A NAIVE BAYES SOURCE CLASSIFIER FOR X-RAY SOURCES

In order to infer a source's classification (or to conclude that it is undetermined), a clearly defined framework for combining whatever observable data are available is required. Note that the observable data may be in conflict for a particular source; for example, the source may have a median X-ray energy most consistent with a young star in Carina but have a J-band flux most consistent with an AGN.

One intuitive framework is a simple decision tree, an ordered network of simple decisions involving the observed data that leads us to a classification decision, or a conclusion that the classification is undetermined. In such a system, real-valued observables (e.g., clustering, median X-ray energy, J-band flux) have to be compared with thresholds in order to construct nodes in the decision network. Choosing such thresholds is a daunting and contentious task, although probabilistic decision tree construction techniques could be tried (Duda et al. 2002, chapter 8).

A more flexible framework introduces metrics—functions of the observations—into discrimination decisions. Such metrics typically involve the offset of an observed datum from reference values that are typical for each class. Metrics can be weighted so that certain source properties trump others; for example, one might believe that detection of X-ray variability should be a more definitive classifier than J-band photometry.

A third approach, which we adopt here, is based on the intuition of maximizing the likelihood of the observed source data with respect to membership in different classes. In this context, a likelihood is simply a joint probability distribution of all the observed data, conditioned on the class to which the source truly belongs,

$$\begin{equation} p(D_1,D_2, \ldots, D_N \,|\, {\rm class}= {\rm H}), \end{equation} \tag{ 1 }$$

where D₁, D₂, ..., D_N are the N observed quantities potentially available for each source. H is the set of mutually exclusive hypotheses.

H1: source is a foreground Galactic field star;

H2: source is a young star in the Carina complex;

H3: source is a background Galactic field star;

H4: source is an extragalactic source.

The likelihood function p(D₁, D₂, ..., D_N | class = H) is a set of four models

$$\begin{eqnarray} &&p(D_1,D_2, \ldots, D_N \,|\, {\rm class}={\rm H}1) \nonumber \\ &&p(D_1,D_2, \ldots, D_N \,|\, {\rm class}={\rm H}2) \nonumber \\ &&p(D_1,D_2, \ldots, D_N \,|\, {\rm class}={\rm H}3) \nonumber \\ &&p(D_1,D_2, \ldots, D_N \,|\, {\rm class}={\rm H}4) \end{eqnarray} \tag{ 2 }$$

that encode our understanding of what combinations of observable data are produced by sources from each of our four parent populations. In principle, these models encapsulate our understanding of the physics of the X-ray and infrared emission of these four classes of objects, plus observational effects such as absorption, instrument response, and survey sensitivities. Formally, these models are N-dimensional joint probability distributions that represent the many physical correlations that exist among the observable source properties. Significant assumptions and informed professional judgments must be introduced to avoid an intractable problem.

We do not view a likelihood approach to the classification problem as inherently superior to a decision tree or set of weighted metrics. However, we find that the likelihood approach helps us to clarify and quantify our beliefs regarding the classification conclusions that should be drawn from the various clues provided by the observations (Section 2), and provides a quantitative means for combining that evidence to make classification decisions.

3.2.1. Likelihoods

We make the common and critical simplification that the joint likelihood in Equation (1) can be approximated by the product of one-dimensional likelihoods. More formally, we assume that the observed properties of a source are statistically independent:

$$\begin{eqnarray} &&p(D_1,D_2, \ldots, D_N \,|\, {\rm class}={\rm H})\nonumber \\ &&\ = p(D_1 \,|\, {\rm class}={\rm H}) \; p(D_2 \,|\, {\rm class}={\rm H}) \cdots p(D_N \,|\, {\rm class}={\rm H}) \nonumber \\ &&\ = \prod _{i=1}^N p(D_i \,|\, {\rm class}={\rm H}). \end{eqnarray} \tag{ 3 }$$

The terms on the right-hand side of Equation (3) have a straightforward interpretation. Each term represents four one-dimensional probability distributions, each representing the distribution of a particular observable property that we expect from each of the four possible source classes. These likelihoods constitute astrophysical models that predict observable quantities for each of the four source populations and are discussed in more detail in Section 4.

For a particular CCCP source, the data D₁, D₂, ..., D_N have specific values obtained from observations. The likelihood should be viewed as a function of H, the source class, which is a discrete variable with four possible values. Thus, the observed data for each source produce four likelihood values defined by Equation (3). The class with the largest likelihood is the maximum likelihood classification of the source.

The assumption of independence is not strictly correct in the X-ray source classification problem. For example, a higher value of median X-ray energy will be somewhat correlated with fainter J-band flux, as both are products of heavier obscuration. Also, for Carina members, the detection of rapid X-ray variability will be correlated with X-ray flux, which itself is linked to pre-main-sequence stellar mass and J magnitude (Telleschi et al. 2007).

As with many multivariate problems, we must treat the case of missing data. When an estimate of a property is not available for a specific X-ray source, we simply choose to omit that term from Equation (3). The missing source property thus plays no role in the classification decision.³

3.2.2. Prior Class Probabilities

Suppose a source were selected from the CCCP catalog, we were told nothing about it whatsoever, and we were asked to assign odds to each of the four classes from which it could have been drawn. The obvious odds we would assign to each contaminant class would be the ratio of the total number of that contaminant predicted by Getman et al. (2011) to the size of the catalog; the odds assigned to the H2 class (member) would then be calculated such that the four odds sum to unity. Classification probabilities such as these—estimated prior to examining the measured properties of individual sources—are commonly referred to as prior probabilities, or "priors" informally.

Our classification problem can be formulated in terms of a single set of prior classification probabilities, applicable to all sources in the catalog, as described above. Under this formulation, the observed position of each source is a measurement (akin to median X-ray energy, J-band flux, etc.), and thus has a likelihood term in Equation (3), as discussed in the Appendix. However, we find it convenient to adopt a different and equivalent formulation in which separate prior classification probabilities are computed for each source, based on the local density of detected sources and on the local densities of expected contaminants. More formally, for each location $\mathbf {r}$ on the sky at which a source is detected we compute a discrete prior distribution for the source class H ∈ { H1, H2, H3, H4}, using the expected local densities of the three contaminants, $\rho _{{\rm H1}}(\mathbf {r}),\rho _{{\rm H3}}(\mathbf {r}),\rho _{{\rm H4}}(\mathbf {r})$, and an estimate of the observed source density, $\rho _{\rm obs}(\mathbf {r})$:

$$\begin{eqnarray} {\rm prior}(\mathbf {r})_{{\rm H1}} &=& \qquad\qquad \rho _{{\rm H}1}\qquad\qquad / \rho _{\rm obs}(\mathbf {r})\! \nonumber \\ {\rm prior}(\mathbf {r})_{{\rm H}2} &=& (\rho _{\rm obs}(\mathbf {r}) - \rho _{{\rm H}1} - \rho _{{\rm H}3} - \rho _{{\rm H}4}) / \rho _{\rm obs}(\mathbf {r}) \\ &=& 1 - \frac{\rho _{{\rm H}1}+\rho _{{\rm H}3}+\rho _{{\rm H}4}}{\rho _{\rm obs}(\mathbf {r})} \nonumber\\ {\rm prior}(\mathbf {r})_{{\rm H}3} &=& \qquad\qquad\rho _{{\rm H}3}\qquad \qquad / \rho _{\rm obs}(\mathbf {r})\! \nonumber \\ {\rm prior}(\mathbf {r})_{{\rm H}4} &=& \qquad\qquad\rho _{{\rm H}4}\qquad \qquad / \rho _{\rm obs}(\mathbf {r}). \!\nonumber \end{eqnarray} \tag{ 4 }$$

In this study, the spatial distributions of the contaminants are expected to be nearly uniform, so ρ _H1, ρ _H3, and ρ _H4 appear above as scalars rather than as position-dependent quantities. The normalizing constant $\rho _{\rm obs}(\mathbf {r})$ is required so that $\sum _{{\rm H}\in \lbrace {\rm H1,H2,H3,H4}\rbrace } {\rm prior}(\mathbf {r}) = 1$.

The spatial prior map for the H2 class, ${\rm prior}(\mathbf {r})_{{\rm H}2}$, representing the overdensity of observed sources above the expected density of contaminants, is shown in Figure 2. In this study, the minimum value of the H2 prior across the field was positive (0.06); if Equation (4) had produced negative or very small values (i.e., the observed source density had been less than the predicted contaminant density) at any location, then we would have imposed a floor on the H2 prior and adjusted the equations above so that the four prior probabilities summed to unity at every location.

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3.2.3. Posterior Probabilities

Bayes' theorem provides a conceptually simple and principled method for combining likelihoods computed from data (Section 3.2.1) and prior probabilities for model parameters (Section 3.2.2) to produce posterior probabilities for the model parameters, conditioned on the data we have observed:

$$\begin{eqnarray} p({\rm H} \,|\, D_1,D_2, \ldots, D_N) & \propto & p(D_1,D_2, \ldots, D_N \,|\, {\rm H}) \; p({\rm H}) \nonumber \\ \mbox{posterior} & \propto & \mbox{likelihood} \times \mbox{prior.} \end{eqnarray} \tag{ 5 }$$

In our classification task we have one model parameter, the discrete classification hypothesis H for a source at location $\mathbf {r}$ with observed source properties D₁, D₂, ..., D_N. The posterior probabilities for the four possible values of H can be written as

$$\begin{eqnarray} {\rm Prob}({\rm class}&=&{\rm H}1 \,|\, \mathbf {r},D_1,D_2, \ldots, D_N)\nonumber\\ &\propto& \prod _{i=1}^N p(D_i \,|\, {\rm class}={\rm H}1) \, {\rm prior}(\mathbf {r})_{{\rm H}1} \nonumber \\ {\rm Prob}({\rm class}&=&{\rm H}2 \,|\, \mathbf {r},D_1,D_2, \ldots,D_N)\nonumber\\ &\propto& \prod _{i=1}^N p(D_i \,|\, {\rm class}={\rm H}2) \, {\rm prior}(\mathbf {r})_{{\rm H}2} \nonumber \\ {\rm Prob}({\rm class}&=&{\rm H}3 \,|\, \mathbf {r},D_1,D_2, \ldots,D_N) \nonumber\\ &\propto& \prod _{i=1}^N p(D_i \,|\, {\rm class}={\rm H}3) \, {\rm prior}(\mathbf {r})_{{\rm H}3} \nonumber \\ {\rm Prob}({\rm class}&=&{\rm H}4 \,|\, \mathbf {r},D_1,D_2, \ldots, D_N)\nonumber\\ &\propto& \prod _{i=1}^N p(D_i \,|\, {\rm class}={\rm H}4) \, {\rm prior}(\mathbf {r})_{{\rm H}4}. \end{eqnarray} \tag{ 6 }$$

The common constant of proportionality in these equations is easily found by requiring that the four posterior probabilities sum to unity, after which they can be interpreted as probabilities.

The classification system we have described is one of the simplest probabilistic procedures for multivariate classification, is widely used, and is commonly referred to as a naive Bayes classifier. The term "naive" refers to the assumption of statistical independence in deriving the likelihood. The naive Bayes classifier is known to perform well in many cases with mild violations of independence (Hand & Yu 2001). The method is described within the context of other procedures for supervised classification of multivariate data sets in textbooks by Ripley (1996), Hastie et al. (2001), and Duda et al. (2002). Recent applications to astronomical problems include works by Bazell & Aha (2001), Norman et al. (2004, Appendix A), Zhang et al. (2004), Picaud et al. (2005), Ptak et al. (2007, Appendix A), Mahabal et al. (2008a, 2008b), and Burnett & Binney (2010).

Broos et al. (2011) and citations therein describe the X-ray, visual, NIR, and MIR measurements available for CCCP sources. From those data, eight observed or physical source properties were derived. Four likelihood functions were assigned to each source property (Equation (3)), representing the distribution of property values that we expect from members of each of the four possible source classes. These eight properties and their likelihoods are described below.

For three properties (median X-ray energy, J-band flux, and X-ray variability) no estimate of the distribution expected from Carina members (the H2 likelihood function) was readily available a priori. These distributions were estimated empirically, using a training set consisting of likely H2 sources within the field of view of the deep HAWK-I infrared survey, which contains all the large clusters in Carina (Preibisch et al. 2011). This region on the sky has the most counterpart information available and has a small fraction of contaminants. The classifier itself was used iteratively to select the training set within this region. Initially, H2 likelihoods for these three properties were not available, and the classifier evaluated membership using only the other five likelihoods and the spatial priors. Once an initial training set of likely H2 sources was identified, classification was re-evaluated (using all eight properties) and the training set was re-defined, proceeding iteratively until a stable training set was defined.

Median X-ray energy. The median X-ray energy (MedianEnergy_t in Broos et al. 2011) distributions assumed for the four classes are shown in Figure 1 (upper panel). The contaminant distributions (H1, H3, H4) were obtained from simulations by Getman et al. (2011). The Carina member distribution (H2) was obtained iteratively from the training set.

J-band flux. J-band photometry was obtained from the SOFI, HAWK-I, and Two Micron All Sky Survey (2MASS) catalogs (Broos et al. 2011). The distributions assumed for the four classes are shown in Figure 1 (lower panel). The contaminant distributions (H1, H3, H4) were obtained from simulations by Getman et al. (2011). The Carina member distribution (H2) was obtained iteratively from the training set.

X-ray variability. In the X-ray catalog, variability is quantified by a p-value⁴ for the no-variability hypothesis, estimated via the Kolmogorov–Smirnov (K-S) statistic (ProbKS_single in Broos et al. 2011). In the classifier, this p-value is discretized into a variability grade with three values, defined in Table 1. Since the contaminant populations (H1, H3, H4) are assumed to be constant on relevant timescales (the null hypothesis), the expected distribution of this grade is by definition obtained from the p-values used to define the flag, as shown in Table 1. For example, the "definitely variable" grade is assigned when a p-value <0.005 is found for the K-S statistic; for constant sources this will occur by chance with a probability of 0.005, and thus the likelihoods for the "definitely variable" observation are assigned that value for the H1, H3, and H4 classes. The Carina member distribution (H2) was obtained iteratively from the training set.

Note that the probabilities shown in Table 1 define proper likelihood functions; for each class hypothesis the sum of the probabilities for all possible observational outcomes (rows) is unity. Because the posterior class probabilities in Equation (6) are normalized to sum to unity, the influence a particular observation has on classification depends on the relative, not the absolute, likelihoods for that observation across the classes. For example, a variability grade of "definitely variable" is interpreted as strong evidence for the H2 class because the H2 likelihood is ∼15 times larger (0.076/0.005) than the other likelihoods. The grade "possibly variable," with a likelihood ratio of ∼4 (0.10/0.045), is interpreted as moderate evidence for the H2 class. Inevitably, the finding of "no evidence" for variability is interpreted as weak evidence against the H2 class because the likelihood ratio (0.82/0.95) is less than unity. This logical, but unintuitive inference might be overlooked in an informal classification framework based on decision trees or metrics.

Visual spectroscopy. Spectral type estimates obtained from visual spectroscopy are available only for a few sources with high apparent brightness (Broos et al. 2011), but are believed to be reliable. We interpreted spectral types AFGKM as conclusive evidence that the source is in the foreground, and spectral types OB as conclusive evidence the source is a Carina member. That policy is represented by the likelihood functions in the D₄ section of Table 2.

4.5 μm photometry. Spitzer/IRAC photometry was obtained from the SpSmith and SpVela catalogs (Broos et al. 2011). We do not use the 4.5 μm flux value itself as a classification property (as was done for J-band flux) because we do not have models for the detailed 4.5 μm flux distribution expected from our four classes. Instead, we choose [4.5] < 13 mag as a definitive selection against H4 because the majority of extragalactic point sources detected by Spitzer are observed to be fainter than this limiting magnitude (Harvey et al. 2007). This criterion is conservative in that AGNs observed through the obscuration produced by the Carina cloud and intervening segment of the Galactic plane are expected to be even fainter. We construct a bright/weak Boolean datum by applying a threshold of 13 mag to the 4.5 μm flux data using the likelihood functions shown in the D₅ section of Table 2. When weak 4.5 μm flux is found, we choose to omit this property (second table row) because we cannot quantify how strongly that finding should be interpreted as weak evidence for the H4 class.

Mid-infrared color–magnitude region. Spitzer/IRAC photometry was obtained from the SpSmith and SpVela catalogs (Broos et al. 2011). We adopt the color–magnitude selection described by Robitaille et al. (2008, Equations (3) and (5)) for the Spitzer GLIMPSE survey to identify intrinsically red mid-infrared stellar sources with dusty circumstellar material. Since this selection is very conservative and no contaminant population is expected to satisfy this property, we interpreted it as conclusive evidence of Carina membership (H2) as shown by the first row of the D₆ section of Table 2. As was the case with the bright 4.5 μm flux property, we are unable to quantify how strongly the lack of a clear MIR excess should weigh against the H2 class, thus we chose to ignore this term in such cases (second row of the D₆ table).

Infrared spectral energy distribution. Spectral energy distribution (SED) modeling using near- and mid-infrared photometry can identify sources exhibiting a robust infrared excess best explained by circumstellar dust in a disk or infalling envelope (Povich et al. 2011). Photometry was obtained from the 2MASS and SpVela catalogs (Broos et al. 2011). We interpret such a detection of an infrared excess as strong but uncertain evidence of Carina membership (H2), as shown by the first row of the D₇ section of Table 2. We used our professional judgment to choose the strength of this evidence for H2, represented by the likelihood ratio 0.90/0.03. As for some other source properties above, we are unable to quantify how strongly the lack of an infrared excess in SED fits should weigh against the H2 class so we chose to ignore this term in such cases (second row of the D₇ table).

Near-infrared color–color region. Although less frequent than in the mid-infrared, protoplanetary disk dust emission sometimes appears as a K-band excess in near-infrared color–color diagrams. JHK-band color data were obtained from the SOFI and 2MASS catalogs (Broos et al. 2011). At the time this analysis was performed, the HAWK-I catalog exhibited small spatially varying systematic uncertainties in colors (Preibisch et al. 2011); to simplify our analysis we omitted HAWK-I data here. For classification purposes, we define K_s-excess objects as those with H−K colors that are 1.5σ redder than the line defining reddened main-sequence photospheres at mass 0.1 M_☉. In addition, J−H colors must be above the locus of classical T Tauri stars (Meyer et al. 1997).

We interpret detection of this K_s-band excess as strong but uncertain evidence of Carina membership (H2), as shown by the first row of the D₈ section of Table 2. As for the SED likelihood, the strength assigned to this evidence is based on our professional judgment. Note that analysis of the infrared excess sources identified via SED fitting reveals that only a minority (<40%) would have been identified via K_s excess (Povich et al. 2011). Lack of K_s excess does not weigh against the H2 class, because stars with disks often do not separate cleanly from reddened, diskless stars in the JHKs color–color diagram; thus, we chose to ignore this term in such cases (second row of the D₈ table).

Uncertainties on observed data tend to dull the effectiveness of any classifier. One way to understand this effect is to imagine that the likelihoods (such as those in Figure 1) for the measured source properties (as opposed to the true values) will broaden and overlap more as the uncertainties rise. An alternate way to represent data uncertainties is to define the likelihoods as distributions for perfect data, and then to evaluate the likelihood for a specific measurement by convolving the likelihood function with a distribution that represents the measurement (e.g., a Gaussian distribution with mean at the measured value and width equal to the estimated uncertainty on the measurement); see, for example, Burnett & Binney (2010) and Norman et al. (2004). In this study, our first foray into Bayesian methods, we have not attempted to formally represent data uncertainties. The resulting distortion to the median X-ray energy likelihood was mitigated by ignoring the 3610 least reliable median energy measurements. Distortion to the J magnitude likelihood was limited by rejecting photometry values flagged as poor quality; the uncertainties on the ∼8600 J magnitudes used in the classification are very small (mean of 0.027 mag). The K-S statistic used to define an X-ray variability likelihood already accounts for the finite number of X-ray events observed. The 4.5 μm likelihood and the mid-infrared color–magnitude likelihood involve selections on MIR photometry that were defined using Spitzer data, presumably with uncertainties comparable to our Spitzer data. The SED modeling process used to identify infrared excess and the NIR color space criteria used to identify K-band excess make use of cataloged photometry uncertainties.

The source-specific priors described in Section 3.2.2 (Equation (4)) represent the classification probabilities that we would adopt if no observations of source properties beyond position were available. The upper panel of Figure 3 shows a histogram for each of these priors. For example, a source appearing near the core of a dense cluster would appear near 1.0 in the histogram of the H2 prior (red), reflecting its near certain membership in the cluster, and would appear near zero in the other histograms (since the four classification probabilities sum to unity for each source). Clearly, there are many such sources, and a classification based on source position alone would be useful. In contrast, no sources are deemed likely to be contaminants based on only their position (i.e., the H1, H3, and H4 histograms are empty above ∼0.5).

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The lower panel of Figure 3 shows posterior class probabilities for all CCCP sources inferred using both the spatial priors (upper panel) and the likelihoods for all observed data, via Equation (6). This graph is the principal result of the naive Bayes probability calculation given in Equation (6) with likelihoods presented in Figure 1 and Tables 1 and 2. Including the data likelihoods allows us to form more definite opinions about individual classifications, as can be seen by the larger number of sources in the lower panel with class probabilities near 1.0 or near zero compared to the upper panel. Now, some individual sources are identified as likely contaminants with large values in the H1, H3, and H4 histograms. The H2 class is ruled out for a few sources with near-zero values in the H2 histogram.

To better understand the influences that the spatial prior and source property likelihoods have on the posterior class probabilities, we repeated the classification calculation nine times, omitting a single term in Equation (6) for each calculation. Figure 4 shows nine histograms, each depicting the change in the H2 posterior probability resulting from omission of the single specified term. The upper panel shows three terms (the spatial prior, J-band flux, and median X-ray energy) that provide significant evidence both for (positive values) and against (negative values) the H2 classification. The middle panel shows two terms (X-ray variability and visual spectroscopy) that provide significant evidence for (positive values) and minor evidence against (negative values) the H2 classification. The lower panel shows four terms (4.5 μm photometry, MIR color–magnitude region, infrared SED models, and NIR color–color region) that, by construction, provide only evidence for the H2 classification. In the histograms corresponding to data likelihood terms, sources are omitted when data are not available.

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Figure 4 shows that spatial location (upper panel, dotted) strongly increases our belief in the H2 classification (Carina member) for thousands of CCCP sources. J-band flux and median X-ray energy (upper panel, black and red) are not decisive for most sources, but do provide key evidence both for and against the H2 class for a significant minority. The other data likelihoods are available for fewer sources, and only rarely provide decisive evidence for Carina membership.

Bayesian posterior probabilities should be regarded as distinct from traditional hypothesis tests (such as goodness-of-fit tests and the K-S test) in which the consistency between the observed data and a single hypothesis/model is quantified. In hypothesis testing, finding a poor consistency between the data and model can rightly be interpreted as evidence that the model (null hypothesis) should be rejected; however, finding good consistency does not in any sense prove the model (null hypothesis) is correct because there may be many other models that well fit the data. In contrast, the formulation of a Bayesian classifier articulates all reasonable alternatives to the hypothesis/model of interest (in our case H2 = "cluster member"), and the relative consistency between the data and each of the models (class hypotheses) is evaluated (by the likelihood terms). The absolute consistency between the data and a model—the concept measured by goodness-of-fit statistics and p-values–plays no explicit role in Bayesian inference. For example, if the data are equally consistent with all the hypotheses/models—whether that consistency is very good or very bad—then the data will have no influence on our beliefs; the posterior class probabilities will be equal to the prior probabilities. To the extent that all reasonable astrophysical classes of X-ray sources have been considered by our model, the resulting posterior probabilities should be interpreted as quantifying on an absolute scale the confidence with which we can infer that the source belongs to a particular class. An additional advantage of Bayesian methods is that they provide a clear mechanism to incorporate a priori information about the quantities of interest. Although posterior probabilities should be interpreted as our confidence in the four class hypotheses, their accuracy will of course depend on the correctness of the likelihood functions and class prior probabilities adopted, and on the choice of source properties considered. In Section 5, we assess the validity of our CCCP source classification.

The calculation of posterior class probabilities for a hypothetical source is shown in Table 3. We imagine that this source lies at a random position in the Carina field, and thus we adopt an H2 class prior probability equal to the median (0.51) of the H2 prior map shown in Figure 2, and adopt the corresponding H1, H3, and H4 priors (0.15, 0.07, and 0.27). These are shown in the top section of the table. The eight source properties observed for this source, D₁, ..., D₈, are shown in the middle table section; for simplicity only four of those produce likelihoods, Prob(D_i | H), in this example. In each row of the table, boldface marks the class most favored by the prior, likelihood term, or posterior reported on that row.

Probability theory does not specify how posterior class probabilities should be used for astrophysical analyses; investigators must make that judgment themselves just as they decide whether 2σ, 3σ, or 4σ is the appropriate indicator of existence in Gaussian detection problems. We choose to adopt a class decision rule that assigns a specific class if the largest posterior probability is more than twice the next-largest posterior probability. When no classification posterior probability stands above the others using this criterion, a source is labeled "unclassified." Table 4 shows the number of sources found to have each outcome of this decision rule. The classification details for each source are given in Table 5, where columns report the spatial prior probabilities, the naive Bayes posterior probabilities, and the class assignment (with "0" representing "unclassified").

The left panel of Figure 5 shows a map of Carina sources color-coded by the choice made by our decision rule. Within the portions of the field exhibiting high source density (gray areas in the background image) the spatial prior provides a strong preference for the H2 class (Carina members), and indeed the majority of sources are classified as such (red symbols). The ∼2000 sources that remain unclassified (yellow symbols) presumably contain a mixture of H1, H2, H3, and H4 sources.

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The right panel illustrates the important role spatial clustering plays in our classifier. Here we repeated the classification with the spatial priors omitted. Ignoring clustering as a classification clue results in large numbers of unclassified sources (yellow symbols) in the known clusters, an obviously unsatisfactory result.

For the convenience of subsequent studies, Table 5 also includes two quantities that characterize the apparent X-ray spectrum in the "total" energy band (0.5–8 keV). Column 11 reproduces the median X-ray energy of the detected events, reported as column MedianEnergy_t in Table 1 in Broos et al. (2011). Column 12 reports an estimate of the lowest-level calibrated photometric quantity that can be used to compare sources: apparent X-ray photon flux (photon cm⁻² s⁻¹) defined (Broos et al. 2010a, Section 7.4)—using three quantities from Table 1 in Broos et al. (2011)—as

$$\begin{eqnarray*} F_{t, {\rm photon}} &\doteq& {\rm NetCounts}\_{\rm t \,/\,MeanEffectiveArea}\_{\rm t\,/}\\ &&{\rm ExposureTimeNominal}. \end{eqnarray*}$$

Broos et al. (2011) use this quantity to study the spatial variation in detection completeness within the survey.

Estimation of intrinsic (absorption-corrected) flux requires two astrophysical inferences or assumptions: the absorbing column and the spectral shape of the emitting plasma. Getman et al. (2010a) describe a set of plasma assumptions that are appropriate for pre-main-sequence stars, a procedure for selecting the best plasma assumption based on apparent X-ray flux in the hard band (2–8 keV), a technique for inferring the absorbing column with uncertainties from the median X-ray energy statistic once a plasma has been assumed, and a set of calibrations for estimating intrinsic flux with uncertainties. Columns 13 and 14 in Table 5 report the resulting absorption and intrinsic X-ray luminosity values (assuming a distance of 2.3 kpc) for sources classified as pre-main sequence i.e., the H2 sample less the massive stars studied by Gagné et al. (2011). Estimates are not available for very weak sources (those with less than two net counts in the hard band or less than three net counts in the total band).

We have also applied the Getman et al. (2010a) procedure to sources reported to be "unclassified" or likely contaminants in Column 10, in anticipation that future studies may conclude that specific members of this group are instead pre-main-sequence stars. To emphasize that this procedure is inappropriate for the current classification, we report the resulting absorption and intrinsic X-ray luminosity estimates in additional columns (not shown in the table stub) rather than in Columns 13 and 14.

Inherent in the formulation of the problem into the particular data likelihoods and class priors described in Section 3.2 is the point of view that the position of a source is not an "observed source property" (one of the D_i quantities), and thus has no corresponding likelihood term. This formulation asserts that we have spatially varying expectations for class fractions (represented by Bayesian priors that are functions of position) based on the clustering properties of the entire catalog, before examining data on individual sources. This formulation is convenient because all spatial clustering information is represented in one term, the "prior" of Equation (6).

However, there is clearly an alternate formulation that represents clustering information in a different but equivalent way. Under this point of view, we assert a single set of class prior probabilities averaged over the entire field,

$$\begin{equation} {\rm prior}_{{\rm H}1} = N1/N_{\rm CCCP} \end{equation} \tag{ A1 }$$

$$\begin{equation} \qquad\quad{\rm prior}_{{\rm H}2} = 1 - \frac{N1+N3+N4}{N_{\rm CCCP}} \end{equation} \tag{ A2 }$$

$$\begin{equation} {\rm prior}_{{\rm H}3} = N3/N_{\rm CCCP} \end{equation} \tag{ A3 }$$

$$\begin{equation} {\rm prior}_{{\rm H}4} = N4/N_{\rm CCCP} \end{equation} \tag{ A4 }$$

N1, N3, and N4 are the number of H1, H3, and H4 contaminants expected (from simulations, in our case); N_CCCP is the number of detected sources. The form of prior _H2 results from the requirement that any set of prior class probabilities must sum to unity.

Under this point of view, source position $\mathbf {r}$ is an observed property. Thus, as for the other observed properties, we define four likelihoods that are probability distributions for source position conditioned on each of the hypotheses (classes):

$$\begin{equation} p(\mathbf {r} \,|\, {\rm class}={\rm H}1) = 1 / A_{\rm CCCP}\qquad \qquad \end{equation} \tag{ A5 }$$

$$\begin{equation} \!\!p(\mathbf {r} \,|\, {\rm class}={\rm H}2) = \frac{\rho _{\rm obs}(\mathbf {r}) - \frac{N1+N3+N4}{A_{\rm CCCP}}}{K} \end{equation} \tag{ A6 }$$

$$\begin{equation} p(\mathbf {r} \,|\, {\rm class}={\rm H}3) = 1 / A_{\rm CCCP}\qquad \qquad \end{equation} \tag{ A7 }$$

$$\begin{equation} p(\mathbf {r} \,|\, {\rm class}={\rm H}4) = 1 / A_{\rm CCCP}. \!\qquad \qquad \end{equation} \tag{ A8 }$$

Expressions (A5), (A7), and (A8) are independent of position $\mathbf {r}$ because we have assumed that the contaminant classes (H1, H3, H4) are spatially uniform; the expressions are normalized by the area of the CCCP field, A_CCCP, so that the integral of each expression across the domain of the position variable $\mathbf {r}$ (the CCCP field of view) is unity. The numerator of Equation (A6) is the expected position-dependent density of H2 sources (Carina members), estimated as the observed position-dependent density of detected sources ($\rho _{\rm obs}(\mathbf {r})$) minus the expected position-independent density of contaminants ($\frac{N1+N3+N4}{A_{\rm CCCP}}$). Since the integral of Equation (A6) across the domain of the position variable $\mathbf {r}$ must be unity, the required normalizing constant is

$$\begin{eqnarray} K &=& \int _{\rm field}\: \bigg [ \rho _{\rm obs}(\mathbf {r}) - \frac{N1+N3+N4}{A_{\rm CCCP}}\bigg ]\,d\mathbf {r} \nonumber \\ &=& \int _{\rm field}\: \rho _{\rm obs}(\mathbf {r})\,d\mathbf {r} \: - \: \frac{N1+N3+N4}{A_{\rm CCCP}} \int _{\rm field}\,d\mathbf {r} \nonumber \\ &=& N_{\rm CCCP} - (N1+N3+N4). \end{eqnarray} \tag{ A9 }$$

The posterior class probabilities arising from this alternate formulation can be shown to be the same as those from our original formulation (Equation (6)).