The probability density function, *f*^{u}(*X*), describing relative use of a habitat with environmental covariates, *X*, is typically defined using:

- (eqn 1)

where *f*^{a}(*X*) gives the relative availability of a habitat composed of environmental covariates *X*, in the study area and the integral applies to all possible combinations of environmental conditions in environmental space *E* (i.e. the multi-dimensional space represented by environmental variables). This specification (eqn 1) can be seen as a weighted distribution (Patil & Rao 1978; Patil 2002; Lele & Keim 2006), where the organism samples the available distribution *f*^{a}(*X*) with a probability proportional to *w*(*X*) in order to obtain the used distribution *f*^{u}(*X*). Hence, in environmental space, *w*(*X*) is proportional to the *ratio* between habitat use and availability. *w*(*X*) is known as the habitat or resource selection function and is most often modelled as an exponential function of covariates, , where *X*^{T} is the transpose of *X* (McDonald, Manly & Raley 1990; Lele & Keim 2006; Johnson *et al*. 2008). Although rarely explicitly stated, eqn 1 assumes that, conditional on *w*(*X*), changes in *f*^{a}(*X*) lead to proportional changes in *f*^{u}(*X*). Assuming the parameters in the habitat selection function are conditional on *f*^{a}(*X*), and *f*^{a}(*X*) is uniform in geographical space (*G*), the likelihood function for the entire dataset can be simplified to:

- (eqn 2)

where the integral of eqn 1, originally defined in environmental space, is replaced by the integral evaluated over all of geographical space; here *A* is the entire study area. This integral can be approximated by evaluation and averaging of *w*(*X*) at random or regular points in geographical space. This alternative specification of the likelihood function illustrates that the habitat selection model fitted in environmental space (see eqn 1), is equivalent to a model which quantifies use in geographical space as a function of the underlying environmental conditions (i.e. eqn 2 – Aarts, Fieberg & Matthiopoulos 2012). An advantage of specifying the likelihood according to eqn 1, however, is that it becomes readily apparent that the estimated habitat selection function *w*(*X*) is *conditional* on what is considered to be available to the organism [i.e. *f*^{a}(*X*)]. As will be illustrated later (see e.g. Figs 1-4), the estimated *w*(*X*) may vary drastically as a function of absolute habitat availability, even though the animal uses the same movement rules to explore and exploit space.

The likelihood function in eqn 2 is equivalent to that of a *conditional inhomogeneous Poisson process* (CIPP – Cressie 1993) and identical to the likelihood used by MaxEnt when an exponential function is used to model *w*(*X*) (Phillips, Anderson & Schapire 2006; Aarts, Fieberg & Matthiopoulos 2012; Renner & Warton 2013). Alternatively, one can fit an Unconditional Inhomogeneous Poisson Process (UIPP) model [Cressie (1993), eqn. 8.5.16, page 655, and Aarts, Fieberg & Matthiopoulos (2012), eqn. 4]. In contrast to the CIPP, the UIPP estimates an intercept, which relates to the mean intensity. In the UIPP likelihood, the integral in the denominator of eqn 2 is exponentiated, and must still be evaluated numerically. In both the (CIPP) and UIPP, the (relative) intensity of the point process is given by *w*(*X*). The two approaches will result in equivalent slope parameters associated with environmental covariates [see Aarts, Fieberg & Matthiopoulos (2012), Appendix A], but the conditional approach cannot estimate an intercept parameter since it cancels from both the numerator and denominator in eqn 2 (Lele & Keim 2006). We fitted models using the unconditional likelihood; such models can easily be fit as a Poisson log-linear model in most statistical software packages using the following numerical trick (Baddeley & Turner 2000; Warton & Shepherd 2010): First, a regular grid with a spatial resolution similar to the environmental data is constructed, an availability point is located at the centre of each grid cell, and for each grid cell (*i*) the total number of used and available locations (*n*_{i}) is calculated. Next, a Poisson log-linear model is fitted to the data, where the value of the response variable is set to 0 for each availability point, but to *n*_{i} for each animal location, and 1/*n*_{i} were specified as ‘prior weights’ for all points in the GLM. For more details see Baddeley & Turner (2000).