Calculation of the posterior distribution of total mortality
(M) given the carcass count, overall detection probability (g),
and prior distribtion; calculation of summary statistics from the
posterior distribution of M, including M* and credibility
intervals.
postM(x, g, prior = "IbinRef", mmax = NA)postM.ab(x, Ba, Bb, prior = "IbinRef", mmax = NULL)
calcMstar(pMgX, alpha)
MCI(pMgX, crlev = 0.95)
The functions postM and postM.ab return the posterior
distributions of \(M | (X, g)\) and \(M | (X, Ba, Bb)\), respectively.
The functions calcMstar and MCI return \(M^*\) value and
credibility interval for the given posterior distribution, pMgX
(which may be the return value of postM or postM.ab) and
\(\alpha\) value or credibility level.
carcass count
overall carcass detection probability
prior distribution of \(M\)
cutoff for prior of M (large max requires large computing resources but does not help in the estimation)
parameters for beta distribution characterizing estimated \(g\)
posterior distribution of \(M\)
credibility level (\(1-\alpha\)) and its complement (\(\alpha\))
The functions postM and postM.ab return the posterior
distributions of \(M|(X, g)\) and \(M|(X, Ba, Bb)\), respectively, where
Ba and Bb are beta distribution parameters for the estimated
detection probability. postM and postM.ab include options to
to specify a prior distribution for \(M\) and a limit for truncating the
prior to disregard implausibly large values of \(M\) and make the
calculations tractable in certain cases where they otherwise might not be.
Use postM when \(g\) is fixed and known; otherwise, use postM.ab
when uncertainty in \(g\) is characterized in a beta distribution with
parameters \(Ba\) and \(Bb\). The non-informative, integrated reference
prior for binomial random variables is the default (prior = "IbinRef").
Other options include "binRef", "IbetabinRef", and "betabinRef", which are
the non-integrated and integrated forms of the binomial and betabinomial
reference priors (Berger et al., 2012). For \(X > 2\), the integrated and
non-integrated reference priors give virtually identical posteriors. However,
the non-integrated priors assign infinite weight to \(m = 0\) and return a
posterior of \(Pr(M = 0| X = 0, \hat{g}) = 1\), implying absolute certainty
that the total number of fatalities was 0 if no carcasses were observed. In
addition, a uniform prior may be specified by prior = "uniform". Alternatively,
a custom prior may be given as a 2-dimensional array with columns for \(m\)
and \(Pr(M = m)\), respectively. The first column (m) must be
sequential integers starting at \(m = 0\). The second column gives the
probabilities associated with \(m\), which must be non-negative and sum to 1.
The named priors ("IbinRef", "binRef", "IbetabinRef",
and "betabinRef") are functions of \(m\) and defined on \(m=0,1,2,...\)
without upper bound. However, the posteriors can only be calculated for a
finite number of \(m\)'s up to a maximum of mmax, which is set by
default to the smallest value of \(m\) such that
\(Pr(X \leq x | m, \hat{g}) < 0.0001\), where \(x\) is the observed
carcass count, or, alternatively, mmax may be specified by the user.