The figure below (AAS 472.09) compares the pdfs for the Poisson intensity (red curves) and the Gaussian equivalent (black curves) for two cases: when the number of counts in the source region is 50 (top) and 8 (bottom) respectively. In both cases a background of 200 counts collected in an area 40x the source area is used. The hatched region represents the 68% equal-tailed interval for the Poisson case, and the solid horizontal line is the ±1σ width of the equivalent Gaussian.

Clearly, for small counts, the support of the Poisson distribution is bounded below at zero, but that of the Gaussian is not. This introduces a visibly large bias in the interval coverage as well as in the normalization properties. Even at high counts, the Poisson is skewed such that larger values are slightly more likely to occur by chance than in the Gaussian case. This skew can be quite critical for marginal results.

Poisson and Gaussian probability densities

No simple IDL code this time; but for reference, the Poisson posterior probability density curves were generated with the PINTofALE routine ppd_src()

For instance, suppose as before, that C counts are observed in a region of the image that overlaps a putative source, and B counts in an adjacent, non-overlapping region that is mostly devoid of the source and which is r times larger in area and exposure than the source region. Further suppose that a fraction f of the source falls in the so-called source region (typically, f~0.9) and a fraction g falls in the background region (we strive to make g~0). Then the observed counts can be written as Poisson realizations of intensities,

C = Poisson(φ_S) ≡ Poisson(f ­ θ_S + θ_B) , and
B = Poisson(φ_B) ≡ Poisson(g ­ θ_S + r ­ θ_B) ,

where the subscripts denote the model intensities in the source (S) or background (B) regions.

The joint probability distribution of the model intensities,

p(φ_S φ_B | C B) dφ_S dφ_B

can be rewritten in terms of the interesting parameters by transforming the variables,

≡ p(θ_S θ_B | C B) J(φ_S φ_B ; θ_S θ_B) d θ_S d θ_B

where J(φ_S φ_B ; θ_S θ_B) is the Jacobian of the coordinate transformation, and thus

= p(θ_S θ_B | C B) (r ­ f – g) d θ_S d θ_B .

The posterior probability distribution of the source intensity, θ_S, can be derived by marginalizing this over the background intensity parameter, θ_B. A number of people have done this calculation in the case f=1,g=0 (e.g., Loredo 1992, SCMA II, p275; see also van Dyk et al. 2001, ApJ 584, 224). The general case is slightly more convoluted, but is still a straightforward calculation (Kashyap et al. 2008, AAS-HEAD 9, 03.02); but more on that another time.

When C counts are observed in a region of the image that overlaps a putative source, and B counts in an adjacent, non-overlapping region that is mostly devoid of the source, the question that is asked is, what is the intensity of a source that might exist in the source region, given that there is also background. Let us say that the source has intensity s, and the background has intensity b in the first region. Further let a fraction f of the source overlap that region, and a fraction g overlap the adjacent, “background” region. Then, if the area of the background region is r times larger, we can solve for s and b and even determine the errors:

Note that the regions do not have to be circular, nor does the source have to be centered in it. As long as the PSF fractions f and g can be calculated, these formulae can be applied. In practice, f is large, typically around 0.9, and the background region is chosen as an annulus centered on the source region, with g~0.

It always comes as a shock to statisticians, but this is not ancient history. We still determine maximum likelihood estimates of source intensities by subtracting out an estimated background and propagate error by the method of moments. To be sure, astronomers are well aware that these formulae are valid only in the high counts regime ( s,C,B>>1, b>0 ) and when the source is well defined ( f~1, g~0 ), though of course it doesn’t stop them from pushing the envelope. This, in fact, is the basis of many standard X-ray source detection algorithms (e.g., celldetect).

Furthermore, it might come as a surprise to many astronomers, but this is also the rationale behind the widely-used wavelet-based source detection algorithm, wavdetect. The Mexican Hat wavelet used with it has a central positive bump, surrounded by a negative annular moat, which is a dead ringer for the source and background regions used here. The difference is that the source intensity is not deduced from the wavelet correlations and the signal-to-noise ratio ( s/sigma_s ) is not used to determine source significance, but rather extensive simulations are used to calibrate it.

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