed, single, reference distribution that does not dep end on the details of the model and data. Question 4: Quantify unknown additional structure and its uncertainties. If one has no good prior knowledge of a simple parametric model for the "mismatch" b etween the data and physical model, how does one estimate shap e and significance of the mismatch? Subtracting the model from the data is no longer a handy estimator even of the mean. Even if the counts are relatively high, when one is interested in low probability events--such as the counts from the unknown additional structure--it is imp ortant to take full account of the skew and long tail of the underlying Poisson distribution. We briefly note that although we have not discussed instrumental "blurring" or "smearing", if the blurring matrix or p oint spread function is assumed to b e known completely, it is straightforward to convolve it with the sky and physical models b efore comparison with the data. Thus, we mainly ignore these issues throughout this article. (The case of uncertainty in instrumental resp onse is b eing addressed in a hierarchical way in a separate effort; see Drake et al. 2006.) 2. 2.1. High Energy Gamma-Ray Bridge Over the Milky Way, circa 1999 Brief Gamma-Ray Astronomy Overview: Matter & Energy in our Galaxy

Gamma-ray emission (and hence detection) is an instrinsically Poisson process (Longair 1992). At very high energies, even after almost a decade of high-resolution satellite observations are summed, there are many regions in the sky with very low counts p er bin (see Figure 1a, Left). Gaussian approximations are not suitable. Detailed physics theory -- such as quantum mechanics -- allows researchers to `decode' the basic physics of distant ob jects via their remotely-detected photons. Across the entire electro-magnetic sp ectrum, from radio (lowest energies) through infrared, optical, UV, Xray, and gamma-rays (highest energies), photons tell us ab out sp ecific processes in the sky. Gamma-ray emission originates, for example, from high-energy cosmic processes including nuclear decay and high energy particles impinging on the magnetic fields, dense material, or lower energy photons. In particular, gamma-rays reveal the whereab outs of extremely energetic particles, in our Milky Way Galaxy and b eyond. Given this, do we understand the production cycle of such highly energetic matter and energy in our Galaxy? Or is there more "stuff " we are missing? From our vantage p oint


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Figure 1. CGRO/EGRET data: a) Left: >1 GeV counts measured over the whole sky over the lifetime of the mission (from HEASARC), plotted in Galactic co ordinates in 0.5 в 0.5 bins. b) Right: corresponding exposure (effective area times time).

two thirds of the way out from the center of the Milky Way Galaxy, how well can we use the information packed in gamma-ray photons to learn ab out these processes? 2.2. The Gamma-Ray Sky

Instruments. These were the questions and techniques we faced in the 1990's, when the Compton Gamma-Ray Observatory (CGRO), the first of NASA's "Great Observatories", surveyed the entire sky for gamma-ray emission. CGRO carried two survey telescop es: COMPTEL (0.5-30 MeV, the nuclear regime); and EGRET (30 MeV-100 GeV). It also carried two non-imaging instruments: BATSE, which focused on gathering signals from the whole sky as a function of time (in order to observe gamma-ray bursts); and OSSE, which concentrated on sp ectra. Of these, the instrumental resp onse of CGRO-EGRET's Spark Chamb er is the simplest, and its particle induced background is the lowest. Hence for this pap er, where we ignore "deconvolution", we used > 1GeV EGRET Spark Chamb er data (real and simulated), binned into 1.5 в 2 bins, a level of resolution where instrumental blurring is minimal. CGRO/EGRET data. In Figure 1a (on the left), we show the >1 GeV counts (p er 0.5 в 0.5 bin) collected by the CGRO/EGRET telescop e during its 9+ years viewing the sky (Nolan et al. 1992, Thompson et al. 1993, Esp osito et al. 1999, Mattox et al. 1992). On the right is the corresp onding EGRET telescop e effective area, in units of cm2 . Visible in the plot of gamma-ray counts are a couple hundred "p oint sources" (Hartman et al. 1999). A numb er are thought to b e spatially unresolved glowing gas from sup ernova remnants (SNRs) (Esp osito et al. 1996). Virtually all of the others are understood to b e compact ob jects such as neutron stars and accreting black holes. These include pulsars (neutron stars with magnetic fields roughly a dozen orders of magnitude greater than that of the Earth) and various kinds of active galaxies. The bulk of the signal in this low-background, low count regime, however, comes from diffuse gas and nearby background photons along the plane of our galaxy (e.g., Cillis and


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Hartman 2005, Elwe 2003, and references therein), glowing as they are impacted by energetic particles called cosmic rays. There were physics-based models available at the time that predicted what the diffuse glow should look like (Bertsch et al. 1993; Hunter et al. 1997, Strong and Moskalenko 1998, Moskalenko, Strong, and Reimer 1998, Moskalenko and Strong 1998). These models of the Galaxy included sup ernovae as sources of energetic cosmic ray particles; the spatially complex gamma-ray glow from energetic particles impinging on gas clouds (Bremmstrahlung and Pion decay); and a broader, smoother comp onent due to these energetic particles "b oosting" the energy of photons from the ubiquitous low energy (microwave and infrared) background (Inverse Compton). The comp onents of these physical models were measured separately. High resolution radio observations were used to map out the gas distribution, and local measurements of cosmic ray particles were used to constrain their energy sp ectra and abundances. With the physical model and CGRO observations in place, a fundamental question was whether the models matched the observed data. 2.3. Imaging Methods for the Whole Sky in Gamma-Rays

CGRO era. In order to answer this question, Dixon et al. (1999) applied the TIPSH (translation invariant Poisson smoothing using Haar wavelets) method of Kolaczyk (1997) and Dixon and Kolaczyk (2000). That is, in Question 4 under sc Problem 4 ab ove, they modeled the "mismatch" b etween the physical model and the CGRO/EGRET all-sky data with a multiscale (b oxy-shap ed Haar wavelet, with cycle-spinning) model. The TIPSH "keep or kill" thresholds were carefully tailored to match the Poisson statistics. At the time, they did find a mismatch that looked rather like a bridge ab ove the plane of the Galaxy. They famously emphasized, however: "The immediate question arises as to the statistical significance of this feature. Though we are able to make rigorous statements ab out the coefficient-wise and level-wise FDR [False Discovery Rate], similar quantification of ob ject-wise significance (e.g., `this blob is significant at the n sigma level') are difficult." The use of multi-scale methods was a notable improvement in modeling unknown diffuse emission, which is exp ected to show b oth the sharp and broad features of the complicated diffuse gas, the photon field, and sea of energetic particles (cosmic rays) from which it arises. Previously, among astrophysicists, other prominent "model-free" or nonparametric (i.e., a model with ab out as many parameters as data bins) methods include Richardson-Lucy (the EM algorithm applied to an unconstrained grid of p oisson parameters; Richardson 1972, Lucy 1974, Shepp and Vardi 1982) and Maximum-Entropy-related methods (such as MaxEnt and successors; Skilling 1989; also Strong 2003). These methods lacked the multiscale representation of b oth diffuse and sharp features. Progress 1: NK. In the ensuing years, several p otential improvements were tried. (See also Willett 2006 and Young 2006, this volume; and the different approaches of, say, Bourdin et al. 2001; Starck, Pantin and Murtagh 2002, Starck and Murtagh 2006 and references therein.) The TIPSH wavelet structure was revamp ed into a more general multiscale structure on certain ratios of exp ected counts to exploit the statistical prop erties of the Poisson


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likelihood and to allow the inclusion of instrument resp onse ("smearing" or blurring) in an elegant (and sp eedy) way (Nowak and Kolaczyk 2000; Kolaczyk and Nowak 2004, 2005; henceforth this method is referred to as NK). Like wavelets (which often use recursive dyadic partitioning), the NK structure breaks an image into four quadrants (or two segments, in 1D) and then successively breaks each of these quadrants into its four quadrants until the inherent pixelization is reached. Unlike wavelets, for which the basis function is typically some sort of differencing function, for NK the basic shap e is a constant intensity block. Further, rather than working with the intensity itself, we consider the ratios of the intensities of each quadrant to the whole (see Willet 2006, this volume). These "split probabilities" are describ ed by prior distributions tuned by smoothing parameters, in analogy to the way that shrinkage (or complexity p enalties) are used to estimate the b est values of coefficients in a wavelet decomp osition. Further, NK is formulated so that the problem is convex and there is only one mode, so good convergence is assured. Finally, as with most wavelet analyses, cycle-spinning is recommended to remove artifacts in the fitted image (Coifman and Donoho, 1995). Still, this used the EM algorithm to get a "b est fit" image, and uncertainty estimates were not available. Progress 2: EMC2. Later, Esch et al. (2004) emb edded the NK multiscale model into a hierachical Bayesian framework. (See Karovska et al., 2005 for applications with b eautiful graphics.) This also allows the smoothing parameters used in NK to b e fit to the data. The range of variation of the tuning parameters is effectively governed by a hyp erparameter that is set ahead of time. In order to fully map out the p osterior distribution of the various model and tuning parameters, Esch et al. (2004) uses a Markov Chain Monte Carlo sampler. This procedure produces not only a "b est fit" image (the mean, rather than the mode), but also a full exploration of the probability space, via the MCMC samples. Hence in principle one has all the information necessary for calculating any summary of the p osterior distribution and to quantify uncertainty in the fitted image. The results of an EMC2 run on a Poisson image are often not as pleasing to the eye as, say, a Gaussian-kernel-smoothed picture, or the related p opular (commercial) package Pixons (Pina and Puetter 1993). However, since the original NK algorithm elegantly factors the log-likelihoood, it is b oth complete and sp eedy. The examples later in the pap er each took ab out 40 minutes to run on a small iBook G4. Further, in theory, one has all the information one needs to understand the uncertainties in the fitted image. Unfortunately, it was not immediately obvious how this information could b e synthesized to answer the questions of direct scientific interest. We emphasize that although we are not discussing it here, NK and EMC2 do have the capacity to "deconvolve" the image, see Esch et al. (2004) and Karovska et al. (2005). (More precisely, EMC2 employs forward-folding of the multiscale model convolved with the instrument resp onse.) New features. During the SAMSI06 thor worked on modularizing, sp eeding In this pap er, we focus particularly on and to fit this comp onent's overall inten additional NK-typ e comp onent. Sp ecial Session on Astrostatistics, the second auup, and adding new features to the EMC2 code. the ability to add a shap ed baseline comp onent sity at the same time as inferring the shap e of an


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The critical question was still: How can we use these methods, in practice, to answer the question Dixon et al. (1999) p osed? How can we b egin to quantify the statistical significance of features in the fitted image and how can we assess goodness of fit of the statistical model? 3. How to Define a `Feature'

Usual thinking. An astrophysicist investigating p ossible unknown diffuse emission (or an unknown sp ectral feature) usually tries to draw a b oundary around it and then tries to estimate its significance. Whether doing simple b ox-car smoothing or more robust Gaussian kernel smoothing (Eb eling, White, and Rangara jan 2006), or even wavelet-based detection of p otential sources at different scales (Freeman et al. 2002, 2003), the usual algorithms can b e thought of as defining a region of `significant' emission by drawing a b oundary around it and calculating its significance. That is, once the b oundary is drawn, one can ask: are there a significant numb er of counts in this new feature, compared to satistical fluctuations exp ected from my "baseline", or known physical model? (See also wavelets in an EM context in Knodelseder et al. 1999). Still, as in Dixon et al. (1999), the Ё researcher is essentially looking at coefficient-wise or level-wise (rather than "ob ject-wise") significance. Even the first uses of EMC2 followed (roughly) this usual thinking (Esch et al. 2004; Karovska et al. 2005), defining the b oundary at the pixel-wise 3 limit. However, the larger question of how to quantify the uncertainty in the b oundary itself remained unanswered. Key breakthrough--It's in there. After much discussion, we realized that the multiscale model itself had already "chosen" the b oundaries. In real astrophysical sources, b oundaries of glowing gas clouds or photon fields are not sharply defined, nor are they smooth or even continuous. No additional step was needed: we could start with the b estfit physical model and add an additional comp onent model via the NK/EMC2 flexible multiscale model. Since we know how to fit this model, we have, in essence, reduced our original two tough problems (Questions 3 and 4 under Problem 4) to one that we already know how to handle (i.e., Question 2, but with a flexible semi-parametric model as our prop osed additional comp onent, rather than a low-dimensional physics-based model). We can quantify the evidence for including the added multiscale comp onent in a simple manner by looking at the p osterior distribution of the total flux from this comp onent using the MCMC sampler, as we have done informally (visually) in Figures 6 and 9. A more computationally intensive but p ossible approach is to run a simulation to calibrate the distribution of the change in log-likelihood with the addition of this comp onent. This strategy is similar to the calibration methods for the LRT promoted by Protossov et al. (2002). In summary, our method starts with the physical model and adds a multiscale model. We aim to investigate whether any statistically significant structure ab ove the supp osed physical model is captured by the multiscale comp onent (detection). This strategy simultaneously gives us a model for the unknown feature's shap e (characterization). Finally, we need a strategy for determining if the original physical model is "good enough". That is,


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Figure 2. Quantifying Nothing--Mo del+Data: a) Left: Physical mo del of all-sky diffuse emission >1 GeV; b) Right: Poisson data simulated from the physical mo del with no extra component. Top and bottom are padded with zeros to make an even 128x128 bins.

we need a procedure that gives a "goodness of fit" measure for our b est-fit physical model and describ es the shap e and significance of the "residuals". For this final ob jective we suggest calibrating the null distribution (for what no added feature looks like) by simulating several datasets from the physical model (our "baseline", or null hyp othesis). On each of these simulated data sets, we run the EMC2+baseline fitting procedure. The baseline corresp onds to the physical model and the multiscale model is used for the added model comp onent or feature. We then compare the joint distribution of the physical model scale factor and the total exp ected counts form the multiscale comp onent for each simulation with that of the real data. If the multiscale comp onent is more imp ortant when fit to the real data than to the simulations, we conclude the new extra comp onent may b e real. 4. Worked Example: Simulations of All-Sky CGRO/EGRET Data

Simulation summary. We b egan with simulated data based on the CGO EGRET summed phase 1-9 exp osure and all-sky count-rates, courtesy of HEASARC.2 We use the output of the physics-based model GALPROP (version fits 49 700405) provided via A. Strong 3 (Strong, Moskalenko, and Reimer 2004; Strong, et al. 2004; 2005); together with the CGRO/EGRET exp osure map of Figure 1b, to simulate several physically realistic maps of the gamma-ray sky >1 GeV. We simulated Poisson datasets that matched the physical model assuming an integration time of ab out two weeks, as well as datasets with extra comp onents and missing pieces. Our procedure is: 1. The physical model is used as a baseline in EMC2.
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cossc.gsfc.nasa.gov/docs/cgro/egret/egret doc.html. www.gamma.mpe.mpg.de/~ aws/aws.html


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Figure 3. Quantifying Nothing--Results: a) Top Left: mean value in each pixel from 900 draws of multiscale mo del (i.e., EMC2 run); b) Top Right: sigma of same; c) Bottom Left: skew; d) Bottom Right: kurtosis.

2. An additional comp onent of unknown shap e is modeled using the NK/EMC2 multiscale model. 3. The normalization of the baseline physical model is fit. 4. If the multiscale model comp onent contains a significant numb er of counts, and/or if the normalization of the baseline is significantly different than one, then we conclude the physical model is not a good enough fit. 5. Significance is assessed by comparison to the p osterior distributions produced by running EMC2 on simulations from the uncontaminated physical model (i.e, the baseline or null model). Measuring nothing (no added feature). In the first simulation study, we demonstrate how this procedure works when our data has no extra comp onent ab ove the physical model (i.e., baseline or null model; Figure 2). Even when started far from the true value, the MCMC sampler converges correctly, showing no significant structure or counts in our multiscale comp onent, see Figure 3. There is no evidence for multiple modes; the inferred mean (and mean/sigma) are very small compared to the counts (and baseline rate); and the skew (and kurtosis) are large, indicating the p osterior distribution of the multiscale count is highly skewed towards zero, as it should b e. In order to spread out the sharply


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Figure 4. Bright Unknown (Discontinuous) Source--Mo del+Data: a) Left: Again, the baseline is the physical model of all-sky diffuse emission >1 GeV; b) Right: Poisson data simulated from the physical mo del with a bright added component. Top and bottom are padded with zeros to make an even 128x128 bins.

p eaked p osterior probability of our two parameters of interest (the baseline scale factor and total exp ected count from the multiscale comp onent) we have plotted them on a log10 scale. Bright discontinuous unknown. The results app ear very different when we add a bright but discontinuous, irregular, and low surface brightness second comp onent to the physical model, see Figure 4. The difference b etween the baseline physical model (Figure 4a) and simulated data (Figure 4b) is visible to the unaided eye. Yet from plots of the mean (and sigma) counts p er pixel (Figure 5a and 5b), one sees the rate p er pixel is not significant. But the overall multiscale comp onent is hugely significant (Figure 6). Again, even when starting far from the true value, the MCMC sampler converges correctly and there is no evidence for multiple modes. The unknown comp onent's shap e--i.e., the difference b etween areas where the the two irregular diffuse comp onents are statistically significant and where they are not--is easily seen in the plots of the mean, sigma, skew, and kurtosis. Again, in order to spread out the sharply p eaked p osterior probability of our two parameters of interest, the baseline scale factor and the total exp ected counts due to the multiscale comp onent, we use the log10 scale. Faint unknown source, baseline shap e mis-sp ecified. In our final simulation, the total difference b etween the model and simulated data is only ab out 102 counts (rather than 103 counts, as ab ove). Further, as careful visual comparison of the baseline (Figure 7a) and the simulated data (Figure 7b) suggests, the Poisson counts were simulated from a distribution with the same spatial shap e as the baseline physical model, but with overall intensity of the Inverse Compton (IC) comp onent suppressed via a p ower-law ( b-2 ) as one moves ab ove the Galactic Center. This roughly represents a realistic mistake in the physics model. We have processed these data two ways: with the baseline scale comp onent fixed; and with it allowed to float. The multiscale comp onent picks up the missing broad glow quite


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Figure 5. Bright Unknown (Discontinuous) Source--Results: a) Top Left: mean value in each pixel from 900 draws of multiscale mo del (i.e., EMC2 run); b) Top Right: sigma of same; c) Bottom Left: skew; d) Bottom Right: kurtosis. Recall these moments display only the shape of the unknown component. Significance is assessed as in Figure 6.

Bright Discontinuous Unknown Log10(Baseline Scale Factor)
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Null (.) vs Bright Unknown (+) Log10(Baseline Scale Factor)
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Figure 6. Calibrating Results--Null vs Bright Unknown Component: Scatter plot of log10 of total number of counts inferred to be in the multscale mo del for each of the draws, versus log10 of the baseline scale factor. a) Left: Bright unknown component; b) Right: Same, compared to results on the baseline (null). Bullets indicate data with no signal (null hypothesis), while `+' indicates results from data with a bright additional component.


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Figure 7. Faint Unknown Source, Baseline Shape Mis-Specicified -- Model+Data: a) Left: Baseline is physical mo del of all-sky diffuse emission >1 GeV, but with intensity distribution of IC component gradually suppressed above the Galactic plane; b) Right: Poisson data simulated from same spatial shape as the physical mo del in Figure 2.

0.5

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Figure 8. Faint Unknown Source, Baseline Shape Mis-Specified--Results, Scale Fixed: a) Left: mean value in each pixel from 900 draws of multiscale mo del (i.e., EMC2 run); b) Right: sigma of same. (The skew and kurtosis follow the same pattern as in the previous example.) Recall these moments display only the shape of the unknown component. Significance is assessed as in Figure 9.


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Null (.) vs Faint Model Mis!Match (+) Log10(Baseline Scale Factor)
+ + ++ +++ + ++ + + + + + ++++++ + + + ++++ ++++ +++++ + + ++ +++ + + + + ++ +++++++++++++++ + + ++ +++++ + +++ ++ +++++++++++++++++ ++ ++ + ++++++++++++++++ ++++++++++++++ ++ + + + +++ ++ + + ++++++ + + +++++++++++++++ ++++++++++++ + +++++++++++++++++++++ ++++ + +++++++ ++ ++ +++ +++ ++++++++++++++++++ +++ + + + ++++++++ + ++++++ + +++ +++++++++++++ + +++ +++++++++++++++++ + + + +++ ++++ +++++ +++ + + ++ + + + ++ + ++ ++++ ++++ ++ +++ ++++++++ ++ + + +++ ++++ + ++ ++ +++++++++ ++++ + + + + ++++++ + + + +++ + +++ + + + +++ 250

Null vs Faint Model Mis!Match

+ 0.03 + ++

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Figure 9. Calibrating results--Null vs Faint Unknown: a) Left -- Baseline scale floats free: Scatter plot of log10 of total number of counts inferred to be in the multscale mo del for each of the draws, versus log10 of the baseline scale factor. Bullets indicate no signal (null hypothesis). A `+' indicates results from data witha faint additional component. b) Right -- Baseline scale fixed (at unity): Histogram of log10 of total number of counts inferred to be in the multiscale mo del for the null (white) and data with faint additional componenent (gray).

easily when the baseline scale factor is fixed; with the brightest parts b eing where the exp osure is greatest (Figure 8). However, interestingly, when the scale factor is allowed to float, the procedure prefers to increase the scale factor rather than put counts into a new comp onent (Figure 9). 5. 5.1. Conclusions: General Principals, and Further Work Brief Mention of Other Metho ds

There are many methods not mentioned here (e.g., Rice, 2006, these proceedings). In a general sense, they share with our prop osed technique an overarching strategy of non- or semi- parametrically looking for regions of fluctuations in the data b eyond those exp ected by chance from the underlying distribution of the data. Our method, however, is sp ecifically tailored to Poisson images and energy sp ectra. We note that other Poisson multiscale (or very flexible) models could also b e used in place of EMC2 (e.g., Bourdin et al. 2001, Starck and Murtagh 2006, and references therein). 5.2. Further Work

The simulations describ ed here, though promising are just the b egining. We hop e to develop methods to quickly calibrate and precisely quantify the improvement offered by the addition multiscale comp onent. Rather than a simple visual insp ection of the p osterior distribution, such methods would offer a precise numerical summary. The p ower of the


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method to detect weak added source is another area that we hop e to explore. (Preliminary tests, however, show that where the total difference b etween our usual baseline physical model--which contains several thousand counts over the whole sky--and the simulated data is a factor of 10 lower than the faintest example shown here, we see no visible detection.) Finally, although the capability for incorp orating an instrument resp onse has b een built into the code, it has not b een extensively tested on these data. There are many practical directions in which we would like to improve the functionality of out method. For example, using all-sky spherical data rather than nested square pixels; allowing for a PSF or instrument resp onse that varies over the field of view; incorp orating simple parametric fitting in the baseline physical model; using the full energy information together with the imaging information; incorp orating uncertainties in the physical model and/or "instrument" model; and including more complex prior physical information (say, high resolution radio data). We exp ect to work on some of these in the near future. 5.3. Soapb ox

Traditionally, physicists receive very little formal training in probability and statistics. Our cultural habit is to think in terms of Gauss-Normal statistics and 2 . ("How many sigma is that result?", we ask of a detection where, say, the exp ected background counts are 0.2 and measured source plus background counts are 2.) At the same time, physicists tend to b e comfortable with likelihood methods, and well-versed in Monte Carlo -- and so have understanding of the basic building blocks for tackling non-Gauss-Normal problems directly, rather than by approximations to older Gauss-Normal statistics, such as 2 . Here, we have tackled Poisson data. It is good as one sp ecific example of the NonGaussian regime. This forces us (astronomers and physicists) to think. We cannot rely on the symmetry and limiting prop erties that make, say, "filtering" or "subtracting" or "correcting" data mathematically/probabilistically/statistically equivalent to a thorough likelihood analysis. Instead we must pay careful attention to the unfolding of the model: astrophysical + instrumental + statistical. We must leave the data alone; our tamp ering will not improve them. In this pap er, we have used hierarchical methods of taking apart and putting together models and data. This enabled us to prop ose a practical solution to two tough longstanding problems for Poisson data: "good-enough" fit and querying the data for evidence of an unpredicted comp onent of unknown shap e. We have at least qualified success in testing a solution. But we argue the larger lesson is: many supp osedly intractable problems can b e analyzed using this same framework of carefully breaking the difficult problem apart into smaller, more manageable pieces. Likelihood-based methods such as Hierarchical Bayes offer ideal tools to this end. Acknowledgments. The authors and CHASC acknowledge supp ort from NSF (DMS 04-06085) and Chandra X-Ray Center. AC acknowledges AISRP NAG5-12082 "Astrostatistical Tools in Python", and the work of her co-investigators (T. Loredo and T. Oliphant). It is a particular pleasure to acknowledge the SAMSI06 Sp ecial Program in Astrostatistics.


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This research has made use of data obtained from the High Energy Astrophysics Science Archive Research Center (HEASARC), provided by NASA's Goddard Space Flight Center.
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