SCR question

[Ashley Cook]

Hello, 

I am using the secr package for my graduate work and I am having difficulty understanding what the maximum likelihood estimation is doing. I'm focusing on proximity detectors and I am specifically looking at equation 2 from this paper...

Density Estimation by Spatially Explicit Capture–Recapture: Likelihood-Based Methods  DOI:10.1007/978-0-387-78151-8_11

In this equation below, I understand the following:

Pr(ωi| θ)= ∫ Pr(ωi | X, θ) dX / ∫ p.(X; θ) dX. 

 Pr(ωi | X, θ) is the probability of the capture history given θ  and the animal's home range center (X). Since the home range center (X) is unknown, the probability of detection is integrated over all possible locations, ∫ Pr(ωi | X, θ) dX, effectively weighting each capture history by the probability of the animal being at that location. The denominator is the probability of the animal being somewhere in the study area, which is calculated by integrating the probability animal being detected at least once (p.) over its possible locations ∫ p.(X; θ) dX.

What I do not understand is it integrating over all the possibilities assuming the animal is equally likely to be anywhere? Or is it integrating over

all the positions and using MLE to estimate where it is? Any clarification on this would be greatly appreciated.

Thank you, 
Ashley Cook

[Murray Efford]

Congratulations for trying to understand this! I can't locate the exact equation you cite in the 2009 paper, but note that the compact description there is for a homogeneous model (i.e. activity centres uniformly distributed in the plane, no weighting for varying density). See Borchers & Efford (2008) for the inhomogeneous (varying density) case. The denominator is the effective sampling area a rather than a probability.

Yes, it's integrating over all positions, but that bypasses the need to infer the actual location.