The state-space modelling approach to capture-recapture models can be split into two alternative approaches: the conditional or the unconditional approach. The focus of this chapter will be on the former approach but it is useful to elucidate the philosophical dif- ferences between the two alternative approaches. The data used to fit the models will be the usual matrix of individual capture histories as described in Section 2.2. The manner
in which the mark-recapture data are conditioned on in the model is the key distinction between the two approaches and leads to two different ways of incorporating the data into the modelling framework.
For the conditional approach, the conditioning is on those capture history patterns that include capture in the current time period. Hence, the capture history associated with an animal determines the state the animal is assigned to. For each time period the model is conditioned on the number of animals that were observed during that time period. That is, the number of animals to have a capture history pattern which includes capture in the current time period is known exactly for each time period. Consequently, there is no error associated with this observation process and there is a direct correspon- dence between elements in the state vector and the observed number of capture histories containing capture in the current period. The unknown elements of the state vector cor- respond to the capture histories that do not contain capture in the current period. Hence, stochasticity enters the model framework through the modelling of the number of animals that are present in the population but remain unobserved in the current time period. Thus, propagating the state vectors through time conditional on the capture history data is what gives rise to the name for the conditional approach. It is possible to categorise the state classification by other covariates (e.g. gender, location, age) in addition to capture history, but for the examples presented in the following chapters the principle variable used to determine state allocation is the observed capture history pattern associated with each animal.
For the unconditional approach there is no conditioning on the matrix of capture his- tory patterns. Consequently, the capture histories are now regarded as observations on the evolving state process and whether an animal is captured or not does not influence its
state. Under this modelling approach stochasticity enters the model through the capture process and the observed capture histories are then regarded as a single random realisation of this stochastic capture process. This is more consistent with the classical Jolly-Seber capture-recapture methods summarised earlier and in recent work on developing state- space models for capture-recapture data (Giminez et al., 2007) this is also the approach taken.
In the standard definition of the three model processes that constitute a state-space model (Eqs (3.3.1a)-(3.3.1c)) the vector of parameters Θ contained all the parameters used in the model for both the state and the observation processes. If θ now denotes the vector of parameters that are used solely in the state process and ψ denotes the vector of parameters that are used solely in the observation process then Θ can be defined as Θ = (θ,ψ). In the standard state-space formulation the observation process (see Chap- ter 3.3.5) models the assumed relationship between the observed data and the states in the population. For a state-space representation of a capture-recapture model (Giminez
et al., 2007) the observation process would be given by the capture process and would
be a function of the probability of capture pt. Thus the parameters relating to capture
would appear in ψ. Under the conditional approach the capture process determines the observed capture histories and, consequently, the state of an animal. Hence, the param- eters relating to the probability of capture will appear in the vector of state parameters θ. However, under the unconditional approach, the parameters relating to capture do appear in the vector of observation model parametersψ.
A further important distinction needs to be made between the conditional and un- conditional approaches with reference to the interpretation of the model structure. By restricting the conditional model to operate on the observed capture histories at each time
point the model inference is restricted to the population that generated this specific set of observed capture histories. The conditional approach can therefore be said to assume a “population model” in which the population is of some fixed but unknown size. The unconditional model allows for stochasticity in the capture process and therefore allows for two conceptually different interpretations of the model inference. A population model may still be assumed in which, as for the conditional case, the population is of some unknown but fixed size and, contrasting with the conditional case, each realisation of the stochastically modelled capture histories corresponds to this fixed population. Al- ternatively a superpopulation model may be assumed in which each realisation of the stochastically modelled capture histories corresponds to a different population. For the superpopulation approach a state process model also needs to be specified as a compo- nent of the likelihood. This then stochastically models the entry probabilities of births into the population for each time period thus determining the possible range of observed capture histories for each realisation of the population. A more detailed discussion of the distinction between population and superpopulation approaches is given in section 6.2.1.