Background Subtraction, the Sequel [Eqn]

As mentioned before, background subtraction plays a big role in astrophysical analyses. For a variety of reasons, it is not a good idea to subtract out background counts from source counts, especially in the low-counts Poisson regime. What Bayesians recommend instead is to set up a model for the intensity of the source and the background and to infer these intensities given the data.

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(φS) ≡ Poisson(f ­ θS + θB) , and
B = Poisson(φB) ≡ Poisson(g ­ θS + r ­ θB)

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

The joint probability distribution of the model intensities,

p(φS φB | C B) dφSB

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

≡ p(θS θB | C B) J(φS φB ; θS θB) d θS d θB

where J(φS φB ; θS θB) is the Jacobian of the coordinate transformation, and thus

= p(θS θB | C B) (r ­ f – g) d θS d θB .

The posterior probability distribution of the source intensity, θS, can be derived by marginalizing this over the background intensity parameter, θB. 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.

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