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STS563 Davide Di Cecco et al.
accordance with the literature on Bayesian capture–recapture, sensible options
include:
i) Jeffreys’ prior, i.e. (N) ∝ 1/N;
ii) a hierarchical Poisson prior: N ∼ Poi(λ,), λ, ∼ Gamma(a, β);
iii) Rissanen’s prior (Rissanen 1983), (N) ∝ 2 − log (N) , where log (N) is the sum
*
∗
of the positive terms in the sequence {log2(N), log2(log2(N)), ...}.
We further assume that N and are a priori independent.

2.2 Missing data: We propose a strategy useful to properly include sources
which do not operate over certain subpopulations (“incomplete lists”). In fact,
if we treat the uncatchable units as sampling zeros, the final population size
estimate would be biased. The idea is to treat the incomplete lists as Missing
at Random (MAR) information, i.e. assuming that, if they could operate on the
whole population, they would retain the same joint distribution as in the
observed subpopulations. In addition, we assume that we can distinguish
whether a unit has not been captured in a list by chance or because it is out
of the scope of that list, i.e., we can divide the population in strata where
different set of lists operates. Then, certain profiles of the captured units are
considered as partially observed, and we develop a data augmentation
algorithm that imputes the complete capture histories using the rest of the
data given the model. We distinguish completely observed capture profiles, y,
from the partially observed capture profiles ymis. In addition, for each stratum,
we have a structural zero z consisting in a different combination of zeros and
missing values. For example, in a 4-lists scenario with 2 strata, one where all lists
operate and one where the first list does not operate, we have the structural
zero n0,0,0,0 in the first strata, and n∗,0,0,0 in the second, where the asterisk denotes
the missing information. Then, our Gibbs algorithm at iteration t + 1 has the
following steps:

(1) we sample the components of (t+1) from their posterior conditional
Dirichlet distributions (constrained or not);
(2) for each observed y and ymis, we randomly divide all the observed
values ny and nymis into the corresponding consistent complete
sequences nx,y according to their conditional probabilities;
(3) if we adopt π(N) ∝ 1/N, it has been demonstrated in Manrique-Vallier
et al (2014) that we can sample all structural zero cells counts nz from
a Negative Multinomial distribution. Otherwise, if we choose an
informative prior for N, we can use a Metropolis-Hasting step to
generate a value for N(t+1) and then conditionally sample the
structural zero cells such that ∑z nz = N − nobs;
(4) for each generated nz, we sample all complete sequences nx,y consistent
with z.

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