#### 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(φ, and_{S}) ≡ Poisson(f θ_{S}+ θ_{B}),

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*. A number of people have done this calculation in the case

_{B}*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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